29,971 research outputs found
Canonical Algebraic Generators in Automata Learning
Many methods for the verification of complex computer systems require the
existence of a tractable mathematical abstraction of the system, often in the
form of an automaton. In reality, however, such a model is hard to come up
with, in particular manually. Automata learning is a technique that can
automatically infer an automaton model from a system -- by observing its
behaviour. The majority of automata learning algorithms is based on the
so-called L* algorithm. The acceptor learned by L* has an important property:
it is canonical, in the sense that, it is, up to isomorphism, the unique
deterministic finite automaton of minimal size accepting a given regular
language. Establishing a similar result for other classes of acceptors, often
with side-effects, is of great practical importance. Non-deterministic finite
automata, for instance, can be exponentially more succinct than deterministic
ones, allowing verification to scale. Unfortunately, identifying a canonical
size-minimal non-deterministic acceptor of a given regular language is in
general not possible: it can happen that a regular language is accepted by two
non-isomorphic non-deterministic finite automata of minimal size. In
particular, it thus is unclear which one of the automata should be targeted by
a learning algorithm. In this thesis, we further explore the issue and identify
(sub-)classes of acceptors that admit canonical size-minimal representatives.Comment: PhD thesi
Canonical Algebraic Generators in Automata Learning
Many methods for the verification of complex computer systems require the existence of a tractable mathematical abstraction of the system, often in the form of an automaton. In reality, however, such a model is hard to come up with, in particular manually. Automata learning is a technique that can automatically infer an automaton model from a system -- by observing its behaviour. The majority of automata learning algorithms is based on the so-called L* algorithm. The acceptor learned by L* has an important property: it is canonical, in the sense that, it is, up to isomorphism, the unique deterministic finite automaton of minimal size accepting a given regular language. Establishing a similar result for other classes of acceptors, often with side-effects, is of great practical importance. Non-deterministic finite automata, for instance, can be exponentially more succinct than deterministic ones, allowing verification to scale. Unfortunately, identifying a canonical size-minimal non-deterministic acceptor of a given regular language is in general not possible: it can happen that a regular language is accepted by two non-isomorphic non-deterministic finite automata of minimal size. In particular, it thus is unclear which one of the automata should be targeted by a learning algorithm. In this thesis, we further explore the issue and identify (sub-)classes of acceptors that admit canonical size-minimal representatives.
In more detail, the contributions of this thesis are three-fold.
First, we expand the automata (learning) theory of Guarded Kleene Algebra with Tests (GKAT), an efficiently decidable logic expressive enough to model simple imperative programs. In particular, we present GL*, an algorithm that learns the unique size-minimal GKAT automaton for a given deterministic language, and prove that GL* is more efficient than an existing variation of L*. We implement both algorithms in OCaml, and compare them on example programs.
Second, we present a category-theoretical framework based on generators, bialgebras, and distributive laws, which identifies, for a wide class of automata with side-effects in a monad, canonical target models for automata learning. Apart from recovering examples from the literature, we discover a new canonical acceptor of regular languages, and present a unifying minimality result.
Finally, we show that the construction underlying our framework is an instance of a more general theory. First, we see that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on a category of subobjects with respect to an epi-mono factorisation system. Second, we explore the abstract theory of generators and bases for algebras over a monad: we discuss bases for bialgebras, the product of bases, generalise the representation theory of linear maps, and compare our ideas to a coalgebra-based approach
Extending the scalars of minimizations
In the classical theory of formal languages, finite state automata allow to recognize the words of a rational subset of where is a set of symbols (or the alphabet). Now, given a semiring (\K,+,.), one can construct \K-subsets of in the sense of Eilenberg , that are alternatively called noncommutative formal power series for which a framework very similar to language theory has been constructed Particular noncommutative formal power series, which are called rational series, are the behaviour of a family of weighted automata (or \K-automata). In order to get an efficient encoding, it may be interesting to point out one of them with the smallest number of states. Minimization processes of \K-automata already exist for \K being:\\ {\bf a)} a field ,\\ {\bf b)} a noncommutative field ,\\ {\bf c)} a PID .\\ When \K is the bolean semiring, such a minimization process (with isomorphisms of minimal objects) is known within the category of deterministic automata. Minimal automata have been proved to be isomorphic in cases {\bf (a)} and {\bf (b)}. But the proof given for (b) is not constructive. In fact, it lays on the existence of a basis for a submodule of \K^n. Here we give an independent algorithm which reproves this fact and an example of a pair of nonisomorphic minimal automata. Moreover, we examine the possibility of extending {\bf (c)}. To this end, we provide an {\em Effective Minimization Process} (or {\em EMP}) which can be used for more general sets of coefficients
Automata, reduced words, and Garside shadows in Coxeter groups
In this article, we introduce and investigate a class of finite deterministic
automata that all recognize the language of reduced words of a finitely
generated Coxeter system (W,S). The definition of these automata is
straightforward as it only requires the notion of weak order on (W,S) and the
related notion of Garside shadows in (W,S), an analog of the notion of a
Garside family. Then we discuss the relations between this class of automata
and the canonical automaton built from Brink and Howlett's small roots. We end
this article by providing partial positive answers to two conjectures: (1) the
automata associated to the smallest Garside shadow is minimal; (2) the
canonical automaton is minimal if and only if the support of all small roots is
spherical, i.e., the corresponding root system is finite.Comment: 21 pages, 7 figures; v2: 23 pages, 8 figures, Remark 3.15 added,
accepted in Journal of Algebra, computational sectio
On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)
Eilenberg correspondence, based on the concept of syntactic monoids, relates
varieties of regular languages with pseudovarieties of finite monoids. Various
modifications of this correspondence related more general classes of regular
languages with classes of more complex algebraic objects. Such generalized
varieties also have natural counterparts formed by classes of finite automata
equipped with a certain additional algebraic structure. In this survey, we
overview several variants of such varieties of enriched automata.Comment: In Proceedings AFL 2014, arXiv:1405.527
Potential of quantum finite automata with exact acceptance
The potential of the exact quantum information processing is an interesting,
important and intriguing issue. For examples, it has been believed that quantum
tools can provide significant, that is larger than polynomial, advantages in
the case of exact quantum computation only, or mainly, for problems with very
special structures. We will show that this is not the case.
In this paper the potential of quantum finite automata producing outcomes not
only with a (high) probability, but with certainty (so called exactly) is
explored in the context of their uses for solving promise problems and with
respect to the size of automata. It is shown that for solving particular
classes of promise problems, even those without some
very special structure, that succinctness of the exact quantum finite automata
under consideration, with respect to the number of (basis) states, can be very
small (and constant) though it grows proportional to in the case
deterministic finite automata (DFAs) of the same power are used. This is here
demonstrated also for the case that the component languages of the promise
problems solvable by DFAs are non-regular. The method used can be applied in
finding more exact quantum finite automata or quantum algorithms for other
promise problems.Comment: We have improved the presentation of the paper. Accepted to
International Journal of Foundation of Computer Scienc
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