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>

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.

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.

Lower Bounds on OBDD Proofs with Several Orders

2021 ◽  
Vol 22 (4) ◽  
pp. 1-30
Author(s):  
Sam Buss ◽  
Dmitry Itsykson ◽  
Alexander Knop ◽  
Artur Riazanov ◽  
Dmitry Sokolov
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Multiparty Communication ◽  
Constant Degree ◽  
Polynomial Size ◽  
Explicit Constant ◽  
Open Question ◽  
System 1 ◽  
Exponential Size

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

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 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 A Tough Nut for Tree Resolution

2000 ◽  
Vol 7 (10) ◽  
Author(s):  
Stefan Dantchev ◽  
Søren Riis
Keyword(s):  
Lower Bound ◽  
Proof Complexity ◽  
Theorem Provers ◽  
Hard Problems ◽  
Tree Resolution

One of the earliest proposed hard problems for theorem provers is<br />a propositional version of the Mutilated Chessboard problem. It is well<br />known from recreational mathematics: Given a chessboard having two<br />diagonally opposite squares removed, prove that it cannot be covered with<br />dominoes. In Proof Complexity, we consider not ordinary, but 2n * 2n<br />mutilated chessboard. In the paper, we show a 2^Omega(n) lower bound for tree resolution.

scholarly journals ON THE PROOF COMPLEXITY OF THE NISAN–WIGDERSON GENERATOR BASED ON A HARD NP ∩ coNP FUNCTION

2011 ◽  
Vol 11 (01) ◽  
pp. 11-27 ◽  
Author(s):  
JAN KRAJÍČEK
Keyword(s):  
Propositional Logic ◽  
Reflection Principle ◽  
Proof Complexity ◽  
Proof System ◽  
Universal Theory ◽  
General Statement ◽  
Disjunction Property ◽  
Proof Systems ◽  
Polynomial Size ◽  
Theory Formula

Let g be a map defined as the Nisan–Wigderson generator but based on an NP ∩ coNP -function f. Any string b outside the range of g determines a propositional tautology τ(g)b expressing this fact. Razborov [27] has conjectured that if f is hard on average for P/poly then these tautologies have no polynomial size proofs in the Extended Frege system EF. We consider a more general Statement (S) that the tautologies have no polynomial size proofs in any propositional proof system. This is equivalent to the statement that the complement of the range of g contains no infinite NP set. We prove that Statement (S) is consistent with Cook' s theory PV and, in fact, with the true universal theory T PV in the language of PV. If PV in this consistency statement could be extended to "a bit" stronger theory (properly included in Buss's theory [Formula: see text]) then Razborov's conjecture would follow, and if TPV could be added too then Statement (S) would follow. We discuss this problem in some detail, pointing out a certain form of reflection principle for propositional logic, and we introduce a related feasible disjunction property of proof systems.

MULTIRESOLUTION FOR SAT CHECKING

2001 ◽  
Vol 10 (04) ◽  
pp. 451-481 ◽  
Author(s):  
PHILIPPE CHATALIC ◽  
LAURENT SIMON
Keyword(s):  
Data Structures ◽  
System A ◽  
High Compression ◽  
Polynomial Size ◽  
Hard Problems ◽  
Exponential Size ◽  
Size Data

This paper presents a system based on new operators for handling sets of propositional clauses compactly represented by means of ZBDDs. The high compression power of such data structures allows efficient encodings of structured instances. A specialized operator for the distribution of sets of clauses is introduced and used for performing multiresolution on clause sets. Cut eliminations between sets of clauses of exponential size may then be performed using polynomial size data structures. The ZRES system, a new implementation of the Davis-Putnam procedure of 1960, solves two hard problems for resolution, that are currently out of the scope of the best SAT provers.

Sign in / Sign up

Export Citation Format

Share Document