6,241 research outputs found
Inflations of geometric grid classes of permutations
All three authors were partially supported by EPSRC via the grant EP/J006440/1.Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than Îș â 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than Îș has a rational generating function. This bound is tight as there are permutation classes with growth rate Îș which have nonrational generating functions.PostprintPeer reviewe
Decision Problems for Deterministic Pushdown Automata on Infinite Words
The article surveys some decidability results for DPDAs on infinite words
(omega-DPDA). We summarize some recent results on the decidability of the
regularity and the equivalence problem for the class of weak omega-DPDAs.
Furthermore, we present some new results on the parity index problem for
omega-DPDAs. For the specification of a parity condition, the states of the
omega-DPDA are assigned priorities (natural numbers), and a run is accepting if
the highest priority that appears infinitely often during a run is even. The
basic simplification question asks whether one can determine the minimal number
of priorities that are needed to accept the language of a given omega-DPDA. We
provide some decidability results on variations of this question for some
classes of omega-DPDAs.Comment: In Proceedings AFL 2014, arXiv:1405.527
Coinductive subtyping for abstract compilation of object-oriented languages into Horn formulas
In recent work we have shown how it is possible to define very precise type
systems for object-oriented languages by abstractly compiling a program into a
Horn formula f. Then type inference amounts to resolving a certain goal w.r.t.
the coinductive (that is, the greatest) Herbrand model of f.
Type systems defined in this way are idealized, since in the most interesting
instantiations both the terms of the coinductive Herbrand universe and goal
derivations cannot be finitely represented. However, sound and quite expressive
approximations can be implemented by considering only regular terms and
derivations. In doing so, it is essential to introduce a proper subtyping
relation formalizing the notion of approximation between types.
In this paper we study a subtyping relation on coinductive terms built on
union and object type constructors. We define an interpretation of types as set
of values induced by a quite intuitive relation of membership of values to
types, and prove that the definition of subtyping is sound w.r.t. subset
inclusion between type interpretations. The proof of soundness has allowed us
to simplify the notion of contractive derivation and to discover that the
previously given definition of subtyping did not cover all possible
representations of the empty type
Decision problems for Clark-congruential languages
A common question when studying a class of context-free grammars is whether
equivalence is decidable within this class. We answer this question positively
for the class of Clark-congruential grammars, which are of interest to
grammatical inference. We also consider the problem of checking whether a given
CFG is Clark-congruential, and show that it is decidable given that the CFG is
a DCFG.Comment: Version 2 incorporates revisions prompted by the comments of
anonymous referees at ICGI and LearnAu
Fragments of first-order logic over infinite words
We give topological and algebraic characterizations as well as language
theoretic descriptions of the following subclasses of first-order logic FO[<]
for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and
Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_2). These
descriptions extend the respective results for finite words. In particular, we
relate the above fragments to language classes of certain (unambiguous)
polynomials. An immediate consequence is the decidability of the membership
problem of these classes, but this was shown before by Wilke and Bojanczyk and
is therefore not our main focus. The paper is about the interplay of algebraic,
topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on
Theoretical Aspects of Computer Science, STACS 200
On the Problem of Computing the Probability of Regular Sets of Trees
We consider the problem of computing the probability of regular languages of
infinite trees with respect to the natural coin-flipping measure. We propose an
algorithm which computes the probability of languages recognizable by
\emph{game automata}. In particular this algorithm is applicable to all
deterministic automata. We then use the algorithm to prove through examples
three properties of measure: (1) there exist regular sets having irrational
probability, (2) there exist comeager regular sets having probability and
(3) the probability of \emph{game languages} , from automata theory,
is if is odd and is otherwise
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