550 research outputs found
Processor Verification Using Efficient Reductions of the Logic of Uninterpreted Functions to Propositional Logic
The logic of equality with uninterpreted functions (EUF) provides a means of
abstracting the manipulation of data by a processor when verifying the
correctness of its control logic. By reducing formulas in this logic to
propositional formulas, we can apply Boolean methods such as Ordered Binary
Decision Diagrams (BDDs) and Boolean satisfiability checkers to perform the
verification.
We can exploit characteristics of the formulas describing the verification
conditions to greatly simplify the propositional formulas generated. In
particular, we exploit the property that many equations appear only in positive
form. We can therefore reduce the set of interpretations of the function
symbols that must be considered to prove that a formula is universally valid to
those that are ``maximally diverse.''
We present experimental results demonstrating the efficiency of this approach
when verifying pipelined processors using the method proposed by Burch and
Dill.Comment: 46 page
Bounded Reachability for Temporal Logic over Constraint Systems
We present CLTLB(D), an extension of PLTLB (PLTL with both past and future
operators) augmented with atomic formulae built over a constraint system D.
Even for decidable constraint systems, satisfiability and Model Checking
problem of such logic can be undecidable. We introduce suitable restrictions
and assumptions that are shown to make the satisfiability problem for the
extended logic decidable. Moreover for a large class of constraint systems we
propose an encoding that realize an effective decision procedure for the
Bounded Reachability problem
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
Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds
Given a formula in quantifier-free Presburger arithmetic, if it has a
satisfying solution, there is one whose size, measured in bits, is polynomially
bounded in the size of the formula. In this paper, we consider a special class
of quantifier-free Presburger formulas in which most linear constraints are
difference (separation) constraints, and the non-difference constraints are
sparse. This class has been observed to commonly occur in software
verification. We derive a new solution bound in terms of parameters
characterizing the sparseness of linear constraints and the number of
non-difference constraints, in addition to traditional measures of formula
size. In particular, we show that the number of bits needed per integer
variable is linear in the number of non-difference constraints and logarithmic
in the number and size of non-zero coefficients in them, but is otherwise
independent of the total number of linear constraints in the formula. The
derived bound can be used in a decision procedure based on instantiating
integer variables over a finite domain and translating the input
quantifier-free Presburger formula to an equi-satisfiable Boolean formula,
which is then checked using a Boolean satisfiability solver. In addition to our
main theoretical result, we discuss several optimizations for deriving tighter
bounds in practice. Empirical evidence indicates that our decision procedure
can greatly outperform other decision procedures.Comment: 26 page
The Small Model Property: How Small Can It Be?
AbstractEfficient decision procedures for equality logic (quantifier-free predicate calculus+the equality sign) are of major importance when proving logical equivalence between systems. We introduce an efficient decision procedure for the theory of equality based on finite instantiations. The main idea is to analyze the structure of the formula and compute accordingly a small domain to each variable such that the formula is satisfiable iff it can be satisfied over these domains. We show how the problem of finding these small domains can be reduced to an interesting graph theoretic problem. This method enabled us to verify formulas containing hundreds of integer and floating point variables that could not be efficiently handled with previously known techniques
The Small Model Property: How Small Can It Be?
AbstractEfficient decision procedures for equality logic (quantifier-free predicate calculus+the equality sign) are of major importance when proving logical equivalence between systems. We introduce an efficient decision procedure for the theory of equality based on finite instantiations. The main idea is to analyze the structure of the formula and compute accordingly a small domain to each variable such that the formula is satisfiable iff it can be satisfied over these domains. We show how the problem of finding these small domains can be reduced to an interesting graph theoretic problem. This method enabled us to verify formulas containing hundreds of integer and floating point variables that could not be efficiently handled with previously known techniques
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