30,647 research outputs found
Symbolic Solving of Extended Regular Expression Inequalities
This paper presents a new solution to the containment problem for extended
regular expressions that extends basic regular expressions with intersection
and complement operators and consider regular expressions on infinite alphabets
based on potentially infinite character sets. Standard approaches deciding the
containment do not take extended operators or character sets into account. The
algorithm avoids the translation to an expression-equivalent automaton and
provides a purely symbolic term rewriting systems for solving regular
expressions inequalities.
We give a new symbolic decision procedure for the containment problem based
on Brzozowski's regular expression derivatives and Antimirov's rewriting
approach to check containment. We generalize Brzozowski's syntactic derivative
operator to two derivative operators that work with respect to (potentially
infinite) representable character sets.Comment: Technical Repor
Better Answers to Real Questions
We consider existential problems over the reals. Extended quantifier
elimination generalizes the concept of regular quantifier elimination by
providing in addition answers, which are descriptions of possible assignments
for the quantified variables. Implementations of extended quantifier
elimination via virtual substitution have been successfully applied to various
problems in science and engineering. So far, the answers produced by these
implementations included infinitesimal and infinite numbers, which are hard to
interpret in practice. We introduce here a post-processing procedure to
convert, for fixed parameters, all answers into standard real numbers. The
relevance of our procedure is demonstrated by application of our implementation
to various examples from the literature, where it significantly improves the
quality of the results
(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
A tropical extremal problem with nonlinear objective function and linear inequality constraints
We consider a multidimensional extremal problem formulated in terms of
tropical mathematics. The problem is to minimize a nonlinear objective
function, which is defined on a finite-dimensional semimodule over an
idempotent semifield, subject to linear inequality constraints. An efficient
solution approach is developed which reduces the problem to that of solving a
linear inequality with an extended set of unknown variables. We use the
approach to obtain a complete solution to the problem in a closed form under
quite general assumptions. To illustrate the obtained results, a
two-dimensional problem is examined and its numerical solution is given.Comment: The 6th WSEAS European Computing Conference (ECC'12), Prague, Czech
Republic, September 24-26, 2012; Advances in Computer Science: Proc. 6th
WSEAS European Computing Conf. (ECC '12), WSEAS Press. ISBN
978-1-61804-126-5; RACES 5, ISSN 1790-510
A constrained tropical optimization problem: complete solution and application example
The paper focuses on a multidimensional optimization problem, which is
formulated in terms of tropical mathematics and consists in minimizing a
nonlinear objective function subject to linear inequality constraints. To solve
the problem, we follow an approach based on the introduction of an additional
unknown variable to reduce the problem to solving linear inequalities, where
the variable plays the role of a parameter. A necessary and sufficient
condition for the inequalities to hold is used to evaluate the parameter,
whereas the general solution of the inequalities is taken as a solution of the
original problem. Under fairly general assumptions, a complete direct solution
to the problem is obtained in a compact vector form. The result is applied to
solve a problem in project scheduling when an optimal schedule is given by
minimizing the flow time of activities in a project under various activity
precedence constraints. As an illustration, a numerical example of optimal
scheduling is also presented.Comment: 20 pages, accepted for publication in Contemporary Mathematic
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