Hierarchic Superposition Revisited

Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory

Hierarchic Superposition Revisited

Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory

Instantiation of SMT problems modulo Integers

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.Comment: Research report, long version of our AISC 2010 pape

07401 Abstracts Collection -- Deduction and Decision Procedures

From 01.10. to 05.10.2007, the Dagstuhl Seminar 07401 ``Deduction and Decision Procedures\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper

Integrating a Global Induction Mechanism into a Sequent Calculus

Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.Comment: 16 page

Combinable Extensions of Abelian Groups

The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination method à la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this non-disjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification

Automatic Generation of Invariants for Circular Derivations in {SUP(LA)} 1

The hierarchic combination of linear arithmetic and firstorder logic with free function symbols, FOL(LA), results in a strictly more expressive logic than its two parts. The SUP(LA) calculus can be turned into a decision procedure for interesting fragments of FOL(LA). For example, reachability problems for timed automata can be decided by SUP(LA) using an appropriate translation into FOL(LA). In this paper, we extend the SUP(LA) calculus with an additional inference rule, automatically generating inductive invariants from partial SUP(LA) derivations. The rule enables decidability of more expressive fragments, including reachability for timed automata with unbounded integer variables. We have implemented the rule in the SPASS(LA) theorem prover with promising results, showing that it can considerably speed up proof search and enable termination of saturation for practically relevant problems

Deduction modulo theory

This paper is a survey on Deduction modulo theor

Disproving in First-Order Logic with Definitions, Arithmetic and Finite Domains

This thesis explores several methods which enable a first-order reasoner to conclude satisfiability of a formula modulo an arithmetic theory. The most general method requires restricting certain quantifiers to range over finite sets; such assumptions are common in the software verification setting. In addition, the use of first-order reasoning allows for an implicit representation of those finite sets, which can avoid scalability problems that affect other quantified reasoning methods. These new techniques form a useful complement to existing methods that are primarily aimed at proving validity. The Superposition calculus for hierarchic theory combinations provides a basis for reasoning modulo theories in a first-order setting. The recent account of ‘weak abstraction’ and related improvements make an mplementation of the calculus practical. Also, for several logical theories of interest Superposition is an effective decision procedure for the quantifier free fragment. The first contribution is an implementation of that calculus (Beagle), including an optimized implementation of Cooper’s algorithm for quantifier elimination in the theory of linear integer arithmetic. This includes a novel means of extracting values for quantified variables in satisfiable integer problems. Beagle won an efficiency award at CADE Automated theorem prover System Competition (CASC)-J7, and won the arithmetic non-theorem category at CASC-25. This implementation is the start point for solving the ‘disproving with theories’ problem. Some hypotheses can be disproved by showing that, together with axioms the hypothesis is unsatisfiable. Often this is relative to other axioms that enrich a base theory by defining new functions. In that case, the disproof is contingent on the satisfiability of the enrichment. Satisfiability in this context is undecidable. Instead, general characterizations of definition formulas, which do not alter the satisfiability status of the main axioms, are given. These general criteria apply to recursive definitions, definitions over lists, and to arrays. This allows proving some non-theorems which are otherwise intractable, and justifies similar disproofs of non-linear arithmetic formulas. When the hypothesis is contingently true, disproof requires proving existence of a model. If the Superposition calculus saturates a clause set, then a model exists, but only when the clause set satisfies a completeness criterion. This requires each instance of an uninterpreted, theory-sorted term to have a definition in terms of theory symbols. The second contribution is a procedure that creates such definitions, given that a subset of quantifiers range over finite sets. Definitions are produced in a counter-example driven way via a sequence of over and under approximations to the clause set. Two descriptions of the method are given: the first uses the component solver modularly, but has an inefficient counter-example heuristic. The second is more general, correcting many of the inefficiencies of the first, yet it requires tracking clauses through a proof. This latter method is shown to apply also to lists and to problems with unbounded quantifiers. Together, these tools give new ways for applying successful first-order reasoning methods to problems involving interpreted theories