336 research outputs found
History-Register Automata
Programs with dynamic allocation are able to create and use an unbounded
number of fresh resources, such as references, objects, files, etc. We propose
History-Register Automata (HRA), a new automata-theoretic formalism for
modelling such programs. HRAs extend the expressiveness of previous approaches
and bring us to the limits of decidability for reachability checks. The
distinctive feature of our machines is their use of unbounded memory sets
(histories) where input symbols can be selectively stored and compared with
symbols to follow. In addition, stored symbols can be consumed or deleted by
reset. We show that the combination of consumption and reset capabilities
renders the automata powerful enough to imitate counter machines, and yields
closure under all regular operations apart from complementation. We moreover
examine weaker notions of HRAs which strike different balances between
expressiveness and effectiveness.Comment: LMCS (improved version of FoSSaCS
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
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Petri net equivalence
Determining whether two Petri nets are equivalent is an interesting problem from both practical and theoretical standpoints. Although it is undecidable in the general case, for many interesting nets the equivalence problem is solvable. This paper explores, mostly from a theoretical point of view, some of the issues of Petri net equivalence, including both reachability sets and languages. Some new definitions of reachability set equivalence are described which allow the markings of some places to be treated identically or ignored, analogous to the Petri net languages in which multiple transitions may be labeled with the same symbol or with the empty string. The complexity of some decidable Petri net equivalence problems is analyzed
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
Decidability Issues for Petri Nets
This is a survey of some decidability results for Petri nets, covering the last three decades. The presentation is structured around decidability of specific properties, various behavioural equivalences and finally the model checking problem for temporal logics
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
Universality Problem for Unambiguous VASS
We study languages of unambiguous VASS, that is, Vector Addition Systems with States, whose transitions read letters from a finite alphabet, and whose acceptance condition is defined by a set of final states (i.e., the coverability language). We show that the problem of universality for unambiguous VASS is ExpSpace-complete, in sheer contrast to Ackermann-completeness for arbitrary VASS, even in dimension 1. When the dimension d ? ? is fixed, the universality problem is PSpace-complete if d ? 2, and coNP-hard for 1-dimensional VASSes (also known as One Counter Nets)
On Functions Weakly Computable by Pushdown Petri Nets and Related Systems
We consider numerical functions weakly computable by grammar-controlled
vector addition systems (GVASes, a variant of pushdown Petri nets). GVASes can
weakly compute all fast growing functions for
, hence they are computationally more powerful than
standard vector addition systems. On the other hand they cannot weakly compute
the inverses or indeed any sublinear function. The proof relies
on a pumping lemma for runs of GVASes that is of independent interest
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
Polynomial Vector Addition Systems With States
The reachability problem for vector addition systems is one of the most difficult and central problems in theoretical computer science. The problem is known to be decidable, but despite intense investigation during the last four decades, the exact complexity is still open. For some sub-classes, the complexity of the reachability problem is known. Structurally bounded vector addition systems, the class of vector addition systems with finite reachability sets from any initial configuration, is one of those classes. In fact, the reachability problem was shown to be polynomial-space complete for that class by Praveen and Lodaya in 2008. Surprisingly, extending this property to vector addition systems with states is open. In fact, there exist vector addition systems with states that are structurally bounded but with Ackermannian large sets of reachable configurations. It follows that the reachability problem for that class is between exponential space and Ackermannian. In this paper we introduce the class of polynomial vector addition systems with states, defined as the class of vector addition systems with states with size of reachable configurations bounded polynomially in the size of the initial ones. We prove that the reachability problem for polynomial vector addition systems is exponential-space complete. Additionally, we show that we can decide in polynomial time if a vector addition system with states is polynomial. This characterization introduces the notion of iteration scheme with potential applications to the reachability problem for general vector addition systems
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