Short Refutations Research Articles

Computational and Proof Complexity of Partial String Avoidability

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

Augmenting the Power of (Partial) MaxSat Resolution with Extension

The refutation power of SAT and MaxSAT resolution is challenged by problems like the soft and hard Pigeon Hole Problem PHP for which short refutations do not exist. In this paper we augment the MaxSAT resolution proof system with an extension rule. The new proof system MaxResE is sound and complete, and more powerful than plain MaxSAT resolution, since it can refute the soft and hard PHP in polynomial time. We show that MaxResE refutations actually subtract lower bounds from the objective function encoded by the formulas. The resulting formula is the residual after the lower bound extraction. We experimentally show that the residual of the soft PHP (once its necessary cost of 1 has been efficiently subtracted with MaxResE) is a concise, easy to solve, satisfiable problem.

On approximating optimal weight “no”-certificates in weighted difference constraint systems

This paper is concerned with the design and analysis of approximation algorithms for the problem of determining the least weight refutation in a weighted difference constraint system. Recall that a difference constraint is a linear constraint of the form $$x_{i}-x_{j} \le b_{ij}$$ and a conjunction of such constraints is called a difference constraint system (DCS). In a weighted DCS (WDCS), a positive weight is associated with each constraint. Every infeasible constraint system has a refutation, which attests to its infeasibility. In the case of a DCS, this refutation is a subset of the input constraints, which when added together produces a contradiction of the form $$0 \le -b$$ , $$b> 0$$ . It follows that every refutation acts as a “no”-certificate. The length of a refutation is the number of constraints used in the derivation of a contradiction. Associated with a DCS $$\mathbf{D: A\cdot x \le b}$$ is its constraint network $$\mathbf{G= \langle V,E, b \rangle }$$ . It is well-known that $$\mathbf{D}$$ is infeasible if and only if $$\mathbf{G}$$ contains a simple, negative cost cycle. Previous research has established that every negative cost cycle of length k in $$\mathbf{G}$$ corresponds exactly to a refutation of $$\mathbf{D}$$ using k constraints. It follows that the shortest refutation of $$\mathbf{D}$$ (i.e., the refutation which uses the fewest number of constraints) corresponds to the length of the shortest negative cycle in $$\mathbf{G}$$ . The constraint network of a WDCS is represented by a constraint network $$\mathbf{G = \langle V, E, b, l \rangle }$$ , where $$\mathbf{l}:\mathbf{E \rightarrow \mathbb {N}}$$ represents a function which associates a positive, integral length with each edge in $$\mathbf{G}$$ . In the case of a WDCS, the weight of a refutation is defined as the sum of the lengths of the edges corresponding to the refutation. The problem of finding the minimum weight refutation in a WDCS is called the weighted optimal length resolution refutation (WOLRR) problem and is known to be NP-hard. In this paper, we describe a pseudo-polynomial time algorithm for the WOLRR problem and convert it into a fully polynomial time approximation scheme (FPTAS).

CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

We consider sets ${\it\Gamma}(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF with a short refutation in extended R, ER, can be easily reduced to an instance of ${\it\Gamma}(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that ${\it\Gamma}(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V_{1}^{1}$. We use the ideas of implicit proofs from Krajíček [J. Symbolic Logic, 69 (2) (2004), 387–397; J. Symbolic Logic, 70 (2) (2005), 619–630] to define from ${\it\Gamma}(0,s,k)$ a nonrelativized NP search problem $i{\it\Gamma}$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V_{2}^{1}$. The reductions are definable in theory $S_{2}^{1}$. We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.

A Short Refutation of Strict Normative Evidentialism

This paper shows that strict evidentialism about normative reasons for belief is inconsistent with taking truth to be the source of normative reasons for belief. It does so by showing that there are circumstances in which one can know what truth requires one to believe, yet still lack evidence for the contents of that belief.

Short propositional refutations for dense random 3CNF formulas

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notable are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Ω(n1.5−ε) clauses (Chvátal and Szemerédi [14], Ben-Sasson and Wigderson [10]). On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with Ω(n2/log⁡n) clauses, shown by Beame et al. [6]. In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are TC0 formulas (i.e., TC0-Frege proofs) for random 3CNF formulas with n variables and Ω(n1.4) clauses.The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek [23]. Since the soundness of these witnesses is verified using spectral techniques, we develop an appropriate way to reason about eigenvectors in propositional systems. To carry out the full argument we work inside weak formal systems of arithmetic and use a general translation scheme to propositional proofs.

Bounded-width QBF is PSPACE-complete

Tree-width and path-width are two well-studied parameters of structures that measure their similarity to a tree and a path, respectively. We show that QBF on instances with constant path-width, and hence constant tree-width, remains PSPACE-complete. This answers a question by Vardi. We also show that on instances with constant path-width and a very slow-growing number of quantifier alternations (roughly inverse-Ackermann many in the number of variables), the problem remains NP-hard. Additionally, we introduce a family of formulas with bounded tree-width that do have short refutations in Q-resolution, the natural generalization of resolution for quantified Boolean formulas.

Parity Games and Propositional Proofs

A propositional proof system is weakly automatizable if there is a polynomial time algorithm that separates satisfiable formulas from formulas that have a short refutation in the system, with respect to a given length bound. We show that if the resolution proof system is weakly automatizable, then parity games can be decided in polynomial time. We give simple proofs that the same holds for depth-1 propositional calculus (where resolution has depth 0) with respect to mean payoff and simple stochastic games. We define a new type of combinatorial game and prove that resolution is weakly automatizable if and only if one can separate, by a set decidable in polynomial time, the games in which the first player has a positional winning strategy from the games in which the second player has a positional winning strategy. Our main technique is to show that a suitable weak bounded arithmetic theory proves that both players in a game cannot simultaneously have a winning strategy, and then to translate this proof into propositional form.

Parameterized Bounded-Depth Frege Is not Optimal

A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results on tree-like Parameterized Resolution---a parameterized version of classical Resolution---and their gap complexity theorem implies lower bounds for that system. The main result of this article significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n Ω(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in Dantchev et al. [2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs.

A Fairly Short Response to a Really Short Refutation

A Fairly Short Response to a Really Short Refutation

A Really Short Refutation of the Pragmatic Theory of Truth

The pragmatic theory of truth (PTT) seeks to illuminate the concept of truth by focusing on concepts like usefulness or adaptivity. However, contrary to common opinion, PTT does not merely face a narrow band of (perhaps) rather artificial counterexamples (as in a case of empirically unfounded but life-extending optimism in a cancer patient); instead, PTT is faced with a vast psychological research literature which suggests that inaccurate beliefs are both (1) pervasive in human beings and nonetheless (2) fully adaptive in many cases. Call this the “pervasive adaptive illusions” (PAI) objection to PTT. According to PAI, the kind of connection drawn by PTT between the beliefs that we (intuitively or pretheoretically) regard as “true” and the beliefs we regard as useful is undercut by hard-nosed empirical work in psychology — work that no empirically-minded pragmatist can ignore. According to PAI, the connection drawn between truth and utility by PTT is subject to a simply overwhelming set of counterexamples (drawn from psychological research, and reviewed below). Thus, PTT is a theory any sensible theorist of truth must reject.

Optimality of size-degree tradeoffs for polynomial calculus

There are methods to turn short refutations in polynomial calculus (Pc) and polynomial calculus with resolution (Pcr) into refutations of low degree. Bonet and Galesi [1999, 2003] asked if such size-degree tradeoffs for Pc [Clegg et al. 1996; Impagliazzo et al. 1999] and Pcr [Alekhnovich et al. 2004] are optimal. We answer this question by showing a polynomial encoding of the graph ordering principle on m variables which requires Pc and Pcr refutations of degree Ω(√ m ). Tradeoff optimality follows from our result and from the short refutations of the graph ordering principle in Bonet and Galesi [1999, 2001]. We then introduce the algebraic proof system Pcr k which combines together polynomial calculus and k-DNF resolution (Res k ). We show a size hierarchy theorem for Pcr k : Pcr k is exponentially separated from Pcr k+1 . This follows from the previous degree lower bound and from techniques developed for Res k . Finally we show that random formulas in conjunctive normal form (3-CNF) are hard to refute in Pcr k .

Hard examples for resolution

Exponential lower bounds are proved for the length-of-resolution refutations of sets of disjunctions constructed from expander graphs, using the method of Tseitin. Since these sets of clauses encode biconditionals, they have short (polynomial-length) refutations in a standard axiomatic formulation of propositional calculus.

Campbell's Refutation of Egoism

In “A Short Refutation of Ethical Egoism,” Richmond Campbell purports to refute “the view that everyone ought (morally) to do what benefits him the most in a given situation.” This is the theory which is “sometimes called impersonal ethical egoism (IEE)” [249). Campbell takes the following proposition as fundamental to his refutation of IEE.I. If an agent ought to do something in a given situation and another agent ought to do something in the same situation, then it is not logically impossible for them to do these things in that situation [250].Granted proposition I, Campbell's refutation would require only:II. There is a situation S where an agent M would benefit most from doing X and another agent N would benefit most from doing what would prevent M from doing X [251].