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34
A Complexity Gap for TreeResolution
 COMPUTATIONAL COMPLEXITY
, 1999
"... It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size treeresolution proofs or is "difficult" i.e requires exponential size treeresolution proofs. It is shown that the class o ..."
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Cited by 16 (2 self)
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It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size treeresolution proofs or is "difficult" i.e requires exponential size treeresolution proofs. It is shown that the class of tautologies which are hard (for treeresolution) is identical to the class of tautologies which are based on combinatorial principles which are violated for infinite sets. Actually it is
A satisfiability tester for nonclausal propositional calculus
 Information and Computation
, 1988
"... An algorithm for satisfiability testing in the propositional calculus with a worst case running time that grows at a rate less than 2 (.25+ε) L is described, where L can be either the length of the input expression or the number of occurrences of literals (i.e., leaves) in it. This represents a new ..."
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Cited by 15 (1 self)
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An algorithm for satisfiability testing in the propositional calculus with a worst case running time that grows at a rate less than 2 (.25+ε) L is described, where L can be either the length of the input expression or the number of occurrences of literals (i.e., leaves) in it. This represents a new upper bound on the complexity of nonclausal satisfiability testing. The performance is achieved by using lemmas concerning assignments and pruning that preserve satisfiability, together with choosing a &quot;good &quot; variable upon which to recur. For expressions in conjunctive normal form, it is shown that an upper bound is 2.128 L.
ZRes: The old DavisPutnam procedure meets ZBDD
 In 17th Intl. Conf. on Automated Deduction (CADE’17), volume 1831 of LNAI
, 2000
"... . ZRes is a propositional prover based on the original procedure of Davis and Putnam, as opposed to its modied version of Davis, Logeman and Loveland, on which most of the current ecient SAT provers are based. On some highly structured SAT instances, such as the well known Pigeon Hole and Urquha ..."
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Cited by 14 (1 self)
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. ZRes is a propositional prover based on the original procedure of Davis and Putnam, as opposed to its modied version of Davis, Logeman and Loveland, on which most of the current ecient SAT provers are based. On some highly structured SAT instances, such as the well known Pigeon Hole and Urquhart problems, both proved hard for resolution, ZRes performs very well and surpasses all classical SAT provers by an order of magnitude. 1 The DP and DLL algorithms Stimulated by hardware progress, many more and more ecient SAT solvers have been designed during the last decade. It is striking that most of the complete solvers are based on the procedure of Davis, Logeman and Loveland (DLL for short) presented in 1962 [11]. The DLL procedure may roughly be described as a backtrack procedure that searches for a model. Each step amounts to the extension of a partial interpretation by choosing an assignment for a selected variable. The success of this procedure is mainly due to its space comp...
Probabilistic Performance of a Heuristic for the Satisfiability Problem
 Discrete Applied Mathematics
, 1986
"... An algorithm for the Satisfiability problem is presented and its probabilistic behavior is analysed when combined with two other algorithms studied earlier. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm dynamic ..."
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Cited by 12 (6 self)
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An algorithm for the Satisfiability problem is presented and its probabilistic behavior is analysed when combined with two other algorithms studied earlier. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm dynamically assigns values to literals appearing in a given instance until a satisfying assignment is found or the algorithm "gives up" without determining whether or not a solution exists. It is shown that if n clauses are constructed independently from r boolean variables where the probability that a variable appears in a clause as a positive literal is p and as a negative literal is p then almost all randomly generated instances of Satisfiability are solved in polynomial time if p ! :4 ln(n)=r or p ? ln(n)=r or p = c ln(n)=r, :4 ! c ! 1 and lim n;r!1 n 1\Gammac =r 1\Gammaffl ! 1 for any ffl ? 0. It is also shown that if p = c ln(n)=r, :4 ! c ! 1 and lim n;r!1 n 1\Gammac =r = 1 then almost ...
Algorithms for SAT/TAUT decision based on various measures
 Information and Computation
, 1999
"... We investigate algorithms deciding propositional tautologies for DNF and coNPcomplete subclasses given by restrictions on the number of occurrences of literals. Especially polynomial use of resolution for reductions in combination with a new combinatorial principle called "Generalized Sign Pr ..."
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We investigate algorithms deciding propositional tautologies for DNF and coNPcomplete subclasses given by restrictions on the number of occurrences of literals. Especially polynomial use of resolution for reductions in combination with a new combinatorial principle called "Generalized Sign Principle" is studied. Upper bounds on time complexity are given with exponential part 2 ff\Delta(F ) where the measure (F ) for a clause set F either is the number n(F ) of variables, the number `(F ) of literal occurrences or the number k(F ) of clauses. ff is called a "power coefficient" for the class of formulas under consideration w.r.t. measure . Power coefficients are derived with the help of a method estimating the size of trees, which is also used to find "good" branching rules. Under natural conditions power coefficients ff; fi; fl for n; k; ` respectively fulfill ff fi fl. We obtain the following power coefficients.  0:1112 for DNF w.r.t. `  0:3334 for DNF w.r.t. k These result...
The Complexity of Automated Reasoning
, 1989
"... This thesis explores the relative complexity of proofs produced by the automatic theorem proving procedures of analytic tableaux, linear resolution, the connection method, tree resolution and the DavisPutnam procedure. It is shown that tree resolution simulates the improved tableau procedure and th ..."
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This thesis explores the relative complexity of proofs produced by the automatic theorem proving procedures of analytic tableaux, linear resolution, the connection method, tree resolution and the DavisPutnam procedure. It is shown that tree resolution simulates the improved tableau procedure and that SLresolution and the connection method are equivalent to restrictions of the improved tableau method. The theorem by Tseitin that the DavisPutnam Procedure cannot be simulated by tree resolution is given an explicit and simplified proof. The hard examples for tree resolution are contradictions constructed from simple Tseitin graphs.
On learning counting functions with queries
 in &quot;7th Annual ACM Conference on Computational Learning Theory, COLT'94
, 1994
"... We investigate the problem of learning disjunctions of counting functions, which are general cases of parity and modulo functions, with equivalence and membership queries. We prove that, for any prime number p, the class of disjunctions of integerweighted counting functions with modulus p over the ..."
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Cited by 4 (2 self)
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We investigate the problem of learning disjunctions of counting functions, which are general cases of parity and modulo functions, with equivalence and membership queries. We prove that, for any prime number p, the class of disjunctions of integerweighted counting functions with modulus p over the domain Zn q (or Zn) for any given integer q 2 is polynomial time learnable using at most n + 1 equivalence queries, where the hypotheses issued by the learner are disjunctions of at most n counting functions with weights from Zp. The result is obtained through learning linear systems over an arbitrary eld. In general a counting function mayhavea composite modulus. We prove that, for any given integer q 2, over the domain Z n 2, the class of readonce disjunctions of Booleanweighted counting functions with modulus q is polynomial time learnable with only one equivalence query, and the class of disjunctions of log log n Booleanweighted counting functions with modulus q is polynomial time learnable. Finally, we present an algorithm for learning graphbased counting functions. 1
Tseitin's Tautologies and Lower Bounds for Nullstellensatz Proofs
 In Proceedings of the 39th IEEE FOCS
, 1998
"... this paper we develop an approach which allows to produce explicitly a system of polynomials of degree 6 and to prove a linear lower bound on the degree of its boolean Nullstellensatz refutation. This approach borrows an idea from [13] to reduce the issue of Nullstellensatz refutations to Thue syste ..."
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Cited by 4 (0 self)
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this paper we develop an approach which allows to produce explicitly a system of polynomials of degree 6 and to prove a linear lower bound on the degree of its boolean Nullstellensatz refutation. This approach borrows an idea from [13] to reduce the issue of Nullstellensatz refutations to Thue systems. First, we introduce and study (see section 1) boolean multiplicative Thue systems (basically, they consist of binomials necessarily containing among them the polynomials X
On Average Case Complexity of SAT for Symmetric Distributions
, 1995
"... We investigate in this paper 'natural' distributions for the satisfiability problem (SAT) of propositional logic, using concepts introduced by [25, 19, 1] to study the averagecase complexity of NPcomplete problems. Gurevich showed that a problem with a flat distribution is not DistNP com ..."
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Cited by 3 (0 self)
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We investigate in this paper 'natural' distributions for the satisfiability problem (SAT) of propositional logic, using concepts introduced by [25, 19, 1] to study the averagecase complexity of NPcomplete problems. Gurevich showed that a problem with a flat distribution is not DistNP complete (for deterministic reductions), unless DEXPTime<F NaN> 6= NEXPTime. We express the known results concerning fixed size and fixed density distributions for CNF in the framework of averagecase complexity and show that all these distributions are flat. We introduce the family of symmetric distributions, which generalizes those mentioned before, and show that bounded symmetric distributions on ordered tuples of clauses (CNFTuples) and on kCNF (sets of kliteralclauses), are flat. This eliminates all these distributions as candidates for 'provably hard' (i.e. DistNP complete) distributions for SAT, if one considers only deterministic reductions. Given the (presumed) naturalness and generality o...