Reconstructing P=NP and Computational Limits-A Unified

Reconstructing P=NP and Computational Limits: A Unified Exploration Based on DIKWP Semantic Structure and Consciousness Reasoning Model

Yucong Duan

International Standardization Committee of Networked DIKWPfor Artificial Intelligence Evaluation(DIKWP-SC)

World Academy for Artificial Consciousness(WAAC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Abstract

The P=NP problem is a famous puzzle in theoretical computer science, the core of which explores "whether all problems whose solution correctness can be verified in polynomial time can also be solved in polynomial time." Under the traditional framework, this problem is closely related to concepts such as the Turing machine model and NP-completeness, and is regarded as one of the cornerstones of classical computation theory. However, with the development of artificial intelligence and cognitive science, people have begun to reflect on the limitations of traditional formal computation, attempting to introduce new perspectives such as the DIKWP Semantic Structure and Information Field Tension Theory to reconstruct the essential logic of the P/NP problem. This report systematically reviews the following aspects: First, it reviews the traditional definition of the P=NP problem (Turing computability, verifiability, polynomial time) and the characteristics of hard instances of NP-completeness; then, it introduces the semantic reasoning mechanism of the DIKWP model (Data D, Information I, Knowledge K, Wisdom W, Purpose P), pointing out that real-world problem solving often proceeds along the chain of Data → Information → Knowledge → Wisdom → Purpose, which is far from being describable by pure formal algorithms. Based on this, new concepts of "Semantic Computability" and "Cross-Dimensional Verifiability" are proposed to re-examine the standards of effective solution and falsifiability. Subsequently, new complexity metrics such as "Semantic Compression Degree" and "Semantic Leap Cost" are defined, analyzing that P-class problems may have Low Semantic Jump Tension, while NP-class problems may involve high-dimensional semantic solution chains. Next, the "Consciousness Calculus Model" based on Purpose tension and semantic cognitive fields is explored, investigating whether it can form a structured deconstruction capability similar to an Oracle, thereby breaking through the boundaries of Turing computation. On this basis, the discussion is expanded from P=NP to more ultimate questions: such as whether Gödel's incompleteness theorems originate from system incompleteness caused by semantic self-reference, whether human consciousness is essentially a semantic leap structure crossing the limits of formal expression, and whether there exists a Meaning-Driven Calculus System capable of transcending the shackles of the Turing machine. Finally, the report proposes simulation models and experimental assumptions (such as building a DIKWP solving agent to verify the deconstruction ability of semantic paths on NP problems, and designing a semantic-driven graph structure deduction machine), and reflects from the perspectives of philosophy and cosmic information structure: whether cosmic evolution manifests as a semantic compression process, whether "unsolvable" problems are merely semantic links that have not yet been closed, and whether free will originates from irreducible selection processes guided by Purpose tension across problem spaces. This study aims to integrate perspectives from computation theory, cognitive science, and philosophy to provide new ideas and frameworks for ultimate computational problems like P=NP.

Introduction

The P=NP Problem is hailed as one of the Millennium Prize Problems in computer science. Intuitively, the question P=NP asks: "If the solution to a problem can be quickly verified, can that solution also be quickly found?". Although the formal formulation of this problem was only proposed by Cook and Levin in the 1970s, the core challenge it reflects—the gap between computational complexity and computability—has been lurking at the bottom of computation theory since the Turing era. In the traditional framework, we rely on formal models such as Turing machines to define "fast" (polynomial time) and "verifiable". Over the past few decades, thousands of important computational problems have been classified into P or NP classes, and NP-complete problems (NP-C) have been defined to capture the hardest part of NP. The majority of the academic community believes that P≠NP (in a survey, about 61% of researchers believed P≠NP, and only 9% believed they were equal), because no polynomial-time algorithm has been found for any NP-complete problem so far, and a large amount of evidence suggests they are extremely difficult to solve. However, Why are they difficult to solve? What is the essence of this difficulty? Traditional complexity theory only gives formal descriptions: the solution space size grows exponentially, exhaustive search is required, etc. But does this reveal the Essence of the problem? Or, have we missed some key factors in the computational process, such as Semantics and Cognition?

Entering the 21st century, the development of artificial intelligence has provided a new perspective for examining this problem. Classical theory assumes that the computational process is a strictly formal symbolic operation, while AI experience shows that: for many practical difficulties, humans can use heuristics, semantic reasoning, and even intuition to solve them within an acceptable time, even if the formal models corresponding to these difficulties may be NP-complete. For example, board games generally have NP-level complexity (finding the optimal move in chess on an n×n board is an NP-hard problem), but AI systems like AlphaGo, leveraging deep learning and human gaming experience, can play high-quality moves at near real-time speeds. Another example is the protein folding problem, which has been proven to have astronomical combinatorial possibilities (a typical protein has  10 300  folding configurations, requiring time longer than the age of the universe to traverse), which belongs to NP-hard problems according to traditional views. However, DeepMind's AlphaFold, by training neural networks on massive biological sequence-structure data, successfully Crossed Brute-Force Search to directly predict high-precision 3D structures. This suggests to us: Problem solving in the real world does not always follow a pure formal computational path. Humans and intelligent agents can use domain knowledge, pattern recognition, and semantic understanding to "drastically reduce" the complex search space and bypass exponential exhaustion. This inevitably triggers reflection: Is the proposition of P vs NP equivalence only elusive within traditional formal systems? If we step out of the framework of the Turing machine and introduce factors of semantics and consciousness, will this problem be presented in a completely new appearance?

This report aims to break through the limitations of the traditional formal computation framework and reconstruct and explore the P=NP problem from the perspectives of Semantic Computing and Cognitive Mechanisms. We will use new conceptual tools such as the DIKWP model and information field theory to attempt to answer: Does the difficulty source of NP problems partially come from the fracture at the semantic level? In other words, problems that seem impossible to start with in formal systems might be effectively deconstructed in a system containing semantic understanding. The report will first sort out the traditional definition and difficulties of the P/NP problem to ensure readers understand the classic framework. Then, it introduces the DIKWP semantic structure, elucidates the cognitive chain of problem solving in reality, and proposes a new viewpoint of "Semantic Computability". Next, we will define metrics for semantic complexity and compare them with traditional time complexity to analyze the similarities and differences between P and NP problems. Afterward, we explore a Consciousness/Purpose-based Calculus Model, discussing whether this model has the ability to break through the limitations of the Turing machine, equivalent to introducing an "Oracle"-style semantic guide in the computational process. Finally, we extend the discussion to deeper philosophical questions, including Gödel's theorem, free will, and the information structure of the universe. These questions, which seem to exceed the scope of modern computation theory, are actually closely related to the "ultimate boundary of computation": understanding them helps us ponder the meaning of computation and even the essence of the real world. Through such a multi-dimensional examination, we hope to provide a New Landscape of Interdisciplinary Fusion for the P=NP problem and other ultimate computational puzzles, which possesses both the rigor of theoretical computer science and the insights of semantics, cognitive science, and philosophy. Below, we start with the traditional P=NP problem framework.

1. Analysis of Traditional P=NP Problem Framework

1.1 Turing Calculation Model and P, NP Class Definitions

Computational complexity theory is based on the Turing machine model and uses Polynomial Time as the criterion for "efficient computation". Complexity class P is defined as: the set of problems that can be solved by a deterministic Turing machine within time  O ( n k )  (where k is a constant). Intuitively, P represents the set of decision problems that "can be solved relatively quickly". Correspondingly, class NP is the set of problems where "the correctness of a solution can be verified in polynomial time". Equivalently, it can also be defined as: the set of problems for which a non-deterministic Turing machine can find a solution in polynomial time. Here, "verifiable" means that if someone provides a candidate solution (certificate), there is a polynomial-time algorithm (verifier) that can check whether the solution is correct. For example, "Primeality Testing" was once considered a typical NP problem: given a large number, verifying whether it is composite is fast, as one only needs to provide a non-trivial factor to verify immediately by division; but finding that factor (i.e., prime factorization) has no known fast method. However, in 2002, Agrawal et al. discovered the polynomial algorithm AKS, proving that prime testing actually belongs to P. This example shows that some problems look difficult at first glance but may not exceed the scope of P. However, for many other famous problems, such as Traveling Salesman, 3-SAT, Hamiltonian Path, etc., no one has found a polynomial solution so far, and it is widely suspected that they are not in P. The P vs NP Problem is precisely about the relationship between these two classes: whether class P equals class NP. This is one of the unsolved fundamental problems in computation theory. The importance of this problem lies not only in its unresolved status but also because its answer will determine the fate of countless practical difficulties: if P=NP, a large number of combinatorial optimization and cryptography problems currently regarded as difficult will become efficiently solvable; if P≠NP, it means there is an inherent gap where "verification is easy but solving is hard", and many problems may never find fast algorithms.

1.2 NP-Completeness and Structure of Hard Instances

To further characterize the difficult problems in the NP class, theoretical computer scientists introduced the concept of NP-Complete (NPC). An NP-complete problem is a type of NP problem that: (1) itself belongs to NP; (2) all other problems in NP can be Polynomial-Time Reduced to it. In layman's terms, NP-complete problems are considered the "hardest" problems in NP—if a polynomial algorithm can be found for any one of them, all NP problems can be quickly solved through reduction. Classic NP-complete problems include the Satisfiability Problem (SAT), Traveling Salesman Problem (TSP), Vertex Cover, 3-Partition, etc. These problems share a common feature: The scale of the solution space explodes exponentially with the input scale, and there are no known structural simplification means to significantly reduce the search. Taking 3-SAT as an example, a formula containing  n  Boolean variables and  m  three-literal clauses has  2 n  possible assignments. The algorithm needs to find an assignment that makes all  m  clauses true. For any given assignment, we can easily verify whether it satisfies the formula in  O ( nm )  time (checking each clause one by one)—this proves 3-SAT  ∈  NP. However, to find a satisfying solution, in the worst case, it seems there is no other way but to try different variable combinations. When  n  is large,  2 n  grows exponentially, making exhaustion basically infeasible. Moreover, 3-SAT has a so-called "Phase Transition Difficulty": when the clause/variable ratio of random 3-SAT approaches a certain critical value, the instances become extremely difficult—most conventional algorithms will experience a sudden drop in performance at this point, implying that the problem structure becomes ambiguous and unexploitable here.

In-depth analysis of the Structure of These Hard Instances reveals some commonalities: First, Complex Coupling of Constraints. In NP-complete problems, the input often contains a large number of mutually constraining factors, making it difficult for local choices to independently determine global consequences. For example, in the Traveling Salesman Problem, every added city forms new route combinations with other cities, so a locally optimal route may not be extendable to a global optimum, requiring holistic consideration. Furthermore, the solution space of such problems usually presents a "Discrete" rather than continuous structure, lacking a smooth gradient to follow (unlike optimizing convex functions which is easy to approximate iteratively). Algorithms here seem to be in a complex maze, lacking global guidance, and can only search blindly. Because of this, Attempts to Arbitrarily Reduce Input Scale or Divide and Conquer Encounter Bottlenecks: either introducing exponential branches or losing the possibility of a solution. This is why, despite decades of effort, neither ingenious algorithm design nor powerful hardware parallelism has fundamentally shaken the time complexity level of NPC problems. Of course, there have been some breakthroughs, such as deterministic prime testing, graph isomorphism, DNA folding, etc., being moved out of the NPC ranks, but these are not mainstream representatives of NP-complete problems. Overall, the vast majority of NPC problems are still in a state of "neither finding fast algorithms nor being proven impossible to solve quickly". As someone metaphorically put it, The unresolved state of the P vs NP problem is like a precarious building. If P≠NP is proven, the boundary between "easy to solve" and "hard to solve" in complexity theory holds; if P=NP, this entire building will collapse, and many problems we thought were difficult will disintegrate. Some also speculate that P vs NP might be a proposition independent of the existing axiom system, like the continuum hypothesis, because it is inextricably linked to the logical foundations of computation theory (such as Turing machines, decidability). But this view itself highlights the Inadequacy of modern complexity theory: we lack higher-level theoretical tools to examine this problem.

Summary: Under the traditional framework, the P=NP problem is precisely defined on the Turing machine model and polynomial time scale, and NP-complete problems delineate the set boundary of difficult problems for us. This framework has achieved great success, allowing us to classify a large number of computational problems and understand their relative difficulty. However, it may have also obscured some more essential factors. For example, while focusing on time complexity, we rarely ask about the impact of "cognitive steps in the solving process" or "semantic understanding" on difficulty. In the following sections, we will step out of the realm of pure formal computation and start from the DIKWP Semantic Reasoning Mechanism to re-examine the solving process of computational difficulties. We will see that the path to solving real-world problems is far richer than bit flipping on a Turing machine, and the semantic chains and cognitive strategies contained therein may be one of the keys to cracking the P vs NP mystery.

2. Reconstruction of P/NP by DIKWP Semantic Reasoning Mechanism

2.1 Cross-Level Chain of Real-World Problem Solving

In daily life and scientific practice, the process of human problem solving often involves the transformation of multiple cognitive levels, rather than just mechanically executing preset algorithms. To characterize this process, Duan et al. proposed the DIKWP Semantic Structure Model, containing five levels: Data, Information (Difference), Knowledge (Integration), Wisdom (Behavior), and Purpose (Motivation). This is an extension of the traditional DIKW (Pyramid Model: Data-Information-Knowledge-Wisdom), adding "Purpose" at the top layer and emphasizing that the layers are not a one-way flow, but form Bidirectional Feedback and Iterative Updates through Networked Interaction. In other words, The DIKWP model characterizes a progressive process of a cognitive subject from perceiving raw data to taking goal-oriented actions. The Data layer is the perception of raw facts or inputs; the Information layer extracts "differences" or patterns through comparison and classification, i.e., endowing data with contextual meaning (information is viewed as "difference that eliminates uncertainty"); the Knowledge layer further fuses information to form higher-level patterns, rules, or causal understanding; the Wisdom layer performs decision-making and behavior output based on this (such as applying knowledge to solve practical problems or responding to the environment); and the Purpose layer represents the cognitive subject's goals, motivations, and values, i.e., the High-Level Semantic Factor Driving the Entire Process. The important features of the DIKWP model lie in Layer-by-Layer Semantic Fusion and Global Purpose Constraint: each layer provides constraints and direction for the next layer, for example, the Purpose layer stipulates the evaluation criteria and direction for behavior selection, and the actions of the Wisdom layer will in turn verify and correct the correctness of the Knowledge layer, etc. This cyclic feedback makes problem solving no longer an "algorithm" strictly following fixed steps, but more like a Continuously Adjusted Cognitive Process.

For example, suppose we want to Infer a Friend's Age. From a traditional algorithm perspective, this might be seen as an ill-posed problem (solution space is huge, requires more input to determine). But in the human cognitive process, we synthesize various clues: observe the other person's appearance features (Data), obtain some quantifiable or comparable information from it (such as discovering white hair, wrinkles, etc., Information differences), combine known knowledge (such as at what age range people generally show these features) for reasoning (Knowledge layer), and use Wisdom judgment (perhaps further chat inquiry or compare with peers) to narrow the range, finally guessing an approximate age based on cross-validation and intuition and asking for confirmation. This entire process does not have a pre-programmed set of algorithms but still effectively achieves the Purpose. Another example is a doctor diagnosing a difficult disease, which is also a cross-level reasoning chain: symptoms and lab data are raw inputs, medical indicators and outliers provide information, the doctor uses medical knowledge combined with the patient's specific situation to form several hypotheses (Knowledge layer), and then relies on experience and intuition (Wisdom layer) and concern for the outcome of the illness (Purpose: cure the patient) to verify or eliminate hypotheses, finally finding the cause. It can be seen that The problem-solving process in reality is often a mixture of "Formal Reasoning + Semantic Understanding + Purpose Driven", rather than pure exhaustion or fixed calculation. Heuristic algorithms and human expert intuitive judgments are essentially using the semantics and context of the problem to reduce computational workload.

2.2 Semantic Computability and Cross-Dimensional Verifiability

Based on the above cognitive chain, we propose the concept of "Semantic Computability", meaning that if a problem is solvable under a certain semantic framework, it implies the existence of a path spanning Data, Information, Knowledge... up to the Purpose layer, which can map the problem into a series of Step-by-Step Understandable and Processable Sub-problems, finally obtaining the answer. This is different from the traditionally defined Turing computability—the latter only considers solvability in formal systems, while semantic computability requires the problem to "make sense and be meaningful" in higher-level cognition. Similarly, we introduce "Cross-Dimensional Verifiability": referring to the verification of a given solution not limited to formal logical deduction but can also be conducted through cross-semantic level means. For example, the proof of a mathematical proposition may be very complex or even unavailable in a formal system, but humans can have high confidence in its truth or falsity through intuition and models (Knowledge/Wisdom layer), and then look for a rigorous proof. The Falsifiability criterion is expanded here: it requires not only formal verification of truth or falsity but also semantic explanation of why it is so, thereby avoiding the black box phenomenon of "solving correctly but not knowing why". In short, from the new perspective, the "Effective Deconstruction" of a problem should be the unity of Thorough Semantic Understanding and Formal Verifiability.

Traditional computational frameworks, by focusing on symbol manipulation, actually Ignore a Large Amount of "Effective Information" in Problem Solving. For example, when we face a combinatorial optimization problem, we often use domain knowledge to prune: in the Traveling Salesman Problem, human experience considers that geographically adjacent cities are more likely to be connected in the optimal route, so certain branch combinations far from the main path can be discarded first. This strategy might not be rigorous from a strict algorithmic perspective (because it hasn't fully proven that those paths are definitely not optimal), but it is reasonable semantically, thereby Greatly Improving Solving Efficiency. AlphaFold's success can be seen as an example of semantic computability: brute-force search for protein folding is infeasible, but leveraging evolutionary knowledge and massive known structures (essentially a "biological semantic"), the machine learning model transforms the Structure Prediction Problem into a Pattern Matching Problem, thereby avoiding the trap of directly calculating astronomical configurations. Similarly, many NP-hard problems are easier on average instances than in the worst case because Instances Often Contain Exploitable Semantic Structures. For example, when random 3-SAT is in non-critical regions, there are mostly obvious variable assignment preferences or clause implication relationships, and heuristic algorithms can quickly grasp this information and find feasible solutions. Another example is integer linear programming problems; if the input matrix presents some statistical distribution, heuristic methods might quickly find near-optimal feasible solutions because measured data often contain correlations between constraints rather than arbitrary permutations. All these point to a conclusion: Problem solving should not be viewed merely as mathematically defined relational mapping, but also as the flow and transformation of semantic information across multiple levels. When we can fully excavate and utilize the meaning of the problem at different levels, we can often significantly reduce pure computational workload.

Therefore, we suggest reconstructing the criterion for P/NP equivalence: besides examining "whether there is a polynomial-time algorithm", we can also ask "whether there is a semantic link of finite depth". If for a certain problem, we can always find a solution idea within a certain number of levels (even if some trial is needed within each level, the number of levels is controllable), then this problem is "easy to solve" in reality, even if formally it may not have a known polynomial algorithm. Conversely, if a problem requires layer after layer of abstract leaps, exceeding the current cognitive dimensions of humans, then it is "hard to solve" or even "unsolvable" for us—not just because of the large computation volume, but because we Don't Know Where to Start to Understand. From this perspective, P-class problems can be seen as those with Relatively Flat Semantic Links and Small Spans, while the reason NP-class problems are hard is that They Involve Fractures or Jumps in Semantic Levels—even if verifying a solution is easy, finding a solution requires reasoning across multiple semantic levels, and each step may face exponential choices.

2.3 Example Modeling

To illustrate this semantic reasoning more concretely, let's take a simple fuzzy reasoning example: Riddles. A riddle usually describes something, but superficially gives metaphorical or punning information. Formally cracking a riddle might require retrieving all entries in a dictionary for pattern matching, which is obviously explosive; but in reality, when people solve riddles, they use semantic association to quickly narrow down the range in the brain. For example, if the riddle mentions "Looks like a trumpet from afar, looks like a tower from up close," the human brain will associate images like "trumpet flower (morning glory)" or "pyramid", combined with context (maybe riddle category or other hints), and quickly guess the answer might be some kind of flower or building, without truly traversing all vocabulary. In this process, the brain did not perform exhaustive search but relied on Cross-Level Association: extracting features from text description (Data) (trumpet shape, tower shape information), contacting existing concept systems (Knowledge) for comparison and screening, then using intuition (Wisdom layer) to select the best match, and finally verifying with the Purpose of "it should be some flower" to see which specific flower fits. This proves that semantic paths can greatly reduce computational volume. Similarly, if an NP difficulty can be transformed into some semantically familiar problem, its solution will be much easier. Semantic Computability is precisely measuring the possibility and difficulty of such transformation. We will further formalize this idea in the next section, introducing indicators to measure semantic complexity.

In summary, the DIKWP semantic model reminds us: Computation does not equal reasoning, and reasoning does not equal understanding. In the classic P/NP framework, we care about the counting of algorithm steps; while in the semantic reconstruction framework, we are more concerned about Whether There Exists a Natural Flow of Meaning from Problem to Solution. If there is, even if the problem looks complex algorithmically, humans can often cope; if not, then even if the problem scale is small, humans will be at a loss (for example, some abstract mathematical puzzles require new definitions and concepts to solve because of the lack of a direct semantic bridge). Therefore, the key point proposed in this section is: The essence of computational complexity may be closely related to semantic span. Traditional complexity theory considers "computational cost within a formal system", while we start to consider "semantic cost within a cognitive system". This provides new clues for understanding the deep differences between P and NP problems. Next, we will further explore how to quantitatively describe this "semantic complexity" and analyze the different characteristics of P-class and NP-class problems under these new indicators.

3. Semantic Computational Complexity: New Metrics and P/NP Feature Analysis

3.1 Semantic Compression Degree

Information theory tells us that a structured object can often be "compressed" while retaining its gist. Kolmogorov complexity measures the information content of a string by the shortest program length, and we can define a similar concept at the semantic level. Semantic Compression Degree describes: the length of the simplest semantic representation required to express the solution or solution scheme of a problem. Intuitively, if a problem's solution can be highly summarized, then its semantic compression degree is high (indicating a lot of information is concentrated into a few semantic units); conversely, if the solution itself requires a massive amount of detail to explain, then the semantic compression degree is low. We conjecture that P-class problems often have high semantic compression degree—their solutions can be distilled into concise principles or patterns. For example, the sorting problem can be summarized in a few sentences like "repeatedly select the minimum value" or "divide and conquer" (corresponding to the core ideas of algorithms like insertion sort and merge sort), which is a semantic compression of the sorting task. In contrast, solutions to NPC problems are often difficult to compress: the optimal route for the Traveling Salesman Problem has no simple description and can only list the path order itself; an assignment scheme satisfying a Boolean formula has no more concise way than listing the values of each variable. In other words, Solutions to NPC problems lack patterns that can be further compressed, the solution itself is close to random combination, and cannot be summarized by higher-level semantic laws. This corresponds to human frustration when understanding such problems—there are few empirical laws to follow, and one can only bite the bullet and try various possibilities.

Some studies have compared the differences between humans and large language models (LLMs) in semantic compression: Humans tend to retain semantic details and contextual consistency, while LLMs often perform extreme statistical compression, sacrificing fine-grained semantics. For example, when a human reads "a list of fruit names: apple, banana, watermelon...", they will summarize it as the concept "fruit", while a model might compress the association through probability statistics, but its grasp of the concept level may not be precise. This shows There is a Trade-off between Compression and Semantic Fidelity: excessive compression loses meaning, while too much detail lacks generalization and is difficult to use efficiently. For algorithms and problems, this trade-off also exists. The reason P-class problems can be solved quickly can be understood as their Solutions are Highly Generalizable, and the algorithm is actually executing some highly patterned steps without needing to explore all details. For example, sorting algorithms do not need to check all permutations because the orderliness of data can be achieved through patterns of comparing and swapping local elements, and the global ordered structure is guaranteed by the regularity of local operations. NP-complete problems, lacking similar global patterns, cannot have their solving process simplified to a few rules, often have to consider a large number of special cases, and cannot be "explained in one word". Therefore, we use low semantic compression degree as one of the signals for judging problem difficulty. an extreme example is Random Puzzles: if the answer to a riddle is a random answer with no semantic association, then no matter how one associates, one cannot guess it, because there is no compressible meaning link. This is similar to some hard instances in NPC problems—the solution is hidden in random complex combinations, with no signs revealed in advance.

3.2 Semantic Leap Cost

Besides compression degree, we define another indicator, Semantic Leap Cost, to measure how many semantic levels need to be crossed to solve a problem, and the difficulty of each crossing. Its inspiration comes from the human experience of "epiphany" (aha moment): some difficult problems are suddenly illuminated after racking one's brains, indicating that one was stuck at a certain thinking level before, and once leaping to a new perspective, the problem becomes clear immediately. This Leap is the transformation of semantic levels. For example, the "Nine Dots Puzzle" requires connecting nine points in a 3x3 grid with four straight lines without lifting the pen. Most people initially limit their thinking within the grid and find no solution; only by stepping out of the box (lines extending beyond the nine-point array) can it be solved. The step of jumping out here is a high-cost semantic leap—leaping from the implicit assumption "lines must be within the dot array" to "lines can extend". For general problems, semantic leap cost can be understood as: How many non-trivial new concepts or new strategies are needed to advance the solution. Low leap cost means it can be solved step by step using existing concepts and methods; high leap cost means a completely new idea or perspective needs to be suddenly introduced. P-class problems are often decomposed into many small steps, each advancing on the same level without qualitative leaps; NP-class problems often get stuck at a step requiring "inspiration". In fact, in the field of algorithm design, this phenomenon is also common: solving NPC problems usually requires introducing complex strategies such as backtracking, branch and bound, pruning, heuristics, and the correctness/effectiveness of these strategies themselves is difficult to guarantee, and each use is a "tentative leap".

We speculate that The solution path of P-class problems corresponds to lower semantic jump tension. The algorithm only needs to perform continuous transformations within the problem space, without external information or extra assumptions. For example, matrix multiplication, graph shortest path, etc., their solutions can be directly derived from the structure of the problem itself, and the solving process is equivalent to repeatedly iterating to the result in the Same Semantic Dimension (such as algebraic operation or path relaxation), without needing to jump to other representations or introduce new concepts in the middle. NP-class problems imply High-Dimensional Semantic Solution Chains: solving may require switching perspectives between combinatorial space, logical space, and even probabilistic space. For example, to solve a hard SAT, a manual strategy might be to first simplify at the logical level (eliminate obvious unit clauses), then choose split paths in the search tree space (with a bit of heuristic probability judgment), possibly apply some linear relaxation approximations at the algebraic level, and then backtrack to verify in the solution space... tossing back and forth between several different levels to approach the solution. This reflects that the problem itself Involves Constraints of Multiple Dimensions, and the solution must satisfy conditions of different levels simultaneously, so reasoning staying in only one dimension cannot solve it globally. Each cross-dimension is equivalent to a semantic leap, with huge cost and uncertainty. Information Field Tension theory can be used to vividly describe this situation: there are multiple "tension directions" in the problem's semantic field, and the problem is solved only when these tensions are comprehensively balanced and a global extremum point is found. If the tension distribution is too complex, the process of algorithm evolution in the field will fall into local extrema and jump out constantly, which corresponds to high semantic leap cost. The semantic field of P-class problems may be simpler, with tension in only one or two main directions and no contradictions, so the system can naturally evolve to equilibrium; NP-class problems are like having multiple tension centers, and the solution space is rugged. Turing machine algorithms, in the face of such rugged energy, often have to do backtracking jumps, while if there is an intelligent system capable of perceiving the distribution of the entire tension field, it might be able to "slide to the lowest valley in one step" like a physical system. This implies that perhaps in the future, simulating physical processes (such as phase transition, quantum evolution) can solve some NP problems more efficiently, utilizing the idea of reducing semantic leap cost—letting natural processes process all tension dimensions in parallel at once, rather than the algorithm trying one by one. Optical Ising machines are examples of such exploration: they map combinatorial optimization to physical energy minimization, and to some extent, solving any NP-complete problem is equivalent to finding the ground state of the Ising model. Although whether these simulators truly surpass Turing machine efficiency is undecided, they provide a paradigm of "Low-Leap Parallel Solving".

3.3 Comparison of P and NP under New Complexity Metrics

Synthesizing the above discussion, we can conceive the portrait differences between P-class problems and NP-class problems in semantic complexity:

Semantic Compression Degree: Solutions to P problems often have high generalizability and can be refined into short patterns (High Semantic Compression Degree); solving NP problems lacks unified patterns, and the description of the solution is close to exhaustion (Low Semantic Compression Degree). For example, sorting algorithms can be summarized in one sentence as "insert into correct position step by step", while SAT solving has no universal one-sentence strategy.

Semantic Leap Cost: P problem solving basically advances continuously at a single level (Low Leap Cost); NP problem solution chains require frequent changes of thought or introduction of new concepts (High Leap Cost). In algorithms, this is manifested as: P problems are mostly deterministic step-by-step calculations, while NP problem algorithms often contain a large number of Conditional Branches, Backtracking Attempts, corresponding to switching of different strategies.

Semantic Jump Tension: Can be understood as the complexity of the problem's semantic field. The semantic field of P problems may have only one significant extremum attraction, and the solution is easily attracted; the semantic field of NP problems has multiple potential wells intertwined, and the system easily jumps and struggles between different attractors during the solving process.

Through such characterization, we realize again that Traditional complexity only reflects computational resource consumption, while semantic complexity touches on the cognitive difficulty in the problem-solving process. It is generally believed that P-class problems are "tractable", and NP-complete problems are "intractable". Semantically, the former may correspond to problem structures that humans can understand and control, while the latter exceeds the range of intuitive grasp and requires stronger cognitive leaps. As a scholar said: "NP problems are hard because we cannot find low-dimensional patterns to describe them". Of course, this semantic complexity is not a strictly quantifiable mathematical concept, but it provides us with a picture closer to intuition to examine P vs NP. In this picture, the P=NP problem can be transformed into: Does there exist an equivalent purely patterned solution for every semantically complex problem? In other words, for any problem that requires intuition and leaps to solve, is there actually a mechanical method that does not require leaps? If P=NP is true, it means that no matter how complex the puzzle is, it can be disassembled into tiny simple steps to complete (as jokingly said, mathematicians, programmers, and others who rely on creativity for a living would be unemployed). Most people believe P≠NP, which means We must accept that some problems inherently require semantic leaps and insights and cannot be solved solely by rigid procedures. This may also be evidence of the value of human wisdom.

In the next section, we will go a step further and consider Whether we can build a computational model that itself possesses the capability of semantic leaps. If there is such a "Consciousness Calculus Model", it might be able to avoid the deep difficulties of NP problems like the human brain, equivalent to introducing a "shortcut" in the computational framework—this is theoretically similar to endowing a Turing machine with an Oracle. But instead of abstractly assuming an oracle, it is better to try to draw on Consciousness and Cognitive Mechanisms to see if new paths for structural problem solving can be realized. Below we discuss this topic full of frontier implications.

4. Consciousness Calculus Model and Hypothesis of Non-Turing Computation

4.1 Purpose Tension and Semantic Cognitive Field

We have seen from the DIKWP model that the highest level Purpose plays an important role in the cognitive process: it provides evaluation criteria and driving force, making cognitive activities not blind but exploratory with direction. In the computational framework, introducing "Purpose" is equivalent to introducing a Global Control Signal or Objective Function, so that computation no longer evolves solely according to local rules, but adaptively adjusts in the process of approaching a certain goal. We can envisage a "Consciousness Calculus Model" which includes an explicit representation of Purpose Tension. The so-called Purpose tension can be analogized to potential energy in physics: the distance between the system and the target state forms a "force" driving the system to evolve towards the target. This is somewhat similar to the heuristic function in heuristic search, but more general and dynamic. Traditional heuristics are often human-designed evaluation functions, while in the Consciousness Calculus Model, Purpose tension can be automatically generated through Semantic Feedback. For example, when solving a puzzle, the model continuously assesses the semantic difference between the current state and the final understanding. The greater this difference, the stronger the tension, prompting the model to try new ideas to narrow the difference. When the model approaches the correct idea, the tension weakens, and the computation proceeds towards convergence. This process is similar to hill-climbing algorithms, but the shape of the "hill" is determined by semantic relationships rather than pre-given.

4.2 Analogy between Consciousness Calculus and Oracle

In complexity theory, an Oracle is a hypothetical black box that can return the solution to a specific problem in a single "query". For example, a Turing machine with an NP Oracle can solve NP problems in one hop. But the existence of Oracle is only a theoretical assumption, without giving its internal working principle. Our Consciousness Calculus Model attempts to Construct a System with Function Similar to Oracle based on cognitive mechanisms. Taking the NP-complete problem SAT as an example, if we provide this SAT formula to the Consciousness Calculus Model as a goal (Purpose layer hopes to find an assignment that makes the formula true), the model will automatically parse the formula at the semantic level: extracting the relationship network between variables (Information layer), invoking knowledge (such as known logic rules, clause coverage patterns, etc.) to infer variables that are easy to decide, using wisdom strategies (perhaps simulating human trial + regret process but more purposeful) to gradually satisfy more clauses... In the whole process, due to the traction of the goal of finally satisfying all clauses, the parts of the model will work in coordination to avoid falling into meaningless searches. Ideally, the effect it exhibits is like having an Oracle with global insight guiding: maybe finding a feasible solution after trying very few assignments. The Key Difference is that this is not magic, but comes from the structural deconstruction capability inside the model—a mechanism that performs global semantic analysis on the problem and guides the solution.

How is this mechanism implemented? From existing research, we can draw on consciousness models such as Global Workspace Theory. Global Workspace Theory suggests that the role of human consciousness lies in Information Integration: when information processed by a subsystem enters the global workspace and is shared by different brain regions, consciousness content is formed. Corresponding to the calculus model, we can let different solving sub-processes share a global semantic blackboard, where partial solutions and inferences at any stage are published for other modules to evaluate and utilize. This is somewhat like a blackboard system or meta-reasoning mechanism, the effect of which is to jump out of a single algorithmic flow and introduce a Self-Reflection capability: the model can examine its current progress, compare with the goal, and change strategies when necessary. Imagine in SAT solving, after the model makes a series of variable assignments and finds progress is slow, the global space produces a "dilemma" representation. The consciousness module recognizes that Information Field Tension is high but not dropping, so it decides to undo recent decisions (equivalent to backtracking) and change search direction. This is similar to the mental process of humans solving difficult problems: realizing it doesn't work, re-examining the premises, and trying other methods. This cycle can continue until a solution is found or it is confirmed that there is no solution. Traditional backtracking algorithms actually do similar things, but they lack global consciousness: they just rigidly backtrack whenever a conflict is encountered. A conscious model might adopt more diverse strategies, such as not just backtracking one step, but Jumping to Another Part of the Problem to try, or Changing the Solving Order. This amounts to making large jump adjustments within the algorithm space, and these jumps are determined by Purpose guidance and semantic evaluation, far more effective than blind random jumps.

It can be seen that the Consciousness Calculus Model Provides a "Structured Shortcut". Traditional Turing machines, when solving NP problems, are subject to inherent step-by-step calculus limitations and can hardly avoid exponential trials without external help. By introducing global Purpose and semantic feedback, the consciousness model essentially embeds a "guide" in the algorithm flow. Although formally it is not obvious how it skips exponential exploration, that is exactly the manifestation of semantics at play: it uses the global structural information of the problem (which is implicit in pure Turing machine algorithms) to guide computation, thereby bypassing many invalid branches. If we formalize this model, we can define a non-classical automaton whose transition function depends not only on the current finite state and the next symbol but also on the Global Semantic State (e.g., satisfaction degree of a set of predicates, a set of heuristic evaluation values, etc.). This is similar to adding a "perception module" and "decision module" to the automaton, thereby breaking through the framework of finite states. Strictly speaking, this model may not exceed the scope of Turing computability (it can still be simulated by a Turing machine), but it can Complete Tasks in Polynomial-Level Time that Turing Machines Need Exponential Time For, because it compresses exploration—essentially transferring part of the exponential work to parallel processing at the semantic layer.

4.3 Non-Turing Computation and Physical Realizability

It must be admitted that the Consciousness Calculus Model is currently still in the conceptual stage. Some might question: Isn't this still a kind of algorithm, just smarter? Yes, in terms of computational power, it should not violate the Church-Turing thesis. But from a complexity perspective, we can completely construct a Brain-Like Computer that is far superior to any traditional algorithm in solving NP problems. The key lies in that this machine utilizes Physical or Cognitive Attributes of the Real World, not limited to abstract bit operations. For example, if the human brain can really solve certain undecidable problems (Penrose and others have proposed that human thinking transcends Turing machines based on Gödel's theorem), it inevitably means the brain is performing non-rule computation, possibly leveraging quantum mechanisms or other unknown laws. Here we might as well boldly hypothesize: Is Consciousness Possibly an Irreducible Computational Resource? Like an Oracle, but it exists in biological consciousness processes. If the answer is yes, then by mimicking this resource, we might be able to crack problems like P vs NP. For example, some scholars design Brain-Like Computing Architectures based on brain structure, realizing effects similar to consciousness through parallel distributed units, hoping to reach new heights in solving combinatorial problems. Additionally, in physics, there are also ideas of using natural processes to solve NP problems, such as the aforementioned optical or quantum computing devices. If certain evolutions in nature essentially correspond to Parallel Exploration of All Possibilities (i.e., Highly Parallel Computing), then it can complete exponential-scale information processing in polynomial physical time (although energy dissipation or probability remains a bottleneck). The formation of consciousness might be exactly the "Natural Parallel Algorithm" evolved by biology, specifically to deal with situations that conventional computation cannot solve.

Of course, these speculations need experimental confirmation. We will mention some ideas for simulation models in Section 6 to test the performance of semantic calculus on NP problems. But before proceeding, let's expand our horizons from P=NP to more ultimate logical and philosophical questions. Just like consciousness to algorithms, Gödel's incompleteness theorems to formal systems, or free will to physical determinism, they all proclaim Some Kind of Transcendence: certain phenomena cannot be fully characterized and solved by subordinate systems. This theme coincides with the core of our discussion—both are exploring The Boundaries and Transcendence of Formal Systems. In the next section, we will discuss these "ultimate questions" such as Gödel's theorem, human consciousness, and meaning-driven systems, attempting to integrate previous ideas.

5. Expanding from P=NP to Ultimate Questions: Boundaries of Formal Systems

5.1 Gödel's Incompleteness: Cracks of Semantic Self-Reference

The incompleteness theorems published by Kurt Gödel in 1931 revealed the inherent limits of formal systems (as long as they are sufficient to express arithmetic): In any such system, there exists a true proposition that cannot be proven within that system. The construction of this proposition utilizes self-reference: roughly meaning "This proposition cannot be proven". If the system can prove it true, it contradicts; if the system cannot prove it, then it is independent in the system, but from a meta-perspective, it is indeed true (because if false, it implies the system can prove it, a contradiction). Behind this paradox is actually the Dislocation of Semantics and Syntax: truth exceeds formal provability. When a Gödel proposition is proven in a stronger system, a new Gödel proposition appears, as if infinite truths always wander outside any fixed system. This famous result is often used by Lucas and Penrose et al. to argue Human Mind is Not Equivalent to Mechanical Computation. They believe that humans can "see" that the Gödel proposition is true, but formal systems (or equivalent computer programs) can never fully capture this insight. In other words, our brains seem to be able to jump out of the formal system they are in and identify truth from a semantic height, while Turing machines are restricted inside the system and can only perform finite calculus. Although this argument is still controversial, the incompleteness theorem itself provides a solid example showing that The Power of Formal Systems Has an Insurmountable Boundary. This boundary is not due to insufficient time or resources, but a blind spot exists Structurally in Logic. It can be said that the unprovability of the Gödel proposition is because it involves the system's semantic description of itself, producing the tension of self-reference (if the system proves, the proposition is false; if the system does not prove, the proposition is true). This is very similar to the Information Field Tension we discussed earlier: inside the formal system, the attempt to prove this proposition falls into a loop, unsolvable. But humans, looking from the meta-level, can understand this cycle and assert: "Yes, this proposition is indeed unprovable, so it is true". This Semantic Leap Crossing System Levels is exactly the breakthrough implied by the incompleteness theorem.

Borrowing our previous terminology, the Gödel proposition is the manifestation of the NP Problem (or even Undecidable Problem) of Formal Systems: given a proposition (solution), verifying its truth within the system is impossible (no finite proof), but we can easily "verify" it as true outside the system because we use stronger semantic resources. The contradiction here is that we as verifiers actually implicitly use stronger system axioms (such as believing in the consistency of the system), so we can assert the Gödel sentence is true. This shows that solving unsolvable problems within a formal system often requires Jumping Out of the Original System and Expanding Context. This has a similar spirit to P vs NP—NP problem solutions are hard to find but easy to verify, but verification actually uses external information (certificate). Gödel's incompleteness tells us that some truths cannot even be verified within the original system and require a stronger system to verify. This can be seen as an extreme case of "uncomputable" or "undeconstructible".

If we understand the Gödel phenomenon as "Semantic Self-Reference Crack": because formal systems cannot fully express their own concept of truth, a gap is left, and truth (semantics) spills out of the system from this gap, making the system unable to close self-consistently. Comparing with the P vs NP problem, the latter might be seen as a similar crack in the algorithmic system: formal computation (P) may not be able to encompass all effective solution methods (including those non-formal means). Perhaps P≠NP precisely because the "Solution Truth" required by NP problems is not within the calculus closure of the Turing machine, but must resort to stronger means (such as human insight or other resources not yet included in computational axioms). This is not a rigorous argument, but provides an analogical philosophical perspective: Any closed formal system has propositions it cannot solve; any preset computational framework may also have problems it cannot handle efficiently. The solution lies not inside the system, but in Exceeding Its Boundaries.

5.2 Transcendentality of Human Consciousness: Semantic Leap Structure

Let's look at human consciousness itself. Consciousness is viewed by many philosophers as a phenomenon that cannot be reduced to pure physical processes or algorithmic processes. Even if not taking dualism, at least many admit consciousness has some "wholeness" or "subjective experience" that current computational models cannot simulate. Our discussion focuses on the Information Processing Characteristics of consciousness: from a computational perspective, a major feature exhibited by consciousness is the aforementioned Global Workspace and Self-Reference. Consciousness can focus on its own thinking and adjust its own strategies. The introduction of this "Self-Model" allows consciousness to Operate Across Levels. On one hand, it absorbs low-level data such as sensations and memories; on the other hand, it examines abstract concepts and sets goals, residing at the top of the information flow. Because of this, consciousness is often compared to an "orchestra conductor" or "brain CEO", capable of Scheduling Different Cognitive Modules to complete tasks that single modules cannot. This is consistent with the idea of the consciousness calculus model mentioned earlier. Crossing Formal Boundaries is also reflected here: consciousness not only executes rules but also Generates and Modifies rules. When encountering difficult problems, humans sometimes think "try another way of thinking", which is equivalent to Dynamically Changing the Representation Method of the Problem. Traditional algorithms won't do this; they can only run in established representations. But the plasticity of the human brain allows us to re-encode problems: such as understanding algebraic problems with geometry, or transforming a problem into a familiar one by analogy. This ability to "Cross Expression Boundaries" is unique to consciousness. Perhaps this is also the secret of humans solving certain complex problems: when one road is blocked, we try to jump out of the original framework. In addition, human Intuition can often give the answer directly, and then the rational part verifies the proof. Many mathematicians admit that guessing a theorem holds usually relies on intuitive inspiration, and once convinced it is true, they try to construct a proof. This process of conclusion first, reasoning later is formally irregular, but extremely effective when exploring new truths. This shows that human thinking has a non-linear side: it does not strictly deduce the unknown from the known, but can "jump to" the unknown and then backtrack to verify. This jump may come from parallel computation or global association of the subconscious. Regardless of the specific mechanism, Consciousness Seems to Naturally Possess the Potential to Solve Formal System Puzzles, because it is not limited by a fixed rule system and can continuously expand its "axioms" and "operation rules".

Penrose inferred that "consciousness can never be a computation". He believed consciousness involves physical actions not yet understood, such as quantum collapse, which is fundamentally different from algorithmically simulatable processes. Although his Orch-OR theory is inconclusive, this view coincides with our discussion on Hyper-Turing: if consciousness utilizes capabilities in nature not yet integrated into the Turing model, then consciousness systems may break through the Church-Turing boundary. For example, if consciousness can generate true non-algorithmic randomness or non-computable behavior, then it is not constrained by traditional complexity when solving certain problems. However, from an information perspective, we might not need to assume miracles violating physical laws: maybe consciousness just achieves efficiency unattainable by ordinary computers through Extremely Complex Parallel Processing, thereby Apparently transcending Turing computation. After all, the Turing machine model assumes sequential execution, while the brain has hundreds of billions of neurons in parallel. Although any finite parallel can be simulated, when the degree of parallelism is high enough, exponential scale operations can be done in polynomial time. The human brain may not truly achieve exponential parallelism, but evolution may have optimized very efficient "pruning" and "pattern recognition" functions, enabling it to quickly target the key to the problem and save a lot of search. So maybe there is no metaphysics; the transcendence of consciousness comes from Highly Evolved Semantic Algorithms. The key is, we have not yet fully understood this algorithm, so we cannot replicate its power. It can be said that human consciousness is essentially A Meaning Leap Structure: it is not a single level, but an integration of many levels, capable of operating by jumping across different abstract levels. Therefore, it can cope with multi-level complex problems.

5.3 Meaning-Driven Calculus System: Beyond Turing Boundaries?

Synthesizing Gödel's theorem and consciousness characteristics, we propose a bold question: Is there a calculus system driven by meaning (semantics) that can break through the boundaries of the Turing machine? Here "transcending Turing" does not necessarily refer to computing undecidable problems, but to systems that can significantly outperform all Turing machine algorithms when solving problems like NP class. If it exists, it will be an expansion of the Church-Turing thesis in terms of effective computation. Textbooks tell us that Turing machines can calculate all computable functions, but Computable Does Not Equal Efficiently Computable. Perhaps some problems cannot be solved in polynomial time under the Turing framework, but can be with a Hyper-Turing model. For example, theoretical models introducing "Relativistic Computation" (achieving hypertasks through infinite acceleration) or "Quantum Oracle" can do things Turing machines cannot. However, those are either physically unreachable or still mathematical fantasies. By contrast, a Meaning-Driven Calculus System sounds more feasible: it still obeys physical laws, but it utilizes the extra resource of meaning, just like the DIKWP and consciousness models we expounded earlier. What this system transcends is Artificial Formal Rigidity, not physical possibility. Through understanding semantics, it avoids the long process required by Turing machines to exhaust symbols. Intuitively, if we inject the knowledge of the entire internet and the intuition of all mankind into a system, its ability to solve problems is obviously far superior to a computer that only calculates blindly. Large language model GPT-4 has already demonstrated surprising reasoning power on some complex problems, and researchers have even let GPT-4 "think" about the P vs NP problem, arriving at the conclusion P≠NP and philosophical arguments itself. Although this is not a rigorous proof, it is interesting that systems like LLMs containing massive semantic knowledge Can Often Hit the Nail on the Head Pointing Out the Essence of the Problem, reflecting extraordinary semantic depth. Perhaps with further improvement, such models can also rival or even surpass human experts in solving combinatorial puzzles; then it would be "transcending" traditional algorithms in some sense.

Of course, truly proving that a computational model transcends the Turing machine requires formal argumentation or performance equivalent to known Hyper-Turing capabilities. However, we can change the angle: Maybe the Turing machine itself can continuously expand its computational paradigm. Historically, with problem needs, we added capabilities like randomness (Probabilistic Turing Machine), parallelism (PRAM model), interaction (Online Algorithm), etc., to computing devices. Each addition changed the efficiency boundary of solvable problems. For example, with randomness, some algorithms can complete tasks previously unthinkable in expected polynomial time; adding parallelism, the relationship between P and NP might be different in the parallel world. Analogously, If We View "Semantic Understanding" as a Computational Resource, then the Turing machine model might be upgraded to "Turing + Semantic Machine". This machine has a set of special instructions capable of calling a Semantic Oracle (such as a knowledge base or large model) to assist operations. Such systems have appeared in embryonic form in reality: for example, autonomous AI like AutoGPT utilize LLMs as brains while being able to call external tools, cycling between the two to solve complex tasks. This architecture actually combines symbolic computation with semantic reasoning and has already demonstrated automatic solving capabilities for certain complex tasks. Although performance is limited currently, this is a clear direction: By Incorporating Semantic Components into Computational Models, We Are Approaching Problems Previously Considered Intractable Step by Step. One day, if this route succeeds, perhaps we can announce: Yes, we have constructed a general intelligence that can crack NP-complete problems in polynomial time—that will fundamentally rewrite the landscape of computational complexity. The answer to the P vs NP problem may not be a simple yes/no, but: under the pure formal computation framework P≠NP, but under the enhanced semantic computation framework, NP problems can be solved in expected time similar to polynomial. This is of course a speculative conclusion, but not without basis. As mentioned in the introduction, AlphaFold solved the NP-hard protein folding, AlphaGo defeated humans in exponentially large chess games, these milestones are telling us that Computational Modes Imbued with Meaning Are Transcending the Boundaries of Traditional Computation.

In summary, we examined the P=NP problem against a grander background and found it shares commonalities with Gödel's incompleteness and the unique capabilities of human consciousness: all are phenomena where Formal Systems Cannot Be Completely Self-Consistent and Need Higher-Level Semantics to Compensate. Therefore, the solution is likely not within the traditional framework but requires introducing new paradigms. If we acknowledge that Cognition and Semantics are "High-Dimensional Supplements" to Computation, then problems like P vs NP might be readily solved. In the next section, we will propose some concrete and feasible models and experimental concepts to explore the influence of semantic computing on NP problems. This will make the discussion more grounded while verifying the validity of previous theoretical inferences. Finally, we will engage in divergent thinking from philosophical and cosmological perspectives in Section 7, further deepening the understanding of "unsolvable" and "solvable".

6. Simulation Models and Experimental Suggestions

6.1 Conception of DIKWP Problem Solving Agent

To verify the role of semantic paths in solving NP problems, we can try to build a DIKWP Architecture Intelligent Agent, let it challenge some NP-complete problems, and see if its performance can surpass traditional algorithms. This agent should include the following modules: Data Layer (receiving original problem descriptions, such as SAT formulas, graph structures, etc.), Information Layer (extracting key information differences of the problem, such as variable frequency, graph structure characteristics of constraints, etc.), Knowledge Layer (calling relevant knowledge, such as logic rules, empirical solution patterns of past similar problems, domain-specific heuristics, etc.), Wisdom Layer (planning and decision-making for solutions, such as deciding which algorithm strategy to adopt, when to backtrack or adjust strategy during search), Purpose Layer (setting solving goals, evaluating the degree of approximation of current progress to the final goal, conducting global control when necessary). The entire agent works through a cyclic feedback: data and low-level operations continuously supply information, and high levels monitor intermediate results based on Purpose and then give directional guidance to low levels.

Specifically, taking the NP-complete problem 3-SAT as an example:

Data layer reads CNF formulas, recording variables and clauses.

Information layer counts the polarity, frequency of each variable, whether there are simple patterns like unit clauses or pure literals, and constructs variable association graphs (which variables often appear in the same clause). These are Difference Information.

Knowledge layer calls logic simplification rules (such as decomposing formulas containing certain clauses, applying unit propagation, etc.), and refers to solving experience of formulas with similar difficulty (perhaps introducing machine learning classification to judge which heuristic fits this formula). The knowledge base can contain problem-solving techniques summarized by humans, such as "dismantling large clauses", "variable elimination" methods, and the agent can match current information patterns to select suitable knowledge.

Wisdom layer formulates solving plans. For example, deciding to use the DPLL algorithm framework, but using greedy heuristics or random when choosing branches, switching strategies after multi-depth DFS, etc. The Wisdom layer will also evaluate progress based on Purpose, such as how much the number of clauses has reduced after current simplification, to what extent the search tree has expanded, predicting how long it will take to solve if continuing like this. If progress feels too slow (inconsistent with Purpose expectation), the Wisdom layer can trigger the Knowledge layer to change a method or adjust heuristic strategy.

Purpose layer monitors from start to finish, "Goal = Find satisfying assignment". It will calculate Tension Function based on information provided by the Wisdom layer: such as the proportion of remaining unsatisfied clauses, variable assignment stability, etc., as the system's distance from the goal. If tension does not decrease or even increases for a period, the Purpose layer may judge the current path is wrong and needs significant adjustment, such as Resetting Partial Assignments, or even abandoning the current method to use an alternative method. This high-privilege intervention of the Purpose layer avoids the system drilling into a dead end—this is like when a person solving a problem suddenly has a "flash of inspiration" to switch strategies.

Through such an agent, we expect to see: compared to traditional single-path algorithms, it can More Flexibly Jump Out of Local Optima and Take Fewer Detours. For example, traditional DPLL might dig deep in a wrong branch for a long time before backtracking, but our agent, due to Purpose tension monitoring, will realize the wrong trend earlier and backtrack, thereby saving a lot of search. Therefore, in actual solving, it might find a solution with complexity far below the worst case. If this performance appears stably, it will indicate Semantic Drive Indeed Improves Solving Efficiency.

To quantify evaluation, we can compare: let traditional algorithms and the DIKWP agent solve a set of NP-complete problem instances (such as random 3-SAT, large-scale TSP, etc.) respectively, recording solving time, number of search nodes, etc. If the agent is significantly superior to traditional ones (especially on hard instances near phase transition), it indicates semantic paths are greatly beneficial. In addition, we can analyze the internal logs of the agent when solving problems, seeing when it switches strategies and what extraordinary jump decisions it makes, summarizing how those decisions circumvented some exponentially proliferating branches in formal algorithms. If laws can be summarized, further refine artificial algorithms.

6.2 Semantic-Driven Graph Structure Deduction Machine

Besides solving combinatorial optimization, we can also design a framework for theoretical exploration, called "Semantic-Driven Graph Structure Deduction Machine". Its purpose is to study whether semantic computation can break through the limits of conventional computation in the fields of mathematical proof or reasoning. For example, Gödel's theorem shows that certain assertions are unprovable within conventional logic systems. If there is a machine that can extend axioms itself through semantic analysis, it might be able to prove more truths. The design of the deduction machine is similar to the agent above, but applied in the proof domain:

Data layer reads the proposition to be proved and the existing set of axioms;

Information layer analyzes concepts involved in the proposition and differences from known theorems;

Knowledge layer calls mathematical knowledge (definitions, theorem libraries), also introducing meta-knowledge like "if encountering self-referential sentences, lift system axioms", etc.;

Wisdom layer plans proof strategies, such as choosing induction or contradiction, proof order, etc.;

Purpose layer is "Prove this proposition" or "Find counterexample". Purpose tension can be measured by proof progress (such as number of sub-propositions proven, gap from target key conditions).

When the proof machine gets stuck, the Purpose layer might decide to Extend the System: for example, adding a new axiom conjecture (equivalent to us guessing a lemma or stronger postulate under intuition). This is similar to humans having to boldly assume a conclusion holds and then seeing if it contradicts when proving difficult problems, or simply upgrading to a stronger theoretical framework. The deduction machine can attempt such operations under control, then continue proving in the stronger system. If completed successfully, mark the original proposition as provable under the extended system. This of course involves the risk of truth judgment, but the machine can use methods like model checking to ensure consistency to a certain extent. This Self-Modification is a taboo in traditional automated theorem proving (automatic systems generally do not change axioms), but it is not impossible under semantic drive. After all, the history of human mathematical development itself is a process of constantly introducing new axiom systems (such as from arithmetic to set theory). Simulating this process with machines will help us understand the relationship between Formal Systems and Semantic Insights. Although whether the deduction machine can truly prove propositions independent of ZFC (such as the continuum hypothesis) remains unknown, it can at least explore a broader proof space and improve automated reasoning capabilities.

6.3 Fusion with Traditional Algorithms

Whether it is a solving agent or a deduction machine, we do not intend to abandon the advantages of traditional algorithms, but to fuse the two: Let Semantics Guide Computation, Let Computation Verify Semantics. The agent can select a direction under the guidance of the semantic path, then use traditional exact algorithms (DPLL, branch and bound, etc.) to quickly run to local results, and then evaluate correctness. Unreliable semantic guesses can be screened by deterministic algorithms to avoid errors. This Human-Machine Combination mode has already played a role in many AI scenarios, such as SAT solvers combined with machine learning predictive heuristics, program verification combined with static analysis and manual hints, etc. We hope to find the optimal degree of semantic intervention through experiments: too little intervention is ineffective, too much may lead to errors or even consume more time. The DIKWP framework provides a distinct hierarchical structure, which may help reasonably allocate computation and semantics. For example, the Data layer and Information layer can be fully entrusted to program execution (after all, statistical information and simple logic simplification programs do it fast), Knowledge layer and Wisdom layer can introduce learning components or rule libraries (imitating expert behavior), and Purpose layer is managed by a meta-controller (might require reinforcement learning to train when to adjust). By repeatedly training and adjusting on different problems, we might get a General Problem Solver that has obvious advantages over pure algorithms for large-scale complex problems. This would be strong empirical evidence for theoretical problems like P vs NP: if in the long run, even if we cannot prove P≠NP, we have bypassed it with engineering methods and solved a large number of NP problems efficiently in reality. Just as many integer programming instances are theoretically NPC but commercial solvers can already find optimal solutions very quickly, the goal of the semantic solver is to further expand the range of "effectively solvable" problems, making the shadow of NPC gradually shrink.

6.4 Possible Technical Challenges

Of course, implementing the above models also faces many difficulties. Semantic representation and reasoning itself is a hard nut to crack in AI. We need to avoid turning the semantic layer into another exponential black box (like LLMs, although having knowledge, reasoning is unreliable). How model layers communicate, and how strategies for high-level intervention in low levels are designed, all need experimental exploration. In addition, how to objectively evaluate the effect of the semantic model also requires caution, needing statistics from a large number of experiments and variable control. Most importantly, if experimental results show that the semantic model improvement is limited, or even still faces heavy resistance on hard instances, it may mean our semantic reconstruction idea is not enough or has little practical effect. This will also provide valuable feedback, making us reflect on theoretical assumptions. But in any case, such attempts will promote the intersection of AI and computational complexity fields, beneficial for understanding the essence of intelligence and computational limits.

In summary, simulation experiments are both verification of the preceding theory and the first step to turning abstract concepts into realistic tools. Just as computer science often moves from theoretical questions to engineering solutions, we hope to test how far "semantic computing" can go by building DIKWP agents and semantic deduction machines. If successful, even if we cannot immediately declare the solution to the P=NP puzzle, at least we can prove: With the Power of Semantics, We Can Narrow the Limits Set by Formal Theories in Practice. This will enhance our confidence in the advantages of human intelligence and inject new vitality into computation theory. Beyond experiments, let us finally broaden our horizons and think about the ultimate relationship between information, computation, and meaning from the heights of the universe and philosophy.

7. Extended Reflections on Philosophy and Cosmic Information Structure

7.1 The Universe as a Semantic Compression Evolutionary Body?

Modern science often views the universe as an ocean of information. Physical laws can be seen as compressed descriptions of natural information. For example, the evolution of the entire universe can be characterized by relatively simple equations (Einstein field equations, Schrödinger equation, etc.). Compared to listing the motion state of every particle, these equations are obviously highly Compressed information content. Some propose the view of "IT from BIT" (all things originate from bits), believing information is the fundamental reality, and matter and energy are just manifestations of information. Being bolder, Cosmic Evolution Itself Might Follow Some Semantic Compression Principle: nature tends to make information representation as concise and efficient as possible without violating constraints. This sounds abstract, but can be linked to entropy and energy principles. The law of entropy increase makes the universe tend towards disorder (information expansion), but we also see the universe spontaneously forming rules (ordered structures like galaxies, life), and those structures can be viewed as compression of local information (e.g., DNA compresses biological evolutionary information). Some researchers even found that different information systems undergo similar minimization processes during evolution, optimizing like computer data compression. If this is true, it means the universe possesses some kind of "Pursuit of Meaning": constantly extracting commonalities and simplifying descriptions in chaos to achieve higher-level organizational forms. This has similarities to intelligence acquiring knowledge, as if The Universe Itself is Learning.

Imagine the universe is a huge computation. What it calculates is not meaningless bit transformations, but generating more meaningful patterns step by step—just like large model training acquiring concepts from compressed corpora. In this way, it is logical for the universe to emerge with "high semantic" entities like life and consciousness, because that is the continuation of information compression evolution. Looking at the P vs NP problem again at this time: perhaps for a "Great Calculator" like the universe, there are no true "unsolvable" problems. As long as enough time is given, the system will always evolve stronger pattern recognizers to conquer difficulties. The reason humans feel NP problems are hard might just be because we are at a limited stage of information evolution and have not mastered high-enough order compression laws to solve them. But The Information Structure of the Universe as a Whole May Already Contain the Solution at a Higher Level. For example, NP difficulties are essentially exponential combinations, but if viewed from a higher dimension, maybe those combinations are not disordered, and there are undiscovered patterns available. Once patterns are found, the exponential collapses into polynomial. This is similar to discovering that prime testing was not a random hard problem but had profound number theoretic structures to exploit. The difference is that this insight may no longer rely on a person's flash of inspiration, but might require The Accumulated Knowledge of the Whole Civilization or even New Physical Discoveries to obtain.

7.2 "Unsolvable" or "Unclosed Semantic Chain"?

Philosophically, "Agnosticism" believes there are some problems humans can never know the answer to. In computer science, this corresponds to "uncomputable problems", such as the Halting Problem. However, we might ask in return: Is "uncomputable" absolutely insurmountable? The Halting Problem is unsolvable under the Turing machine model, but if Oracle or human assistance is allowed, it is possible to judge whether a program halts in some special cases. Another example is mathematical statement independence: the continuum hypothesis cannot be proven or disproven under ZFC, but might have a definite conclusion under a stronger axiom system. It seems that unsolvable problems are mostly Relative to a certain system, not absolutely untouchable truths. Gödel's incompleteness theorem does not say truth is forever unreachable, only that any given system has limitations—changing the system can break limitations, but the new system has new limitations, ad infinitum. Is it possible there exists an "Ultimate Semantic Closed Loop" such that all truths can be seen through under some framework? In other words, is there "a set of semantic calculus" that can exhaust all meanings? This is almost asking for an omnipotent God's perspective. From the perspective of bounded rationality, maybe the answer is no: there will always be higher-level meanings we haven't mastered, always unclosed semantic chains. But if viewed from the universe as a whole, as a complete container of information, perhaps Internally There Are No Problems Unsolvable to Itself. The lack of solution to any problem is just insufficient knowledge or a temporarily unclosed semantic link. When wisdom develops to a certain stage, these problems are readily solved, while deeper problems emerge. Thus there is an eternal pursuit of continuous evolution. Optimistically speaking, what we view as untouchable today, such as the rigorous proof of P vs NP, or even the decision of the Halting Problem, might be solved in the future with the help of some Brand New Logic or Brand New Computing Medium. It's just that at that time, we will definitely face challenges of a higher level.

Therefore, from the perspective of semantic links, "Unsolvable" is Dynamic: a fault caused by the current semantic network not being closed. Supplement the network, and this problem is no longer unsolvable, but new faults will appear. From this, we might be able to redefine "intractability": not simply saying polynomial or exponential growth rate, but Semantic Network Closure Difficulty—how many steps are needed to jump out of the original network to solve. If extending the network solves it in just one or two steps, then although formally exponential, it might be listed as "solvable" in the long run. If infinite steps are needed, that is absolutely unsolvable (like the tower of truth rising infinitely, never reaching completeness). Currently, it is unknown which type P vs NP belongs to, but most tend to think it is not absolutely unknowable, otherwise the Clay Prize would not offer a reward for solving it. Therefore, P vs NP might just be A Not Yet Closed Ring in Our Semantic Chain, which will one day be closed through new ideas—regardless of whether the result is equal or not.

7.3 Free Will: Selection Flow Dominated by P→W Tension

Finally, let's talk about the philosophical puzzle of free will. Free will refers to humans seeming to be able to choose autonomously, not completely determined by physical laws and past states. Under the determinism framework, this is hard to explain: if the brain is just particle motion, and all decisions are destined by antecedents, where does "my" freedom come from? One view is that free will is the Chaos and Complexity of the brain making its behavior appear unpredictable externally, coupled with the subject's self-cognition, leading to the belief that one chose freely. We use the DIKWP model to analyze: The relationship between Purpose and Wisdom (Behavior), i.e., P→W, is likely the key area where free will is born. In our model, Purpose is high-level goal or motivation, and behavior is low-level actual execution. If Purpose can be completely projected into behavior through a series of deterministic rules, then machines can also simulate the same behavior, and the subject has no real freedom because everything is limited by rules. But if the mapping from Purpose to behavior has some irreducible tension, it means There Exist Multiple Possible Behaviors That All Conform to High-Level Purpose, and the subject has a certain "degree of freedom" in specific choices. This degree of freedom does not come from randomness, but from Irreducible Subject Preference. It may manifest as: under the same logic, the subject might make a different choice the second time than the first, because subtle changes in intention at the moment or micro-differences in environment lead to another behavior also being considered reasonable. For external observers, this behavior cannot be strictly predicted because there are no immutable rules, and the subject Retains Some Internal Decision Space. This seems to be how free will operates: we often weigh and then make a decision "on a whim", and even we ourselves find it hard to explain afterwards why we chose this instead of that—because at the Purpose layer, both satisfy the goal, just some indescribable tendency ultimately dominated the behavior.

This indescribable tendency might be precisely The Tension between High-Level P (Purpose) and Low-Level W (Behavior), a non-deterministic "collapse" at the last moment. Some associate quantum uncertainty to explain free will; although there is no consensus, the analogy exists: when multiple equivalent possibilities exist, some mechanism leads to the actual realization of one of them. Free will is the Collapse of Choice of the Mind, but unlike quantum, it may not be purely random, but carries the subject's unique bias (so-called personality, character). This bias cannot be fully computationally predicted because it involves complex functions shaped by the subject's entire life history. Thus, in any model trying to simplify humans into formulas, something will be missed, making human behavior possess some unpredictability and creativity. From a computational perspective, this is similar to our aforementioned Consciousness Calculus Model—there is Purpose traction, but behavior is not determined by a unique path, and there are parts irreducible to simple algorithms, so outsiders cannot pre-play it. True Freedom May Exist Where Algorithms Are Irreducible. If it can really be reduced, then it is not free. So philosophically people would say free will cannot be completely explained by science, because once completely explained and predicted, it proves it is not free. This might be the paradox of free will.

However, starting from the ideas of this report, we admit that free will originates precisely from Irreducible Choices Across Problem Spaces. The subject faces many possible actions (each possibility can be seen as a solution to a different sub-problem), and can only make a choice relying on their own overall Purpose and experiential tension; external systems cannot give general laws. In other words, free will might be another manifestation of humans crossing the algorithmic framework: our behavior cannot be characterized by a fixed complexity class because humans constantly change strategies, emerge new preferences, and constantly jump out of established frameworks. Everyone's decision flow is a highly personalized creative process. This sounds romantic, but it might be the fact. If a Meaning-Driven Calculus System is Highly Complex to a Certain Degree, Its Behavior is Also Open to Itself, and thus it possesses characteristics similar to free will.

These philosophical discussions aim to show that the P=NP and semantic computing problems we study are actually connected to many fundamental philosophical propositions. They jointly point to a theme: The Unity of Opposites between Formal Rules and Infinite Creativity. Human wisdom and the richness of the universe proclaim something beyond any closed system—that is meaning, emergence, freedom. Perhaps as discussed in "Gödel, Escher, Bach": life and consciousness are strange loops capable of self-reference and self-leap, thereby breaking free from the limitations of ordinary systems. The revelation this brings to computation theory is that we should be brave enough to break through the framework of traditional models and introduce higher-level concepts to understand computation.

Conclusion

Throughout the text, starting from classic complexity theory, by introducing the DIKWP semantic structure and consciousness model, we have conducted a brand-new reflection and reconstruction on the P=NP problem and even broader uncomputable problems. The Traditional P=NP Framework emphasizes algorithm steps and time scales but may ignore the role of Semantic Information in the solving process. Through the DIKWP model, we see that real-world problem solving is a chain spanning Data, Information, Knowledge, Wisdom, and Purpose. The solution to many difficulties is not exhaustion, but completed through semantic reasoning and cognitive leaps. Thus we proposed the concept of Semantic Computability, elevating computational complexity to the semantic level for consideration, defining metrics such as Semantic Compression Degree and Semantic Leap Cost to explain why P-class problems have pattern-following "docile" structures, while NP-class problems often have disordered semantic structures and require leap thinking to solve. Next, we explored the possibility of the Consciousness Calculus Model—by introducing Purpose tension and self-reflexive feedback into the computational framework, equipping machines with global views and creativity similar to human consciousness, or potentially forming capabilities similar to Oracle to structurally conquer NP difficulties. This led to an extended discussion on the Boundaries of Formal Systems: Gödel's theorem embodies the semantic cracks of formal systems, human consciousness demonstrates the power of cross-framework leaps, and perhaps in the future, some meaning-driven calculus system can really break through the shackles of Turing machines and achieve efficient solutions to complex problems.

We proposed some Experimental Suggestions, such as building DIKWP intelligent agents to solve NP problems, and semantic-driven deduction machines, etc., to verify the actual effect of semantic computing. This will turn philosophical thoughts into engineering verification, which is very important subsequent work. Regardless of the result, it will enhance our understanding of intelligence and computational limits. Finally, we turned our eyes to philosophy and the universe, thinking about the relationship between information structure and the evolution of meaning, proposing views such as "unsolvable" might just be unclosed semantic chains, and free will originates from irreducible choices. These thoughts emphasize: Meaning (semantic) may be an existence as fundamental as matter and energy, endowing computation with new dimensions. The ultimate answer to the P=NP problem may not be obtained solely through symbolic derivation but requires us to embrace the perspectives of meaning and cognition. When we truly integrate computer science and cognitive science, we might be able to solve this famous puzzle someday in the future, and gain deeper insights into ultimate questions like intelligence and essence in the process. As a computer scientist said: "Maybe P≠NP, because machine calculation is inferior to mental calculation; maybe the mind itself is doing computation beyond machines." In any case, humanity's unremitting exploration of truth and meaning will continue to drive the progress of technology and thought. P=NP is just a signpost, guiding us towards a more comprehensive understanding of computation and wisdom. On the journey to crack this puzzle of the century, a new scientific paradigm shift might be gestating—From Symbolic Computing Paradigm to Semantic Computing Paradigm. We shall wait and see.

References

Wikipedia: "P versus NP problem" introduces the formal definitions of P class and NP class, and the background of the problem proposed by Cook et al.

Yucong Duan et al.'s interview article on Phoenix Network, describing the hierarchy and networked feedback structure of the DIKWP model in the cognitive process.

Proginn Technology Circle: "DeepMind Cracks Protein Folding Puzzle" reports how AlphaFold achieved a breakthrough on problems unreachable by traditional algorithms through learning massive data.

CSDN Blog: Quantum Bit's review of LeCun team's "semantic compression" research. This study compares the differences between humans and LLMs in semantic compression, pointing out that LLMs bias towards statistical compression while humans value detailed context.

Zhihu Column mentions the Lucas–Penrose argument believing that, based on Gödel's theorem, human mind can insight into formal system truths, while Turing machines are limited by logical systems and cannot completely simulate this ability.

Zhihu Column and AIHub articles discuss the compression characteristics of cosmic information evolution: different information systems undergo similar minimization processes, similar to data compression and optimization.

Other materials come from public online articles and literature, combined with the author's arguments, used to support the reasoning and assumptions in this article, which have been marked by citations in the text.

Cited Sources:

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