59 research outputs found
The complexity of presburger arithmetic with bounded quantifier alternation depth
AbstractIt is shown how the method of Fischer and Rabin can be extended to get good lower bounds for Presburger arithmetic with a bounded number of quantifier alternations. In this case, the complexity is one exponential lower than in the unbounded case. This situation is typical for first order theories
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
On Presburger arithmetic extended with non-unary counting quantifiers
We consider a first-order logic for the integers with addition. This logic
extends classical first-order logic by modulo-counting, threshold-counting, and
exact-counting quantifiers, all applied to tuples of variables. Further, the
residue in modulo-counting quantifiers is given as a term. Our main result
shows that satisfaction for this logic is decidable in two-fold exponential
space. If only threshold- and exact-counting quantifiers are allowed, we prove
an upper bound of alternating two-fold exponential time with linearly many
alternations. This latter result almost matches Berman's exact complexity of
first-order logic without counting quantifiers.
To obtain these results, we first translate threshold- and exact-counting
quantifiers into classical first-order logic in polynomial time (which already
proves the second result). To handle the remaining modulo-counting quantifiers
for tuples, we first reduce them in doubly exponential time to modulo-counting
quantifiers for single elements. For these quantifiers, we provide a quantifier
elimination procedure similar to Reddy and Loveland's procedure for first-order
logic and analyse the growth of coefficients, constants, and moduli appearing
in this process. The bounds obtained this way allow to replace quantification
in the original formula to integers of bounded size which then implies the
first result mentioned above.
Our logic is incomparable with the logic considered recently by Chistikov et
al. They allow more general counting operations in quantifiers, but only unary
quantifiers. The move from unary to non-unary quantifiers is non-trivial,
since, e.g., the non-unary version of the H\"artig quantifier results in an
undecidable theory
On the Complexity of Quantified Integer Programming
Quantified integer programming is the problem of deciding assertions of the form Q_k x_k ... forall x_2 exists x_1 : A * x >= c where vectors of variables x_k,..,x_1 form the vector x, all variables are interpreted over N (alternatively, over Z), and A and c are a matrix and vector over Z of appropriate sizes. We show in this paper that quantified integer programming with alternation depth k is complete for the kth level of the polynomial hierarchy
Unary Pushdown Automata and Straight-Line Programs
We consider decision problems for deterministic pushdown automata over a
unary alphabet (udpda, for short). Udpda are a simple computation model that
accept exactly the unary regular languages, but can be exponentially more
succinct than finite-state automata. We complete the complexity landscape for
udpda by showing that emptiness (and thus universality) is P-hard, equivalence
and compressed membership problems are P-complete, and inclusion is
coNP-complete. Our upper bounds are based on a translation theorem between
udpda and straight-line programs over the binary alphabet (SLPs). We show that
the characteristic sequence of any udpda can be represented as a pair of
SLPs---one for the prefix, one for the lasso---that have size linear in the
size of the udpda and can be computed in polynomial time. Hence, decision
problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP
can be converted in logarithmic space into a udpda, and this forms the basis
for our lower bound proofs. We show coNP-hardness of the ordered matching
problem for SLPs, from which we derive coNP-hardness for inclusion. In
addition, we complete the complexity landscape for unary nondeterministic
pushdown automata by showing that the universality problem is -hard, using a new class of integer expressions. Our techniques have
applications beyond udpda. We show that our results imply -completeness for a natural fragment of Presburger arithmetic and coNP lower
bounds for compressed matching problems with one-character wildcards
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