Reflections on Proof Complexity and Counting Principles

2021 ◽  
pp. 497-520
Author(s):  
Noah Fleming ◽  
Toniann Pitassi
Keyword(s):  
Proof Complexity

scholarly journals Computational and Proof Complexity of Partial String Avoidability

2021 ◽  
Vol 13 (1) ◽  
pp. 1-25
Author(s):  
Dmitry Itsykson ◽  
Alexander Okhotin ◽  
Vsevolod Oparin
Keyword(s):  
Lower Bounds ◽  
Circuit Complexity ◽  
Conjunctive Normal Form ◽  
Proof Complexity ◽  
Proof Systems ◽  
Finite Set ◽  
Propositional Proof Systems ◽  
Classical Proof ◽  
Exponential Size ◽  
Infinite String

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.

scholarly journals Parameterized Proof Complexity

2011 ◽  
Vol 20 (1) ◽  
pp. 51-85 ◽  
Author(s):  
Stefan Dantchev ◽  
Barnaby Martin ◽  
Stefan Szeider
Keyword(s):  
Proof Complexity

scholarly journals The Impact of Heterogeneity and Geometry on the Proof Complexity of Random Satisfiability

2021 ◽  
pp. 42-53
Author(s):  
Thomas Bläsius ◽  
Tobias Friedrich ◽  
Andreas Göbel ◽  
Jordi Levy ◽  
Ralf Rothenberger
Keyword(s):  
Proof Complexity ◽  
The Impact

scholarly journals Proof Complexity of Modal Resolution

2021 ◽  
Author(s):  
Sarah Sigley ◽  
Olaf Beyersdorff
Keyword(s):  
Artificial Intelligence ◽  
Automated Reasoning ◽  
Proof Complexity ◽  
Pigeonhole Principle ◽  
Analytic Tableaux ◽  
Frege Systems ◽  
International Joint ◽  
Resolution Proofs ◽  
Exponential Size ◽  
Sixth International

AbstractWe investigate the proof complexity of modal resolution systems developed by Nalon and Dixon (J Algorithms 62(3–4):117–134, 2007) and Nalon et al. (in: Automated reasoning with analytic Tableaux and related methods—24th international conference, (TABLEAUX’15), pp 185–200, 2015), which form the basis of modal theorem proving (Nalon et al., in: Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI’17), pp 4919–4923, 2017). We complement these calculi by a new tighter variant and show that proofs can be efficiently translated between all these variants, meaning that the calculi are equivalent from a proof complexity perspective. We then develop the first lower bound technique for modal resolution using Prover–Delayer games, which can be used to establish “genuine” modal lower bounds for size of dag-like modal resolution proofs. We illustrate the technique by devising a new modal pigeonhole principle, which we demonstrate to require exponential-size proofs in modal resolution. Finally, we compare modal resolution to the modal Frege systems of Hrubeš (Ann Pure Appl Log 157(2–3):194–205, 2009) and obtain a “genuinely” modal separation.

Chapter 31. Quantified Boolean Formulas

2021 ◽  
Author(s):  
Olaf Beyersdorff ◽  
Mikoláš Janota ◽  
Florian Lonsing ◽  
Martina Seidl
Keyword(s):  
Performance Characteristics ◽  
Proof Complexity ◽  
Proof Systems ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
Clause Learning ◽  
Dependency Schemes

Solvers for quantified Boolean formulas (QBF) have become powerful tools for tackling hard computational problems from various application domains, even beyond the scope of SAT. This chapter gives a description of the main algorithmic paradigms for QBF solving, including quantified conflict driven clause learning (QCDCL), expansion-based solving, dependency schemes, and QBF preprocessing. Particular emphasis is laid on the connections of these solving approaches to QBF proof systems: Q-Resolution and its variants in the case of QCDCL, expansion QBF resolution calculi for expansion-based solving, and QRAT for preprocessing. The chapter also surveys the relations between the various QBF proof systems and results on their proof complexity, thereby shedding light on the diverse performance characteristics of different solving approaches that are observed in practice.

scholarly journals INCOMPLETENESS IN THE FINITE DOMAIN

2017 ◽  
Vol 23 (4) ◽  
pp. 405-441 ◽  
Author(s):  
PAVEL PUDLÁK
Keyword(s):  
Computational Complexity ◽  
Proof Complexity ◽  
Finite Domain ◽  
Semantic Concept ◽  
Syntactic Complexity ◽  
Special Cases ◽  
Incompleteness Theorems ◽  
High Computational Complexity

AbstractMotivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyondNP≠coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be special cases of such general statements and we want to formalize and fully understand these statements. Roughly speaking, we are trying to connect syntactic complexity, by which we mean the complexity of sentences and strengths of the theories in which they are provable, with the semantic concept of complexity of the computational problems represented by these sentences.We have introduced the most fundamental conjectures in our earlier works [27, 33–35]. Our aim in this article is to present them in a more systematic way, along with several new conjectures, and prove new connections between them and some other statements studied before.

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