154 research outputs found
The complexity of disjunctive linear Diophantine constraints.
We study the Constraint Satisfaction Problem CSP( A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete
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
On Deciding Linear Arithmetic Constraints Over p-adic Integers for All Primes
Given an existential formula Φ of linear arithmetic over p-adic integers together with valuation constraints, we study the p-universality problem which consists of deciding whether Φ is satisfiable for all primes p, and the analogous problem for the closely related existential theory of Büchi arithmetic. Our main result is a coNEXP upper bound for both problems, together with a matching
lower bound for existential Büchi arithmetic. On a technical level, our results are obtained from analysing properties of a certain class of p-automata, finite-state automata whose languages encode
sets of tuples of natural numbers
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
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 Büchi 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 Büchi arithmetic, obtain some derived results, and further discuss some open problems
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
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
A decidable policy language for history-based transaction monitoring
Online trading invariably involves dealings between strangers, so it is
important for one party to be able to judge objectively the trustworthiness of
the other. In such a setting, the decision to trust a user may sensibly be
based on that user's past behaviour. We introduce a specification language
based on linear temporal logic for expressing a policy for categorising the
behaviour patterns of a user depending on its transaction history. We also
present an algorithm for checking whether the transaction history obeys the
stated policy. To be useful in a real setting, such a language should allow one
to express realistic policies which may involve parameter quantification and
quantitative or statistical patterns. We introduce several extensions of linear
temporal logic to cater for such needs: a restricted form of universal and
existential quantification; arbitrary computable functions and relations in the
term language; and a "counting" quantifier for counting how many times a
formula holds in the past. We then show that model checking a transaction
history against a policy, which we call the history-based transaction
monitoring problem, is PSPACE-complete in the size of the policy formula and
the length of the history. The problem becomes decidable in polynomial time
when the policies are fixed. We also consider the problem of transaction
monitoring in the case where not all the parameters of actions are observable.
We formulate two such "partial observability" monitoring problems, and show
their decidability under certain restrictions
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