Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests

We first propose algorithms for checking language equivalence of finite automata over a large alphabet. We use symbolic automata, where the transition function is compactly represented using a (multi-terminal) binary decision diagrams (BDD). The key idea consists in computing a bisimulation by exploring reachable pairs symbolically, so as to avoid redundancies. This idea can be combined with already existing optimisations, and we show in particular a nice integration with the disjoint sets forest data-structure from Hopcroft and Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an algebraic theory that can be used for verification in various domains ranging from compiler optimisation to network programming analysis. This theory is decidable by reduction to language equivalence of automata on guarded strings, a particular kind of automata that have exponentially large alphabets. We propose several methods allowing to construct symbolic automata out of KAT expressions, based either on Brzozowski's derivatives or standard automata constructions. All in all, this results in efficient algorithms for deciding equivalence of KAT expressions

Operations on Boolean and Alternating Finite Automata

We examine the complexity of basic regular operations on languages represented by Boolean and alternating finite automata. We get tight upper bounds m+n and m+n+1 for union, intersection, and difference, 2^m+n and 2^m+n+1 for concatenation, 2^n+n and 2^n+n+1 for square, m and m+1 for left quotient, 2^m and 2^m+1 for right quotient. We also show that in both models, the complexity of complementation and symmetric difference is n and m+n, respectively, while the complexity of star and reversal is 2^n. All our witnesses are described over a unary or binary alphabets, and whenever we use a binary alphabet, it is always optimal.Comment: In Proceedings AFL 2023, arXiv:2309.0112

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

The Complexity of Codiagnosability for Discrete Event and Timed Systems

In this paper we study the fault codiagnosis problem for discrete event systems given by finite automata (FA) and timed systems given by timed automata (TA). We provide a uniform characterization of codiagnosability for FA and TA which extends the necessary and sufficient condition that characterizes diagnosability. We also settle the complexity of the codiagnosability problems both for FA and TA and show that codiagnosability is PSPACE-complete in both cases. For FA this improves on the previously known bound (EXPTIME) and for TA it is a new result. Finally we address the codiagnosis problem for TA under bounded resources and show it is 2EXPTIME-complete.Comment: 24 pages

Coalgebra Learning via Duality

Automata learning is a popular technique for inferring minimal automata through membership and equivalence queries. In this paper, we generalise learning to the theory of coalgebras. The approach relies on the use of logical formulas as tests, based on a dual adjunction between states and logical theories. This allows us to learn, e.g., labelled transition systems, using Hennessy-Milner logic. Our main contribution is an abstract learning algorithm, together with a proof of correctness and termination

Revisiting Synthesis for One-Counter Automata

We study the (parameter) synthesis problem for one-counter automata with parameters. One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables called parameters. The synthesis problem for such automata asks whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some omega-regular property. Lechner showed that (the complement of) the problem can be encoded in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment, called AERPADPLUS, is unfortunately undecidable. Nevertheless, by a careful re-encoding of the problem into a decidable restriction of AERPADPLUS, (ii) we prove that the synthesis problem is decidable in general and in N2EXP for several fixed omega-regular properties. Finally, (iii) we give a polynomial-space algorithm for the special case of the problem where parameters can only be used in tests, and not updates, of the counter

Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

Two-Way Automata Making Choices Only at the Endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply LNL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.Comment: 23 page