An Automata Theoretic Approach to the Zero-One Law for Regular Languages: Algorithmic and Logical Aspects

A zero-one language L is a regular language whose asymptotic probability converges to either zero or one. In this case, we say that L obeys the zero-one law. We prove that a regular language obeys the zero-one law if and only if its syntactic monoid has a zero element, by means of Eilenberg's variety theoretic approach. Our proof gives an effective automata characterisation of the zero-one law for regular languages, and it leads to a linear time algorithm for testing whether a given regular language is zero-one. In addition, we discuss the logical aspects of the zero-one law for regular languages.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

Simultaneous Finite Automata: An Efficient Data-Parallel Model for Regular Expression Matching

Automata play important roles in wide area of computing and the growth of multicores calls for their efficient parallel implementation. Though it is known in theory that we can perform the computation of a finite automaton in parallel by simulating transitions, its implementation has a large overhead due to the simulation. In this paper we propose a new automaton called simultaneous finite automaton (SFA) for efficient parallel computation of an automaton. The key idea is to extend an automaton so that it involves the simulation of transitions. Since an SFA itself has a good property of parallelism, we can develop easily a parallel implementation without overheads. We have implemented a regular expression matcher based on SFA, and it has achieved over 10-times speedups on an environment with dual hexa-core CPUs in a typical case.Comment: This paper has been accepted at the following conference: 2013 International Conference on Parallel Processing (ICPP- 2013), October 1-4, 2013 Ecole Normale Suprieure de Lyon, Lyon, Franc

Words-to-Letters Valuations for Language Kleene Algebras with Variable Complements

We investigate the equational theory of Kleene algebra terms with variable complements -- (language) complement where it applies only to variables -- w.r.t. languages. While the equational theory w.r.t. languages coincides with the language equivalence (under the standard language valuation) for Kleene algebra terms, this coincidence is broken if we extend the terms with complements. In this paper, we prove the decidability of some fragments of the equational theory: the universality problem is coNP-complete, and the inequational theory t <= s is coNP-complete when t does not contain Kleene-star. To this end, we introduce words-to-letters valuations; they are sufficient valuations for the equational theory and ease us in investigating the equational theory w.r.t. languages. Additionally, we prove that for words with variable complements, the equational theory coincides with the word equivalence.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence

It is well known that the length of a beta-reduction sequence of a simply typed lambda-term of order k can be huge; it is as large as k-fold exponential in the size of the lambda-term in the worst case. We consider the following relevant question about quantitative properties, instead of the worst case: how many simply typed lambda-terms have very long reduction sequences? We provide a partial answer to this question, by showing that asymptotically almost every simply typed lambda-term of order k has a reduction sequence as long as (k-1)-fold exponential in the term size, under the assumption that the arity of functions and the number of variables that may occur in every subterm are bounded above by a constant. To prove it, we have extended the infinite monkey theorem for strings to a parametrized one for regular tree languages, which may be of independent interest. The work has been motivated by quantitative analysis of the complexity of higher-order model checking