787 research outputs found
A Congruence-based Perspective on Automata Minimization Algorithms
In this work we use a framework of finite-state automata constructions based on equivalences over words to provide new insights on the relation between well-known methods for computing the minimal deterministic automaton of a language
Minimization via duality
We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object
Reasoning about Regular Properties: A Comparative Study
Several new algorithms for deciding emptiness of Boolean combinations of
regular languages and of languages of alternating automata (AFA) have been
proposed recently, especially in the context of analysing regular expressions
and in string constraint solving. The new algorithms demonstrated a significant
potential, but they have never been systematically compared, neither among each
other nor with the state-of-the art implementations of existing
(non)deterministic automata-based methods. In this paper, we provide the first
such comparison as well as an overview of the existing algorithms and their
implementations. We collect a diverse benchmark mostly originating in or
related to practical problems from string constraint solving, analysing LTL
properties, and regular model checking, and evaluate collected implementations
on it. The results reveal the best tools and hint on what the best algorithms
and implementation techniques are. Roughly, although some advanced algorithms
are fast, such as antichain algorithms and reductions to IC3/PDR, they are not
as overwhelmingly dominant as sometimes presented and there is no clear winner.
The simplest NFA-based technology may be actually the best choice, depending on
the problem source and implementation style. Our findings should be highly
relevant for development of these techniques as well as for related fields such
as string constraint solving
Regular Methods for Operator Precedence Languages
The operator precedence languages (OPLs) represent the largest known subclass of the context-free languages which enjoys all desirable closure and decidability properties. This includes the decidability of language inclusion, which is the ultimate verification problem. Operator precedence grammars, automata, and logics have been investigated and used, for example, to verify programs with arithmetic expressions and exceptions (both of which are deterministic pushdown but lie outside the scope of the visibly pushdown languages). In this paper, we complete the picture and give, for the first time, an algebraic characterization of the class of OPLs in the form of a syntactic congruence that has finitely many equivalence classes exactly for the operator precedence languages. This is a generalization of the celebrated Myhill-Nerode theorem for the regular languages to OPLs. As one of the consequences, we show that universality and language inclusion for nondeterministic operator precedence automata can be solved by an antichain algorithm. Antichain algorithms avoid determinization and complementation through an explicit subset construction, by leveraging a quasi-order on words, which allows the pruning of the search space for counterexample words without sacrificing completeness. Antichain algorithms can be implemented symbolically, and these implementations are today the best-performing algorithms in practice for the inclusion of finite automata. We give a generic construction of the quasi-order needed for antichain algorithms from a finite syntactic congruence. This yields the first antichain algorithm for OPLs, an algorithm that solves the ExpTime-hard language inclusion problem for OPLs in exponential time
A Quasiorder-Based Perspective on Residual Automata
In this work, we define a framework of automata constructions based on quasiorders over words to provide new insights on the class of residual automata. We present a new residualization operation and a generalized double-reversal method for building the canonical residual automaton for a given language. Finally, we use our framework to offer a quasiorder-based perspective on NL^*, an online learning algorithm for residual automata. We conclude that quasiorders are fundamental to residual automata as congruences are to deterministic automata
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