19,190 research outputs found
Decision Problems For Convex Languages
In this paper we examine decision problems associated with various classes of
convex languages, studied by Ang and Brzozowski (under the name "continuous
languages"). We show that we can decide whether a given language L is prefix-,
suffix-, factor-, or subword-convex in polynomial time if L is represented by a
DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the
case that a regular language is not convex, we prove tight upper bounds on the
length of the shortest words demonstrating this fact, in terms of the number of
states of an accepting DFA. Similar results are proved for some subclasses of
convex languages: the prefix-, suffix-, factor-, and subword-closed languages,
and the prefix-, suffix-, factor-, and subword-free languages.Comment: preliminary version. This version corrected one typo in Section
2.1.1, line
Mean-payoff Automaton Expressions
Quantitative languages are an extension of boolean languages that assign to
each word a real number. Mean-payoff automata are finite automata with
numerical weights on transitions that assign to each infinite path the long-run
average of the transition weights. When the mode of branching of the automaton
is deterministic, nondeterministic, or alternating, the corresponding class of
quantitative languages is not robust as it is not closed under the pointwise
operations of max, min, sum, and numerical complement. Nondeterministic and
alternating mean-payoff automata are not decidable either, as the quantitative
generalization of the problems of universality and language inclusion is
undecidable.
We introduce a new class of quantitative languages, defined by mean-payoff
automaton expressions, which is robust and decidable: it is closed under the
four pointwise operations, and we show that all decision problems are decidable
for this class. Mean-payoff automaton expressions subsume deterministic
mean-payoff automata, and we show that they have expressive power incomparable
to nondeterministic and alternating mean-payoff automata. We also present for
the first time an algorithm to compute distance between two quantitative
languages, and in our case the quantitative languages are given as mean-payoff
automaton expressions
Constraint Satisfaction and Semilinear Expansions of Addition over the Rationals and the Reals
A semilinear relation is a finite union of finite intersections of open and
closed half-spaces over, for instance, the reals, the rationals, or the
integers. Semilinear relations have been studied in connection with algebraic
geometry, automata theory, and spatiotemporal reasoning. We consider semilinear
relations over the rationals and the reals. Under this assumption, the
computational complexity of the constraint satisfaction problem (CSP) is known
for all finite sets containing R+={(x,y,z) | x+y=z}, <=, and {1}. These
problems correspond to expansions of the linear programming feasibility
problem. We generalise this result and fully determine the complexity for all
finite sets of semilinear relations containing R+. This is accomplished in part
by introducing an algorithm, based on computing affine hulls, which solves a
new class of semilinear CSPs in polynomial time. We further analyse the
complexity of linear optimisation over the solution set and the existence of
integer solutions.Comment: 22 pages, 1 figur
The Hardness of Finding Linear Ranking Functions for Lasso Programs
Finding whether a linear-constraint loop has a linear ranking function is an
important key to understanding the loop behavior, proving its termination and
establishing iteration bounds. If no preconditions are provided, the decision
problem is known to be in coNP when variables range over the integers and in
PTIME for the rational numbers, or real numbers. Here we show that deciding
whether a linear-constraint loop with a precondition, specifically with
partially-specified input, has a linear ranking function is EXPSPACE-hard over
the integers, and PSPACE-hard over the rationals. The precise complexity of
these decision problems is yet unknown. The EXPSPACE lower bound is derived
from the reachability problem for Petri nets (equivalently, Vector Addition
Systems), and possibly indicates an even stronger lower bound (subject to open
problems in VAS theory). The lower bound for the rationals follows from a novel
simulation of Boolean programs. Lower bounds are also given for the problem of
deciding if a linear ranking-function supported by a particular form of
inductive invariant exists. For loops over integers, the problem is PSPACE-hard
for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of
natural numbers as invariants.Comment: In Proceedings GandALF 2014, arXiv:1408.5560. I thank the organizers
of the Dagstuhl Seminar 14141, "Reachability Problems for Infinite-State
Systems", for the opportunity to present an early draft of this wor
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