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Lower bounds for resolution and cutting plane proofs and monotone computations, (1997)

by P Pudlák
Venue:Journal of Symbolic Logic,
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Interpolation and SAT-based model checking

by K. L. Mcmillan , 2003
"... Abstract. We consider a fully SAT-based method of unbounded symbolic model checking based on computing Craig interpolants. In benchmark studies using a set of large industrial circuit verification instances, this method is greatly more efficient than BDD-based symbolic model checking, and compares f ..."
Abstract - Cited by 285 (11 self) - Add to MetaCart
Abstract. We consider a fully SAT-based method of unbounded symbolic model checking based on computing Craig interpolants. In benchmark studies using a set of large industrial circuit verification instances, this method is greatly more efficient than BDD-based symbolic model checking, and compares favorably to some recent SAT-based model checking methods on positive instances. 1
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... model. In particular, given a partition of a set of clauses into a pair of subsets (A, B), and a proof by resolution that the clauses are unsatisfiable, we can generate an interpolant in linear time =-=[20]-=-. An interpolant [11] for the pair (A, B) is a formula P with the following properties: – A implies P , – P ∧ B is unsatisfiable, and – P refers only to the common variables of A and B. Using interpol...

Lazy Satisfiability Modulo Theories

by Roberto Sebastiani - JOURNAL ON SATISFIABILITY, BOOLEAN MODELING AND COMPUTATION 3 (2007) 141€“224 , 2007
"... Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a first-order formula with respect to some decidable first-order theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingl ..."
Abstract - Cited by 189 (50 self) - Add to MetaCart
Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a first-order formula with respect to some decidable first-order theory T (SMT (T)). These problems are typically not handled adequately by standard automated theorem provers. SMT is being recognized as increasingly important due to its applications in many domains in different communities, in particular in formal verification. An amount of papers with novel and very efficient techniques for SMT has been published in the last years, and some very efficient SMT tools are now available. Typical SMT (T) problems require testing the satisfiability of formulas which are Boolean combinations of atomic propositions and atomic expressions in T, so that heavy Boolean reasoning must be efficiently combined with expressive theory-specific reasoning. The dominating approach to SMT (T), called lazy approach, is based on the integration of a SAT solver and of a decision procedure able to handle sets of atomic constraints in T (T-solver), handling respectively the Boolean and the theory-specific components of reasoning. Unfortunately, neither the problem of building an efficient SMT solver, nor even that
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... of equality, linear rational arithmetic, arrays, and finite sets [52, 103, 120, 159]. In propositional logic, interpolants can be computed from resolution proofs using a simple method due to Pudlák =-=[136]-=-. For theories T with the quantifier-free inter30 Clark Barrett and Cesare Tinelli polation property, which guarantees the existence of quantifier-free interpolants for any T-unsatisfiable pair A,B of...

Finding Hard Instances of the Satisfiability Problem: A Survey

by Stephen A. Cook, David G. Mitchell , 1997
"... . Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case ..."
Abstract - Cited by 127 (1 self) - Add to MetaCart
. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.

An interpolating theorem prover

by K. L. Mcmillan - In TACAS , 2004
"... Abstract. We present a method of deriving Craig interpolants from proofs in the quantifier-free theory of linear inequality and uninterpreted function symbols, and an interpolating theorem prover based on this method. The prover has been used for predicate refinement in the Blast software model chec ..."
Abstract - Cited by 101 (11 self) - Add to MetaCart
Abstract. We present a method of deriving Craig interpolants from proofs in the quantifier-free theory of linear inequality and uninterpreted function symbols, and an interpolating theorem prover based on this method. The prover has been used for predicate refinement in the Blast software model checker, and can also be used directly for model checking infinite-state systems, using interpolation-based image approximation. 1
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... and refers only to variables common to A and B. If A and B are propositional formulas, and we are given a refutation of A∧B by resolution steps, we can derive an interpolant for (A,B) in linear time =-=[5,12]-=-. This fact has been exploited in a method of over-approximate image computation based on interpolation [7]. This provides a complete symbolic method of model checking finite-state systems with respec...

Some Consequences of Cryptographical Conjectures for . . .

by Jan Krajícek , Pavel Pudlák , 1995
"... We show that there is a pair of disjoint NP-sets, whose disjointness is provable in S 1 2 and which cannot be separated by a set in P=poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 1 2 admittin ..."
Abstract - Cited by 70 (14 self) - Add to MetaCart
We show that there is a pair of disjoint NP-sets, whose disjointness is provable in S 1 2 and which cannot be separated by a set in P=poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 1 2 admitting an NP -definition of primes about which it can prove that no number satisfying the definition is composite. As a corollary we obtain that the Extended Frege (EF) proof system does not admit feasible interpolation theorem unless the RSA cryptosystem is not secure, and that an extension of EF by tautologies p (p primes), formalizing that p is not composite, as additional axioms does not admit feasible interpolation theorem unless factoring and the discrete logarithm are in P=poly . The NP 6= coNP conjecture is equivalent to the statement that no propositional proof system (as defined in [6]) admits polynomial size proofs of all tautologies. However, only for few proof systems occur...

Space Bounds for Resolution

by Juan Luis Esteban, Jacobo Torán , 2000
"... We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition [11] the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer ..."
Abstract - Cited by 65 (3 self) - Add to MetaCart
We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition [11] the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows to relate the space needed in a resolution proof of a formula to other well studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula, and as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n \Gamma c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general Pigeonhole Principle with n holes and more than n pigeons, need space n + 1 independently of the number of pigeons. Since a matching space upper bound of n + 1 for these formulas exist, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations.

The Lengths of Proofs

by Pavel Pudlák , 1998
"... ..."
Abstract - Cited by 57 (3 self) - Add to MetaCart
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Applications of Craig interpolants in model checking

by K L Mcmillan - In Proceedings of TACAS 2005 [TAC05
"... Abstract. A Craig interpolant for a mutually inconsistent pair of formulas (A, B) is a formula that is (1) implied by A, (2) inconsistent with B, and (3) expressed over the common variables of A and B. An interpolant can be efficiently derived from a refutation of A ∧ B, for certain theories and pr ..."
Abstract - Cited by 50 (0 self) - Add to MetaCart
Abstract. A Craig interpolant for a mutually inconsistent pair of formulas (A, B) is a formula that is (1) implied by A, (2) inconsistent with B, and (3) expressed over the common variables of A and B. An interpolant can be efficiently derived from a refutation of A ∧ B, for certain theories and proof systems. We will discuss a number of applications of this concept in finite-and infinite-state model checking.
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...utation of A ∧ B, for certain theories and proof systems. For example, interpolants can be derived from resolution proofs in propositional logic, and for systems of linear inequalities over the reals =-=[8, 14]-=-. These methods have been recently been extended [10] to combine linear inequalities with uninterpreted function symbols, and to deal with integer models. One key aspect of these procedures is that th...

Pseudorandom Generators Hard for k-DNF Resolution and Polynomial Calculus. Unpublished

by Alexander A Razborov , 2003
"... Abstract A pseudorandom generator Gn : {0, 1} n → {0, 1} m is hard for a propositional proof system P if (roughly speaking) P cannot efficiently prove the statement Gn(x1, . . . , xn) = b for any string b ∈ {0, 1} m . We present a func- ) generator which is hard for Res(ε log n); here Res(k) is the ..."
Abstract - Cited by 50 (4 self) - Add to MetaCart
Abstract A pseudorandom generator Gn : {0, 1} n → {0, 1} m is hard for a propositional proof system P if (roughly speaking) P cannot efficiently prove the statement Gn(x1, . . . , xn) = b for any string b ∈ {0, 1} m . We present a func- ) generator which is hard for Res(ε log n); here Res(k) is the propositional proof system that extends Resolution by allowing k-DNFs instead of clauses. As a direct consequence of this result, we show that whenever t ≥ n 2 , every Res(ε log t) proof of the principle ¬Circuitt(fn) (asserting that the circuit size of a Boolean function fn in n variables is greater than t) must have size exp(t Ω(1) ). In particular, Res(log log N ) (N ∼ 2 n is the overall number of propositional variables) does not possess efficient proofs of NP ⊆ P/poly. Similar results hold also for the system PCR (the natural common extension of Polynomial Calculus and Resolution) when the characteristic of the ground field is different from 2. As a byproduct, we also improve on the small restriction switching lemma due to Segerlind, Buss and Impagliazzo by removing a square root from the final bound. This in particular implies that the (moderately) weak pigeonhole principle PHP 2n n is hard for Res(ε log n/ log log n).
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...but still natural and believable This task will be referred to as proving conditional lower bounds (for the proof system P ). It should be remarked in this respect that NP 6= co-NP implies lower bounds for any propositional proof system whatsoever. Therefore, purely computational above refers to the demand that the assumption itself should speak only about computations and should not attempt to restrict the power of proofs even in a disguised form. One extremely exciting and, in a sense, model approach to this task was gradually developed in the sequence of papers [Raz95b], [BPR97], [Kra97a], [Pud97] and finally became known as the Efficient Interpolation Property (EIP in what follows). EIP was shown to be true for some weak proof systems and it was also remarked that for every proof system (be it weak or strong) EIP implies conditional lower bounds. Unfortunately, it turned out rather soon PSEUDORANDOM GENERATORS 3 [KP98], [BPR00] that neither Frege nor Extended Frege have Efficient Interpolation modulo (somewhat ironically) hardness assumptions of the same sort that are needed to prove conditional lower bounds for proof systems with EIP. This omnipresent hardness assumption is nothing o...

On Interpolation and Automatization for Frege Systems

by Maria Luisa Bonet, Toniann Pitassi, Ran Raz , 2000
"... The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the ..."
Abstract - Cited by 49 (8 self) - Add to MetaCart
The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC 0 -Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC 0 -Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial size TC 0 -Frege. As a corollary, we obtain that TC 0 -Frege as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integ...
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