10 research outputs found
Efficient Interpolant Generation in Satisfiability Modulo Theories
The problem of computing Craig Interpolants for propositional (SAT) formulas has recently received a lot of interest, mainly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be more naturally addressed in the framework of Satisfiability Modulo Theories, SMT. Although {some} works have addressed the topic of generating interpolants in SMT, the techniques and tools that are currently available have some limitations, and their performance still does not exploit the full power of current state-of-the-art SMT solvers. In this paper we try to close this gap. We present several techniques for interpolant generation in SMT which overcome the limitations of the current generators mentioned above, and which take full advantage of state-of-the-art SMT technology. These novel techniques can lead to substantial performance improvements wrt. the currently available tools. We support our claims with an extensive experimental evaluation of our implementation of the proposed techniques in the MathSAT SMT solver
Splitting Proofs for Interpolation
We study interpolant extraction from local first-order refutations. We
present a new theoretical perspective on interpolation based on clearly
separating the condition on logical strength of the formula from the
requirement on the com- mon signature. This allows us to highlight the space of
all interpolants that can be extracted from a refutation as a space of simple
choices on how to split the refuta- tion into two parts. We use this new
insight to develop an algorithm for extracting interpolants which are linear in
the size of the input refutation and can be further optimized using metrics
such as number of non-logical symbols or quantifiers. We implemented the new
algorithm in first-order theorem prover VAMPIRE and evaluated it on a large
number of examples coming from the first-order proving community. Our
experiments give practical evidence that our work improves the state-of-the-art
in first-order interpolation.Comment: 26th Conference on Automated Deduction, 201
Interpolation in local theory extensions
In this paper we study interpolation in local extensions of a base theory. We
identify situations in which it is possible to obtain interpolants in a
hierarchical manner, by using a prover and a procedure for generating
interpolants in the base theory as black-boxes. We present several examples of
theory extensions in which interpolants can be computed this way, and discuss
applications in verification, knowledge representation, and modular reasoning
in combinations of local theories.Comment: 31 pages, 1 figur
Quantifier-Free Interpolation of a Theory of Arrays
The use of interpolants in model checking is becoming an enabling technology
to allow fast and robust verification of hardware and software. The application
of encodings based on the theory of arrays, however, is limited by the
impossibility of deriving quantifier- free interpolants in general. In this
paper, we show that it is possible to obtain quantifier-free interpolants for a
Skolemized version of the extensional theory of arrays. We prove this in two
ways: (1) non-constructively, by using the model theoretic notion of
amalgamation, which is known to be equivalent to admit quantifier-free
interpolation for universal theories; and (2) constructively, by designing an
interpolating procedure, based on solving equations between array updates.
(Interestingly, rewriting techniques are used in the key steps of the solver
and its proof of correctness.) To the best of our knowledge, this is the first
successful attempt of computing quantifier- free interpolants for a variant of
the theory of arrays with extensionality
Efficient Generation of Craig Interpolants in Satisfiability Modulo Theories
The problem of computing Craig Interpolants has recently received a lot of
interest. In this paper, we address the problem of efficient generation of
interpolants for some important fragments of first order logic, which are
amenable for effective decision procedures, called Satisfiability Modulo Theory
solvers.
We make the following contributions.
First, we provide interpolation procedures for several basic theories of
interest: the theories of linear arithmetic over the rationals, difference
logic over rationals and integers, and UTVPI over rationals and integers.
Second, we define a novel approach to interpolate combinations of theories,
that applies to the Delayed Theory Combination approach.
Efficiency is ensured by the fact that the proposed interpolation algorithms
extend state of the art algorithms for Satisfiability Modulo Theories. Our
experimental evaluation shows that the MathSAT SMT solver can produce
interpolants with minor overhead in search, and much more efficiently than
other competitor solvers.Comment: submitted to ACM Transactions on Computational Logic (TOCL