113 research outputs found
Approaching Arithmetic Theories with Finite-State Automata
The automata-theoretic approach provides an elegant method for deciding linear arithmetic theories. This approach has recently been instrumental for settling long-standing open problems about the complexity of deciding the existential fragments of BĂĽchi arithmetic and linear arithmetic over p-adic fields. In this article, which accompanies an invited talk, we give a high-level exposition of the NP upper bound for existential BĂĽchi arithmetic, obtain some derived results, and further discuss some open problems
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
Integer Vector Addition Systems with States
This paper studies reachability, coverability and inclusion problems for
Integer Vector Addition Systems with States (ZVASS) and extensions and
restrictions thereof. A ZVASS comprises a finite-state controller with a finite
number of counters ranging over the integers. Although it is folklore that
reachability in ZVASS is NP-complete, it turns out that despite their
naturalness, from a complexity point of view this class has received little
attention in the literature. We fill this gap by providing an in-depth analysis
of the computational complexity of the aforementioned decision problems. Most
interestingly, it turns out that while the addition of reset operations to
ordinary VASS leads to undecidability and Ackermann-hardness of reachability
and coverability, respectively, they can be added to ZVASS while retaining
NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure
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
Tightening the Complexity of Equivalence Problems for Commutative Grammars
We show that the language equivalence problem for regular and context-free
commutative grammars is coNEXP-complete. In addition, our lower bound
immediately yields further coNEXP-completeness results for equivalence problems
for communication-free Petri nets and reversal-bounded counter automata.
Moreover, we improve both lower and upper bounds for language equivalence for
exponent-sensitive commutative grammars.Comment: 21 page
Presburger arithmetic, rational generating functions, and quasi-polynomials
Presburger arithmetic is the first-order theory of the natural numbers with
addition (but no multiplication). We characterize sets that can be defined by a
Presburger formula as exactly the sets whose characteristic functions can be
represented by rational generating functions; a geometric characterization of
such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the
free variables in a Presburger formula, we can define a counting function g(p)
to be the number of solutions to the formula, for a given p. We show that every
counting function obtained in this way may be represented as, equivalently,
either a piecewise quasi-polynomial or a rational generating function. Finally,
we translate known computational complexity results into this setting and
discuss open directions.Comment: revised, including significant additions explaining computational
complexity results. To appear in Journal of Symbolic Logic. Extended abstract
in ICALP 2013. 17 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
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
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