1,708 research outputs found
On Generalized Records and Spatial Conjunction in Role Logic
We have previously introduced role logic as a notation for describing
properties of relational structures in shape analysis, databases and knowledge
bases. A natural fragment of role logic corresponds to two-variable logic with
counting and is therefore decidable. We show how to use role logic to describe
open and closed records, as well the dual of records, inverse records. We
observe that the spatial conjunction operation of separation logic naturally
models record concatenation. Moreover, we show how to eliminate the spatial
conjunction of formulas of quantifier depth one in first-order logic with
counting. As a result, allowing spatial conjunction of formulas of quantifier
depth one preserves the decidability of two-variable logic with counting. This
result applies to two-variable role logic fragment as well. The resulting logic
smoothly integrates type system and predicate calculus notation and can be
viewed as a natural generalization of the notation for constraints arising in
role analysis and similar shape analysis approaches.Comment: 30 pages. A version appears in SAS 200
Finding polynomial loop invariants for probabilistic programs
Quantitative loop invariants are an essential element in the verification of
probabilistic programs. Recently, multivariate Lagrange interpolation has been
applied to synthesizing polynomial invariants. In this paper, we propose an
alternative approach. First, we fix a polynomial template as a candidate of a
loop invariant. Using Stengle's Positivstellensatz and a transformation to a
sum-of-squares problem, we find sufficient conditions on the coefficients.
Then, we solve a semidefinite programming feasibility problem to synthesize the
loop invariants. If the semidefinite program is unfeasible, we backtrack after
increasing the degree of the template. Our approach is semi-complete in the
sense that it will always lead us to a feasible solution if one exists and
numerical errors are small. Experimental results show the efficiency of our
approach.Comment: accompanies an ATVA 2017 submissio
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Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation
A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined
From Boolean Equalities to Constraints
Although functional as well as logic languages use equality to discriminate between logically different cases, the operational meaning of equality is different in such languages. Functional languages reduce equational expressions to their Boolean values, True or False, logic languages use unification to check the validity only and fail otherwise. Consequently, the language Curry, which amalgamates functional and logic programming features, offers two kinds of equational expressions so that the programmer has to distinguish between these uses. We show that this distinction can be avoided by providing an analysis and transformation method that automatically selects the appropriate operation. Without this distinction in source programs, the language design can be simplified and the execution of programs can be optimized. As a consequence, we show that one kind of equational expressions is sufficient and unification is nothing else than an optimization of Boolean equality
Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials
This paper analyzes to what extent it is possible to efficiently reduce the
number of clauses in NP-hard satisfiability problems, without changing the
answer. Upper and lower bounds are established using the concept of
kernelization. Existing results show that if NP is not contained in coNP/poly,
no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT
with d literals per clause, to equivalent instances with bits for
any e > 0. For the Not-All-Equal SAT problem, a compression to size
exists. We put these results in a common framework by analyzing
the compressibility of binary CSPs. We characterize constraint types based on
the minimum degree of multivariate polynomials whose roots correspond to the
satisfying assignments, obtaining (nearly) matching upper and lower bounds in
several settings. Our lower bounds show that not just the number of
constraints, but also the encoding size of individual constraints plays an
important role. For example, for Exact Satisfiability with unbounded clause
length it is possible to efficiently reduce the number of constraints to n+1,
yet no polynomial-time algorithm can reduce to an equivalent instance with
bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the
previous version did NOT satisfy requirement 4 of lemma 18 (the proof of
Claim 20 was incorrect
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 First-Order Theory of Sets with Cardinality Constraints is Decidable
We show that the decidability of the first-order theory of the language that
combines Boolean algebras of sets of uninterpreted elements with Presburger
arithmetic operations. We thereby disprove a recent conjecture that this theory
is undecidable. Our language allows relating the cardinalities of sets to the
values of integer variables, and can distinguish finite and infinite sets. We
use quantifier elimination to show the decidability and obtain an elementary
upper bound on the complexity.
Precise program analyses can use our decidability result to verify
representation invariants of data structures that use an integer field to
represent the number of stored elements.Comment: 18 page
(Un)Decidability Results for Word Equations with Length and Regular Expression Constraints
We prove several decidability and undecidability results for the
satisfiability and validity problems for languages that can express solutions
to word equations with length constraints. The atomic formulas over this
language are equality over string terms (word equations), linear inequality
over the length function (length constraints), and membership in regular sets.
These questions are important in logic, program analysis, and formal
verification. Variants of these questions have been studied for many decades by
mathematicians. More recently, practical satisfiability procedures (aka SMT
solvers) for these formulas have become increasingly important in the context
of security analysis for string-manipulating programs such as web applications.
We prove three main theorems. First, we give a new proof of undecidability
for the validity problem for the set of sentences written as a forall-exists
quantifier alternation applied to positive word equations. A corollary of this
undecidability result is that this set is undecidable even with sentences with
at most two occurrences of a string variable. Second, we consider Boolean
combinations of quantifier-free formulas constructed out of word equations and
length constraints. We show that if word equations can be converted to a solved
form, a form relevant in practice, then the satisfiability problem for Boolean
combinations of word equations and length constraints is decidable. Third, we
show that the satisfiability problem for quantifier-free formulas over word
equations in regular solved form, length constraints, and the membership
predicate over regular expressions is also decidable.Comment: Invited Paper at ADDCT Workshop 2013 (co-located with CADE 2013
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