103 research outputs found
The Containment Problem for Unambiguous Register Automata
We investigate the complexity of the containment problem "Does L(A)subseteq L(B) hold?", where B is an unambiguous register automaton and A is an arbitrary register automaton. We prove that the problem is decidable and give upper bounds on the computational complexity in the general case, and when B is restricted to have a fixed number of registers
Bidimensional Linear Recursive Sequences and Universality of Unambiguous Register Automata
We study the universality and inclusion problems for register automata over
equality data. We show that the universality and the inclusion problems can be
solved with 2-EXPTIME complexity when the input automata are without guessing
and unambiguous, improving on the currently best-known 2-EXPSPACE upper bound
by Mottet and Quaas. When the number of registers of both automata is fixed, we
obtain a lower EXPTIME complexity, also improving the EXPSPACE upper bound from
Mottet and Quaas for fixed number of registers. We reduce inclusion to
universality, and then we reduce universality to the problem of counting the
number of orbits of runs of the automaton. We show that the orbit-counting
function satisfies a system of bidimensional linear recursive equations with
polynomial coefficients (linrec), which generalises analogous recurrences for
the Stirling numbers of the second kind, and then we show that universality
reduces to the zeroness problem for linrec sequences. While such a counting
approach is classical and has successfully been applied to unambiguous finite
automata and grammars over finite alphabets, its application to register
automata over infinite alphabets is novel. We provide two algorithms to decide
the zeroness problem for bidimensional linear recursive sequences arising from
orbit-counting functions. Both algorithms rely on techniques from linear
non-commutative algebra. The first algorithm performs variable elimination and
has elementary complexity. The second algorithm is a refined version of the
first one and it relies on the computation of the Hermite normal form of
matrices over a skew polynomial field. The second algorithm yields an EXPTIME
decision procedure for the zeroness problem of linrec sequences, which in turn
yields the claimed bounds for the universality and inclusion problems of
register automata.Comment: full version of the homonymous paper to appear in the proceedings of
STACS'2
Language Inclusion for Boundedly-Ambiguous Vector Addition Systems Is Decidable
We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASSes) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general the problem of language equivalence is undecidable even for one-dimensional VASSes, thus to get decidability we investigate restricted subclasses. On one hand we show that the problem of language inclusion of a VASS in k-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand we prove that the language equivalence problem is Ackermann-hard already for deterministic VASSes. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems
A Linear-Time Nominal ?-Calculus with Name Allocation
Logics and automata models for languages over infinite alphabets, such as Freeze LTL and register automata, serve the verification of processes or documents with data. They relate tightly to formalisms over nominal sets, such as nondetermininistic orbit-finite automata (NOFAs), where names play the role of data. Reasoning problems in such formalisms tend to be computationally hard. Name-binding nominal automata models such as {regular nondeterministic nominal automata (RNNAs)} have been shown to be computationally more tractable. In the present paper, we introduce a linear-time fixpoint logic Bar-?TL} for finite words over an infinite alphabet, which features full negation and freeze quantification via name binding. We show by a nontrivial reduction to extended regular nondeterministic nominal automata that even though Bar-?TL} allows unrestricted nondeterminism and unboundedly many registers, model checking Bar-?TL} over RNNAs and satisfiability checking both have elementary complexity. For example, model checking is in 2ExpSpace, more precisely in parametrized ExpSpace, effectively with the number of registers as the parameter
The many facets of string transducers
Regular word transductions extend the robust notion of regular languages from a qualitative to a quantitative reasoning. They were already considered in early papers of formal language theory, but turned out to be much more challenging. The last decade brought considerable research around various transducer models, aiming to achieve similar robustness as for automata and languages. In this paper we survey some older and more recent results on string transducers. We present classical connections between automata, logic and algebra extended to transducers, some genuine definability questions, and review approaches to the equivalence problem
Equivalence of finite-valued streaming string transducers is decidable
In this paper we provide a positive answer to a question left open by Alur and and Deshmukh in 2011 by showing that equivalence of finite-valued copyless streaming string transducers is decidable
Logics with rigidly guarded data tests
The notion of orbit finite data monoid was recently introduced by Bojanczyk
as an algebraic object for defining recognizable languages of data words.
Following Buchi's approach, we introduce a variant of monadic second-order
logic with data equality tests that captures precisely the data languages
recognizable by orbit finite data monoids. We also establish, following this
time the approach of Schutzenberger, McNaughton and Papert, that the
first-order fragment of this logic defines exactly the data languages
recognizable by aperiodic orbit finite data monoids. Finally, we consider
another variant of the logic that can be interpreted over generic structures
with data. The data languages defined in this variant are also recognized by
unambiguous finite memory automata
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