[Paper Review] Complexity Considerations, cSAT Lower Bound

Radosław Hofman|ArXiv.org|Apr 4, 2007

Complexity and Algorithms in Graphs6 references3 citations

TL;DR

This paper argues that the Boolean satisfiability problem (cSAT) has an exponential lower bound under deterministic computation, implying P ≠ NP. By analyzing completeness in first-order Boolean logic and showing that optimal derivations are inherently exponential, the author constructs a non-deterministic polynomial-time algorithm (NDTM) to solve cSAT, thereby proving NP-completeness and concluding that P ≠ NP, while also addressing relativization via oracle distinctions between deterministic and non-deterministic models.

ABSTRACT

This article discusses completeness of Boolean Algebra as First Order Theory in Goedel's meaning. If Theory is complete then any possible transformation is equivalent to some transformation using axioms, predicates etc. defined for this theory. If formula is to be proved (or disproved) then it has to be reduced to axioms. If every transformation is deducible then also optimal transformation is deducible. If every transformation is exponential then optimal one is too, what allows to define lower bound for discussed problem to be exponential (outside P). Then we show algorithm for NDTM solving the same problem in O(n^c) (so problem is in NP), what proves that P eq NP. Article proves also that result of relativisation of P=NP question and oracle shown by Baker-Gill-Solovay distinguish between deterministic and non-deterministic calculation models. If there exists oracle A for which P^A=NP^A then A consists of infinite number of algorithms, DTMs, axioms and predicates, or like NDTM infinite number of simultaneous states.

Motivation & Objective

• To establish a lower bound on the complexity of solving the Boolean satisfiability problem (cSAT) under deterministic computation.
• To analyze the completeness of Boolean algebra as a first-order theory in Gödel's sense and its implications for derivational complexity.
• To demonstrate that optimal derivations in such a system are inherently exponential, implying the problem is not in P.
• To show that the same problem is solvable in polynomial time by a non-deterministic Turing machine (NDTM), placing it in NP.
• To clarify the distinction between deterministic and non-deterministic computation models through oracle relativization, as per Baker-Gill-Solovay.

Proposed method

• Analyzes the completeness of first-order Boolean logic, asserting that any valid transformation must be derivable from axioms and predicates.
• Demonstrates that if all derivations are exponential in length, then even the optimal derivation must be exponential, establishing a lower bound.
• Constructs a non-deterministic Turing machine (NDTM) that solves cSAT in O(n^c) time, proving the problem is in NP.
• Uses the existence of such an NDTM to contrast with the exponential lower bound under deterministic models.
• Applies the Baker-Gill-Solovay relativization framework to show that oracles distinguishing P and NP must encode infinite sets of axioms, predicates, or states.
• Relies on logical completeness and derivational equivalence to argue that no shorter derivation can exist in the deterministic case.

Experimental results

Research questions

• RQ1 Is the Boolean satisfiability problem (cSAT) solvable in deterministic polynomial time, given that all derivations in its logical system are exponential?
• RQ2 Can the completeness of first-order Boolean logic be used to derive a lower bound on the complexity of cSAT?
• RQ3 Does the existence of a polynomial-time NDTM solution for cSAT imply that P ≠ NP, under the assumption that deterministic derivations are inherently exponential?
• RQ4 How does the relativization of the P vs. NP question distinguish between deterministic and non-deterministic computation models?
• RQ5 What structural properties must an oracle A possess for P^A = NP^A to hold, given the logical and computational constraints of the system?

Key findings

• The paper establishes an exponential lower bound for solving cSAT under deterministic computation, implying the problem is not in P.
• It proves that any optimal derivation in the given first-order Boolean logic framework must be exponential in length, due to the exponential nature of all possible derivations.
• A non-deterministic Turing machine (NDTM) is constructed that solves cSAT in O(n^c) time, confirming the problem is in NP.
• The existence of such an NDTM, combined with the exponential lower bound for deterministic computation, leads to the conclusion that P ≠ NP.
• The paper shows that any oracle A for which P^A = NP^A must encode an infinite number of algorithms, axioms, or non-deterministic states, highlighting a fundamental distinction between deterministic and non-deterministic models.
• The analysis confirms that the Baker-Gill-Solovay relativization result is consistent with the conclusion that P ≠ NP, as oracles distinguishing the classes must be highly complex.