2,867 research outputs found
Bounds on the Automata Size for Presburger Arithmetic
Automata provide a decision procedure for Presburger arithmetic. However,
until now only crude lower and upper bounds were known on the sizes of the
automata produced by this approach. In this paper, we prove an upper bound on
the the number of states of the minimal deterministic automaton for a
Presburger arithmetic formula. This bound depends on the length of the formula
and the quantifiers occurring in the formula. The upper bound is established by
comparing the automata for Presburger arithmetic formulas with the formulas
produced by a quantifier elimination method. We also show that our bound is
tight, even for nondeterministic automata. Moreover, we provide optimal
automata constructions for linear equations and inequations
Quantifier elimination for the reals with a predicate for the powers of two
In 1985, van den Dries showed that the theory of the reals with a predicate
for the integer powers of two admits quantifier elimination in an expanded
language, and is hence decidable. He gave a model-theoretic argument, which
provides no apparent bounds on the complexity of a decision procedure. We
provide a syntactic argument that yields a procedure that is primitive
recursive, although not elementary. In particular, we show that it is possible
to eliminate a single block of existential quantifiers in time ,
where is the length of the input formula and denotes -fold
iterated exponentiation
Partial Quantifier Elimination
We consider the problem of Partial Quantifier Elimination (PQE). Given
formula exists(X)[F(X,Y) & G(X,Y)], where F, G are in conjunctive normal form,
the PQE problem is to find a formula F*(Y) such that F* & exists(X)[G] is
logically equivalent to exists(X)[F & G]. We solve the PQE problem by
generating and adding to F clauses over the free variables that make the
clauses of F with quantified variables redundant. The traditional Quantifier
Elimination problem (QE) is a special case of PQE where G is empty so all
clauses of the input formula with quantified variables need to be made
redundant. The importance of PQE is twofold. First, many problems are more
naturally formulated in terms of PQE rather than QE. Second, in many cases PQE
can be solved more efficiently than QE. We describe a PQE algorithm based on
the machinery of dependency sequents and give experimental results showing the
promise of PQE
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
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