arXiv subjects
Publications and source records attributed to Nicolas Peltier.
Two distinct algorithms are presented to extract (schemata of) resolution proofs from closed tableaux for propositional schemata. The first one handles the most efficient version of the tableau calculus but generates very complex derivations (denoted by rather elaborate rewrite systems). The second one has the advantage that much simpler systems can be obtained, however the considered proof procedure is less efficient.
This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed.
Many decision procedures for SMT problems rely more or less implicitly on an instantiation of the axioms of the theories under consideration, and differ by making use of the additional properties of each theory, in order to increase efficiency. We present a new technique for devising complete instantiation schemes on SMT problems over a combination of linear arithmetic with another theory T. The method consists in first instantiating the arithmetic part of the formula, and then getting rid of the remaining variables in the problem by using an instantiation strategy which is complete for T. We provide examples evidencing that not only is this technique generic (in the sense that it applies to a wide range of theories) but it is also efficient, even compared to state-of-the-art instantiation schemes for specific theories.
Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated schemata", allow to express such patterns. Schemata extend propositional logic with indexed propositions, e.g. P_i, P_i+1, P_1, and with generalized connectives, e.g. /ı=1..n or i=1..n (called "iterations") where n is an (unbound) integer variable called a "parameter". The expressive power of iterated schemata is strictly greater than propositional logic: it is even out of the scope of first-order logic. We define a proof procedure, called DPLL*, that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the DPLL procedure. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable so DPLL* does not terminate in general. Still, we prove that it terminates for schemata of a syntactic subclass called "regularly nested". This is the first non trivial class for which DPLL* is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form /ı=1..n (/\j=1..n ...).