489 research outputs found
Buffered Simulation Games for B\"uchi Automata
Simulation relations are an important tool in automata theory because they
provide efficiently computable approximations to language inclusion. In recent
years, extensions of ordinary simulations have been studied, for instance
multi-pebble and multi-letter simulations which yield better approximations and
are still polynomial-time computable.
In this paper we study the limitations of approximating language inclusion in
this way: we introduce a natural extension of multi-letter simulations called
buffered simulations. They are based on a simulation game in which the two
players share a FIFO buffer of unbounded size. We consider two variants of
these buffered games called continuous and look-ahead simulation which differ
in how elements can be removed from the FIFO buffer. We show that look-ahead
simulation, the simpler one, is already PSPACE-hard, i.e. computationally as
hard as language inclusion itself. Continuous simulation is even EXPTIME-hard.
We also provide matching upper bounds for solving these games with infinite
state spaces.Comment: In Proceedings AFL 2014, arXiv:1405.527
Regular Cost Functions, Part I: Logic and Algebra over Words
The theory of regular cost functions is a quantitative extension to the
classical notion of regularity. A cost function associates to each input a
non-negative integer value (or infinity), as opposed to languages which only
associate to each input the two values "inside" and "outside". This theory is a
continuation of the works on distance automata and similar models. These models
of automata have been successfully used for solving the star-height problem,
the finite power property, the finite substitution problem, the relative
inclusion star-height problem and the boundedness problem for monadic-second
order logic over words. Our notion of regularity can be -- as in the classical
theory of regular languages -- equivalently defined in terms of automata,
expressions, algebraic recognisability, and by a variant of the monadic
second-order logic. These equivalences are strict extensions of the
corresponding classical results. The present paper introduces the cost monadic
logic, the quantitative extension to the notion of monadic second-order logic
we use, and show that some problems of existence of bounds are decidable for
this logic. This is achieved by introducing the corresponding algebraic
formalism: stabilisation monoids.Comment: 47 page
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Boundedness in languages of infinite words
We define a new class of languages of -words, strictly extending
-regular languages.
One way to present this new class is by a type of regular expressions. The
new expressions are an extension of -regular expressions where two new
variants of the Kleene star are added: and . These new
exponents are used to say that parts of the input word have bounded size, and
that parts of the input can have arbitrarily large sizes, respectively. For
instance, the expression represents the language of infinite
words over the letters where there is a common bound on the number of
consecutive letters . The expression represents a similar
language, but this time the distance between consecutive 's is required to
tend toward the infinite.
We develop a theory for these languages, with a focus on decidability and
closure. We define an equivalent automaton model, extending B\"uchi automata.
The main technical result is a complementation lemma that works for languages
where only one type of exponent---either or ---is used.
We use the closure and decidability results to obtain partial decidability
results for the logic MSOLB, a logic obtained by extending monadic second-order
logic with new quantifiers that speak about the size of sets
Church Synthesis on Register Automata over Linearly Ordered Data Domains
Register automata are finite automata equipped with a finite set of registers in which they can store data, i.e. elements from an unbounded or infinite alphabet. They provide a simple formalism to specify the behaviour of reactive systems operating over data ?-words. We study the synthesis problem for specifications given as register automata over a linearly ordered data domain (e.g. (?, ?) or (?, ?)), which allow for comparison of data with regards to the linear order. To that end, we extend the classical Church synthesis game to infinite alphabets: two players, Adam and Eve, alternately play some data, and Eve wins whenever their interaction complies with the specification, which is a language of ?-words over ordered data. Such games are however undecidable, even when the specification is recognised by a deterministic register automaton. This is in contrast with the equality case, where the problem is only undecidable for nondeterministic and universal specifications.
Thus, we study one-sided Church games, where Eve instead operates over a finite alphabet, while Adam still manipulates data. We show they are determined, and deciding the existence of a winning strategy is in ExpTime, both for ? and ?. This follows from a study of constraint sequences, which abstract the behaviour of register automata, and allow us to reduce Church games to ?-regular games. Lastly, we apply these results to the transducer synthesis problem for input-driven register automata, where each output data is restricted to be the content of some register, and show that if there exists an implementation, then there exists one which is a register transducer
The logical strength of Büchi's decidability theorem
We study the strength of axioms needed to prove various results related to automata on infinite words and Büchi's theorem on the decidability of the MSO theory of (N, less_or_equal). We prove that the following are equivalent over the weak second-order arithmetic theory RCA: 1. Büchi's complementation theorem for nondeterministic automata on infinite words, 2. the decidability of the depth-n fragment of the MSO theory of (N, less_or_equal), for each n greater than 5, 3. the induction scheme for Sigma^0_2 formulae of arithmetic. Moreover, each of (1)-(3) is equivalent to the additive version of Ramsey's Theorem for pairs, often used in proofs of (1); each of (1)-(3) implies McNaughton's determinisation theorem for automata on infinite words; and each of (1)-(3) implies the "bounded-width" version of König's Lemma, often used in proofs of McNaughton's theorem
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