scholarly journals Hard Examples for Common Variable Decision Heuristics

2020 ◽  
Vol 34 (02) ◽  
pp. 1652-1659
Author(s):  
Marc Vinyals
Keyword(s):  
Negative Answer ◽  
Proof System ◽  
Decision Heuristics ◽  
Exponential Time ◽  
Polynomial Size ◽  
Resolution Proofs ◽  
Resolution Proof ◽  
Common Variable

The CDCL algorithm for SAT is equivalent to the resolution proof system under a few assumptions, one of them being an optimal non-deterministic procedure for choosing the next variable to branch on. In practice this task is left to a variable decision heuristic, and since the so-called VSIDS decision heuristic is considered an integral part of CDCL, whether CDCL with a VSIDS-like heuristic is also equivalent to resolution remained a significant open question.We give a negative answer by building a family of formulas that have resolution proofs of polynomial size but require exponential time to decide in CDCL with common heuristics such as VMTF, CHB, and certain implementations of VSIDS and LRB.

scholarly journals Seeking Practical CDCL Insights from Theoretical SAT Benchmarks

2018 ◽  
Author(s):  
Jan Elffers ◽  
Jesús Giráldez-Cru ◽  
Stephan Gocht ◽  
Jakob Nordström ◽  
Laurent Simon
Keyword(s):  
Systematic Study ◽  
Boolean Satisfiability ◽  
Proof System ◽  
Sat Solvers ◽  
Proof Search ◽  
Wide Range ◽  
Clause Learning ◽  
Resolution Proof ◽  
Shed Light

Over the last decades Boolean satisfiability (SAT) solvers based on conflict-driven clause learning (CDCL) have developed to the point where they can handle formulas with millions of variables. Yet a deeper understanding of how these solvers can be so successful has remained elusive. In this work we shed light on CDCL performance by using theoretical benchmarks, which have the attractive features of being a) scalable, b) extremal with respect to different proof search parameters, and c) theoretically easy in the sense of having short proofs in the resolution proof system underlying CDCL. This allows for a systematic study of solver heuristics and how efficiently they search for proofs. We report results from extensive experiments on a wide range of benchmarks. Our findings include several examples where theory predicts and explains CDCL behaviour, but also raise a number of intriguing questions for further study.

scholarly journals Augmenting the Power of (Partial) MaxSat Resolution with Extension

2020 ◽  
Vol 34 (02) ◽  
pp. 1561-1568 ◽  
Author(s):  
Javier Larrosa ◽  
Emma Rollon
Keyword(s):  
Lower Bound ◽  
Objective Function ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Proof System ◽  
Extension Rule ◽  
Resolution Proof

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.

Resolution in the Domain of Strongly Finite Logics

1990 ◽  
Vol 13 (3) ◽  
pp. 333-351
Author(s):  
Zbigniew Stachniak ◽  
Peter O’Hearn
Keyword(s):  
Propositional Logic ◽  
Proof System ◽  
Support Strategies ◽  
Resolution Rule ◽  
Resolution Proof

In this paper the notion of a resolution counterpart of a propositional logic is introduced and studied. This notion is based on a generalization of the resolution rule of J.A. Robinson. It is shown that for every strongly finite logic a refutationally complete nonclausal resolution proof system can be constructed and that the completeness of such systems is preserved with respect to the polarity and set of support strategies.

scholarly journals A Complexity Gap for Tree-Resolution

1999 ◽  
Vol 6 (29) ◽  
Author(s):  
Søren Riis
Keyword(s):  
Predicate Logic ◽  
Proof Complexity ◽  
Polynomial Size ◽  
Combinatorial Principles ◽  
Resolution Proofs ◽  
Symmetrical Group ◽  
Tree Resolution ◽  
Single Proposition ◽  
Exponential Size ◽  
Recent Idea

<p>It is shown that any sequence  psi_n of tautologies which expresses the<br />validity of a fixed combinatorial principle either is "easy" i.e. has polynomial<br />size tree-resolution proofs or is "difficult" i.e requires exponential<br />size tree-resolution proofs. It is shown that the class of tautologies which<br />are hard (for tree-resolution) is identical to the class of tautologies which<br />are based on combinatorial principles which are violated for infinite sets.<br />Actually it is shown that the gap-phenomena is valid for tautologies based<br />on infinite mathematical theories (i.e. not just based on a single proposition).<br />We clarify the link between translating combinatorial principles (or<br />more general statements from predicate logic) and the recent idea of using<br /> the symmetrical group to generate problems of propositional logic.<br />Finally, we show that it is undecidable whether a sequence  psi_n (of the<br />kind we consider) has polynomial size tree-resolution proofs or requires<br />exponential size tree-resolution proofs. Also we show that the degree of<br />the polynomial in the polynomial size (in case it exists) is non-recursive,<br />but semi-decidable.</p><p>Keywords: Logical aspects of Complexity, Propositional proof complexity,<br />Resolution proofs.</p><p> </p>

Implicit proofs

2004 ◽  
Vol 69 (2) ◽  
pp. 387-397 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Proof System ◽  
Bounded Arithmetic ◽  
Polynomial Size ◽  
Arithmetic Theory ◽  
General Method

Abstract.We describe a general method how to construct from a prepositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described “implicitly” by polynomial size circuits.As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory and hence, in particular, polynomially simulates the quantified prepositional calculus G and the -consequences of proved with one use of exponentiation. Furthermore, the soundness of iEF is not provable in . An iteration of the construction yields a proof system corresponding to T2 + Exp and, in principle, to much stronger theories.

scholarly journals Examining Fragments of the Quantified Propositional Calculus

2008 ◽  
Vol 73 (3) ◽  
pp. 1051-1080 ◽  
Author(s):  
Steven Perron
Keyword(s):  
Polynomial Time ◽  
Propositional Calculus ◽  
Proof System ◽  
Polynomial Size ◽  
Propositional Proof ◽  
Polynomial Time Hierarchy

AbstractWhen restricted to proving formulas, the quantified propositional proof system is closely related to the theorems of Buss's theory . Namely, has polynomial-size proofs of the translations of theorems of , and proves that is sound. However, little is known about when proving more complex formulas. In this paper, we prove a witnessing theorem for similar in style to the KPT witnessing theorem for . This witnessing theorem is then used to show that proves is sound with respect to formulas. Note that unless the polynomial-time hierarchy collapses is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi, p-simulates with respect to all quantified propositional formulas. We finish by proving that S2 can be axiomatized by plus axioms stating that the cut-free version of is sound. All together this shows that the connection between and does not extend to more complex formulas.

Structured pigeonhole principle, search problems and hard tautologies

2005 ◽  
Vol 70 (2) ◽  
pp. 619-630 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Proof Complexity ◽  
Proof System ◽  
Search Problem ◽  
Ramsey Property ◽  
Relational Structures ◽  
Proof Systems ◽  
Polynomial Size ◽  
Hashing Function ◽  
The Universe ◽  
Study Behaviour

AbstractWe consider exponentially large finite relational structures (with the universe {0, 1}n) whose basic relations are computed by polynomial size (nO(1)) circuits. We study behaviour of such structures when pulled back by P/poly maps to a bigger or to a smaller universe. In particular, we prove that:1. If there exists a P/poly map g: {0, 1}n → {0, 1}m, n < m, iterable for a proof system then a tautology (independent of g) expressing that a particular size n set is dominating in a size 2n tournament is hard for the proof system.2. The search problem WPHP. decoding RSA or finding a collision in a hashing function can be reduced to finding a size m homogeneous subgraph in a size 22m graph.Further we reduce the proof complexity of a concrete tautology (expressing a Ramsey property of a graph) in strong systems to the complexity of implicit proofs of implicit formulas in weak proof systems.

scholarly journals Extended ASP Tableaux and rule redundancy in normal logic programs

2008 ◽  
Vol 8 (5-6) ◽  
pp. 691-716 ◽  
Author(s):  
MATTI JÄRVISALO ◽  
EMILIA OIKARINEN
Keyword(s):  
Answer Set Programming ◽  
Proof System ◽  
Logic Programs ◽  
Interesting Insight ◽  
Extension Rule ◽  
Normal Logic ◽  
Relationship Of ◽  
The Relationship ◽  
Resolution Proof ◽  
Answer Set

AbstractWe introduce an extended tableau calculus for answer set programming (ASP). The proof system is based on the ASP tableaux defined in the work by Gebser and Schaub (Tableau calculi for answer set programming. In Proceedings of the 22nd International Conference on Logic Programming (ICLP 2006), S. Etalle and M. Truszczynski, Eds. Lecture Notes in Computer Science, vol. 4079. Springer, 11–25) with an added extension rule. We investigate the power of Extended ASP Tableaux both theoretically and empirically. We study the relationship of Extended ASP Tableaux with the Extended Resolution proof system defined by Tseitin for sets of clauses, and separate Extended ASP Tableaux from ASP Tableaux by giving a polynomial-length proof for a family of normal logic programs {Φn} for which ASP Tableaux has exponential-length minimal proofs with respect to n. Additionally, Extended ASP Tableaux imply interesting insight into the effect of program simplification on the lengths of proofs in ASP. Closely related to Extended ASP Tableaux, we empirically investigate the effect of redundant rules on the efficiency of ASP solving.

Frege proof system and TNC°

1998 ◽  
Vol 63 (2) ◽  
pp. 709-738
Author(s):  
Gaisi Takeuti
Keyword(s):  
Modus Ponens ◽  
Free Variable ◽  
Proof System ◽  
Order System ◽  
Equational Logic ◽  
Separation Problem ◽  
Conservative Extension ◽  
First Order ◽  
Polynomial Size ◽  
Extension Axiom

A Frege proof systemFis any standard system of prepositional calculus, e.g., a Hilbert style system based on finitely many axiom schemes and inference rules. An Extended Frege systemEFis obtained fromFas follows. AnEF-sequence is a sequence of formulas ψ1, …, ψκsuch that eachψiis either an axiom ofF, inferred from previous ψuand ψv(= ψu→ ψi) by modus ponens or of the formq↔ φ, whereqis an atom occurring neither in φ nor in any of ψ1,…,ψi−1. Suchq↔ φ, is called an extension axiom andqa new extension atom. AnEF-proof is anyEF-sequence whose last formula does not contain any extension atom. In [12], S. A. Cook and R. Reckhow proved that the pigeonhole principlePHPhas a simple polynomial sizeEF-proof and conjectured thatPHPdoes not admit polynomial sizeF-proof. In [5], S. R. Buss refuted this conjecture by furnishing polynomial sizeF-proof forPHP. Since then the important separation problem of polynomial sizeFand polynomial sizeEFhas not shown any progress.In [11], S. A. Cook introduced the systemPV, a free variable equational logic whose provable functional equalities are ‘polynomial time verifiable’ and showed that the metamathematics ofFandEFcan be developed inPVand the soundness ofEFproved inPV. In [3], S. R. Buss introduced the first order systemand showed thatis essentially a conservative extension ofPV. There he also introduced a second order system(BD).

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