1,228 research outputs found
Partial Quantifier Elimination
We consider the problem of Partial Quantifier Elimination (PQE). Given
formula exists(X)[F(X,Y) & G(X,Y)], where F, G are in conjunctive normal form,
the PQE problem is to find a formula F*(Y) such that F* & exists(X)[G] is
logically equivalent to exists(X)[F & G]. We solve the PQE problem by
generating and adding to F clauses over the free variables that make the
clauses of F with quantified variables redundant. The traditional Quantifier
Elimination problem (QE) is a special case of PQE where G is empty so all
clauses of the input formula with quantified variables need to be made
redundant. The importance of PQE is twofold. First, many problems are more
naturally formulated in terms of PQE rather than QE. Second, in many cases PQE
can be solved more efficiently than QE. We describe a PQE algorithm based on
the machinery of dependency sequents and give experimental results showing the
promise of PQE
An exactly solvable random satisfiability problem
We introduce a new model for the generation of random satisfiability
problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt
and Zecchina, which is a variant of the famous K-SAT model: it is extended to
q-state variables and relates to a different choice of the statistical
ensemble. The model has an exactly solvable statistic: the critical exponents
and scaling functions of the SAT/UNSAT transition are calculable at zero
temperature, with no need of replicas, also with exact finite-size corrections.
We also introduce an exact duality of the model, and show an analogy of
thermodynamic properties with the Random Energy Model of disordered spin
systems theory. Relations with Error-Correcting Codes are also discussed.Comment: 31 pages, 1 figur
New results on rewrite-based satisfiability procedures
Program analysis and verification require decision procedures to reason on
theories of data structures. Many problems can be reduced to the satisfiability
of sets of ground literals in theory T. If a sound and complete inference
system for first-order logic is guaranteed to terminate on T-satisfiability
problems, any theorem-proving strategy with that system and a fair search plan
is a T-satisfiability procedure. We prove termination of a rewrite-based
first-order engine on the theories of records, integer offsets, integer offsets
modulo and lists. We give a modularity theorem stating sufficient conditions
for termination on a combinations of theories, given termination on each. The
above theories, as well as others, satisfy these conditions. We introduce
several sets of benchmarks on these theories and their combinations, including
both parametric synthetic benchmarks to test scalability, and real-world
problems to test performances on huge sets of literals. We compare the
rewrite-based theorem prover E with the validity checkers CVC and CVC Lite.
Contrary to the folklore that a general-purpose prover cannot compete with
reasoners with built-in theories, the experiments are overall favorable to the
theorem prover, showing that not only the rewriting approach is elegant and
conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page
Pushing the envelope of Optimization Modulo Theories with Linear-Arithmetic Cost Functions
In the last decade we have witnessed an impressive progress in the
expressiveness and efficiency of Satisfiability Modulo Theories (SMT) solving
techniques. This has brought previously-intractable problems at the reach of
state-of-the-art SMT solvers, in particular in the domain of SW and HW
verification. Many SMT-encodable problems of interest, however, require also
the capability of finding models that are optimal wrt. some cost functions. In
previous work, namely "Optimization Modulo Theory with Linear Rational Cost
Functions -- OMT(LAR U T )", we have leveraged SMT solving to handle the
minimization of cost functions on linear arithmetic over the rationals, by
means of a combination of SMT and LP minimization techniques. In this paper we
push the envelope of our OMT approach along three directions: first, we extend
it to work also with linear arithmetic on the mixed integer/rational domain, by
means of a combination of SMT, LP and ILP minimization techniques; second, we
develop a multi-objective version of OMT, so that to handle many cost functions
simultaneously; third, we develop an incremental version of OMT, so that to
exploit the incrementality of some OMT-encodable problems. An empirical
evaluation performed on OMT-encoded verification problems demonstrates the
usefulness and efficiency of these extensions.Comment: A slightly-shorter version of this paper is published at TACAS 2015
conferenc
Learning for Dynamic subsumption
In this paper a new dynamic subsumption technique for Boolean CNF formulae is
proposed. It exploits simple and sufficient conditions to detect during
conflict analysis, clauses from the original formula that can be reduced by
subsumption. During the learnt clause derivation, and at each step of the
resolution process, we simply check for backward subsumption between the
current resolvent and clauses from the original formula and encoded in the
implication graph. Our approach give rise to a strong and dynamic
simplification technique that exploits learning to eliminate literals from the
original clauses. Experimental results show that the integration of our dynamic
subsumption approach within the state-of-the-art SAT solvers Minisat and Rsat
achieves interesting improvements particularly on crafted instances
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