18,470 research outputs found
What's Decidable About Sequences?
We present a first-order theory of sequences with integer elements,
Presburger arithmetic, and regular constraints, which can model significant
properties of data structures such as arrays and lists. We give a decision
procedure for the quantifier-free fragment, based on an encoding into the
first-order theory of concatenation; the procedure has PSPACE complexity. The
quantifier-free fragment of the theory of sequences can express properties such
as sortedness and injectivity, as well as Boolean combinations of periodic and
arithmetic facts relating the elements of the sequence and their positions
(e.g., "for all even i's, the element at position i has value i+3 or 2i"). The
resulting expressive power is orthogonal to that of the most expressive
decidable logics for arrays. Some examples demonstrate that the fragment is
also suitable to reason about sequence-manipulating programs within the
standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl
On Deciding Local Theory Extensions via E-matching
Satisfiability Modulo Theories (SMT) solvers incorporate decision procedures
for theories of data types that commonly occur in software. This makes them
important tools for automating verification problems. A limitation frequently
encountered is that verification problems are often not fully expressible in
the theories supported natively by the solvers. Many solvers allow the
specification of application-specific theories as quantified axioms, but their
handling is incomplete outside of narrow special cases.
In this work, we show how SMT solvers can be used to obtain complete decision
procedures for local theory extensions, an important class of theories that are
decidable using finite instantiation of axioms. We present an algorithm that
uses E-matching to generate instances incrementally during the search,
significantly reducing the number of generated instances compared to eager
instantiation strategies. We have used two SMT solvers to implement this
algorithm and conducted an extensive experimental evaluation on benchmarks
derived from verification conditions for heap-manipulating programs. We believe
that our results are of interest to both the users of SMT solvers as well as
their developers
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
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
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