2,783 research outputs found
An Instantiation-Based Approach for Solving Quantified Linear Arithmetic
This paper presents a framework to derive instantiation-based decision
procedures for satisfiability of quantified formulas in first-order theories,
including its correctness, implementation, and evaluation. Using this framework
we derive decision procedures for linear real arithmetic (LRA) and linear
integer arithmetic (LIA) formulas with one quantifier alternation. Our
procedure can be integrated into the solving architecture used by typical SMT
solvers. Experimental results on standardized benchmarks from model checking,
static analysis, and synthesis show that our implementation of the procedure in
the SMT solver CVC4 outperforms existing tools for quantified linear
arithmetic
Satisfiability in multi-valued circuits
Satisfiability of Boolean circuits is among the most known and important
problems in theoretical computer science. This problem is NP-complete in
general but becomes polynomial time when restricted either to monotone gates or
linear gates. We go outside Boolean realm and consider circuits built of any
fixed set of gates on an arbitrary large finite domain. From the complexity
point of view this is strictly connected with the problems of solving equations
(or systems of equations) over finite algebras.
The research reported in this work was motivated by a desire to know for
which finite algebras there is a polynomial time algorithm that
decides if an equation over has a solution. We are also looking for
polynomial time algorithms that decide if two circuits over a finite algebra
compute the same function. Although we have not managed to solve these problems
in the most general setting we have obtained such a characterization for a very
broad class of algebras from congruence modular varieties. This class includes
most known and well-studied algebras such as groups, rings, modules (and their
generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie
algebras), lattices (and their extensions like Boolean algebras, Heyting
algebras or other algebras connected with multi-valued logics including
MV-algebras).
This paper seems to be the first systematic study of the computational
complexity of satisfiability of non-Boolean circuits and solving equations over
finite algebras. The characterization results provided by the paper is given in
terms of nice structural properties of algebras for which the problems are
solvable in polynomial time.Comment: 50 page
The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case
The constraint satisfaction probem (CSP) is a well-acknowledged framework in
which many combinatorial search problems can be naturally formulated. The CSP
may be viewed as the problem of deciding the truth of a logical sentence
consisting of a conjunction of constraints, in front of which all variables are
existentially quantified. The quantified constraint satisfaction problem (QCSP)
is the generalization of the CSP where universal quantification is permitted in
addition to existential quantification. The general intractability of these
problems has motivated research studying the complexity of these problems under
a restricted constraint language, which is a set of relations that can be used
to express constraints.
This paper introduces collapsibility, a technique for deriving positive
complexity results on the QCSP. In particular, this technique allows one to
show that, for a particular constraint language, the QCSP reduces to the CSP.
We show that collapsibility applies to three known tractable cases of the QCSP
that were originally studied using disparate proof techniques in different
decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan 1979), Quantified
Horn-SAT (Karpinski, Kleine B\"{u}ning, and Schmitt 1987), and Quantified
Affine-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals
common structure among these cases, which are describable by constraint
languages over a two-element domain. In addition to unifying these known
tractable cases, we study constraint languages over domains of larger size
Fundamental group of simple -algebras with unique trace
We introduce the fundamental group of a unital simple
-algebra with a unique normalized trace. We compute fundamental groups
of several nuclear or non-nuclear -algebras .
K-theoretical obstruction enables us to compute the fundamental group easily.
Our study is essentially based on the computation of Picard groups by Kodaka.Comment: 11 page
Notes on Lie symmetry group methods for differential equations
Fundamentals on Lie group methods and applications to differential equations
are surveyed. Many examples are included to elucidate their extensive
applicability for analytically solving both ordinary and partial differential
equations.Comment: 85 Pages. expanded and misprints correcte
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