ω-automaton

About: ω-automaton is a research topic. Over the lifetime, 2299 publications have been published within this topic receiving 68468 citations. The topic is also known as: stream automaton & ω-automata.

Logical simulation of physical systems by classical automata

Abstract: The simulation of the behavior of physical systems on the coarsest level, namely, restoring the logical structure of propositions about the system is suggested. To realize this simulation, the special class of finite automata, called normalized, is introduced. Relevant graph-theoretical techniques are considered.

Automata to Recognize Finite and Infinite Words with at Least Two Factorizations on a Given Finite Set

Abstract: This paper is about the recognition of finite and infinite words with at least two factorizations on a finite language. We implement the construction of the graph of delays for a finite language L which can be view as an automaton that accept words and ω-words that are ambiguously covered by L.

A Free Energy Foundation of Semantic Similarity in Automata and Languages

Abstract: This paper develops a free energy theory from physics including the variational principles for automata and languages and also provides algorithms to compute the energy as well as efficient algorithms for estimating the nondeterminism in a nondeterministic finite automaton. This theory is then used as a foundation to define a semantic similarity metric for automata and languages. Since automata are a fundamental model for all modern programs while languages are a fundamental model for the programs’ behaviors, we believe that the theory and the metric developed in this paper can be further used for real-word programs as well.

On the Representation of Finite Automata

Abstract: We give an unique string representation, up to isomorphism, for initially connected deterministic finite automata (ICDFA’s) with n states over an alphabet of k symbols. We show how to generate all these strings for each n and k, and how its enumeration provides an alternative way to obtain the exact number of ICDFA’s. 1 Motivation In symbolic manipulation environments for finite automata, it is important to have an adequate representation of automata and, dependent upon their use, several representations may be available. For example, for testing if two finite automata are isomorphic objects or for (random) generation of automata, the representation must be compact and somehow canonical. In the FAdo project [MR05a, fad] a canonical form is used to test if two minimal DFA’s are isomorphic (i.e are the same up to renaming of states). In this paper we prove the correctness of that representation and show how it can be used for the exact enumeration and generation of initially connected deterministic finite automata (ICDFA). The problem of enumeration of finite automata was considered by several authors since early 1960s, in particular see Harrison [Har65], Robinson [Rob85], Harary and Palmer [HP67] and Liskovets [Lis69] amongst many others. A survey may be found in Domaratzki et al. [DKS02]. More recently, several authors examined related problems. Domaratzki et al. [DKS02] studied the enumeration of distinct languages accepted by finite automata with n states; Nicaud [Nic99], Champarnaud and Paranthoen [CP05, Par04] and Bassino and Nicaud [BN] analysed several aspects of the average behaviour of regular languages; Liskovets [Lis03] and Domaratzki [Dom04] gave (exact and asymptotic) enumerations of acyclic DFA’s and of finite languages. The paper is organised as follows. In the next section, we review some basic notions and introduce some notation. Section 3 describes a string representation for deterministic finite automata that is unique up to isomorphism for initially connected deterministic finite automata. Section 4 presents an efficient method to generate those strings. Section 5 shows how their enumeration provides an upper bound and the exact value for the number of ICDFA’s. Section 6 and Appendix A report some implementation issues and final remarks. 2 Preliminaries We first recall some basic notions from automata theory and formal languages, that can be found in standard books [HMU00]. An alphabet Σ is a nonempty set of symbols. A string over Σ is a finite sequence of symbols of Σ. The empty string is denoted by ǫ. The set Σ is the set of all strings over Σ. A language L is a subset of Σ. The density of a language L over Σ, ρL(n), is the number of strings of length n that are in L, i.e., ρL(n) = |L ∩ Σ |. A regular expression (r.e.) α over Σ represents a language L(α) ⊆ Σ and is inductively defined by: ∅, ǫ and σ ∈ Σ are a r.e., where L(∅) = ∅, L(ǫ) = {ǫ} and L(σ) = {σ}; if α1 and α2 are r.e., (α1+α2), (α1α2) and α ⋆ 1 are r.e., respectively with L((α1+α2)) = L(α1)∪L(α2), L((α1α2)) = L(α1)L(α2) and L(α1 ) = L(α1) . In this paper, we will use regular expressions to represent descriptions of finite automata. A deterministic finite automaton (DFA) A is a quintuple (Q,Σ, δ, q0, F ) where Q is a finite set of states, Σ is the alphabet, δ : Q× Σ → Q is the transition function, q0 the initial state and F ⊆ Q the set of final states. We assume that the transition function is total, so we consider only complete DFA’s. The size of a DFA 2 is the number of its states, |Q|. Normally, we are not interested in the labels of the states and we can represent them by an integer 0 ≤ i < |Q|. The transition function δ extends naturally to Σ: for all q ∈ Q, if x = ǫ then δ(q, ǫ) = q; if x = yσ then δ(q, x) = δ(δ(q, y), σ). A DFA is initially connected1 (ICDFA) if for each state q ∈ Q there exists a string x ∈ Σ such that δ(q0, x) = q. Two DFA’s A = (Q,Σ, δ, q0, F ) and A ′ = (Q,Σ, δ, q 0, F ) are called isomorphic (by states) if there exists a bijection f : Q → Q such that f(q0) = q ′ 0 and for all σ ∈ Σ and q ∈ Q, f(δ(q, σ)) = δ(f(q), σ). Furthermore, for all q ∈ Q, q ∈ F if and only if f(q) ∈ F . The language accepted by a DFA A is L(A) = {x ∈ Σ | δ(q0, x) ∈ F}. Two DFA are equivalent if they accept the same language. Obviously, two isomorphic automata are equivalent, but two non-isomorphic automata may be equivalent. A DFA A is minimal if there is no DFA A with fewer states equivalent to A. Trivially a minimal DFA is an ICDFA. Minimal DFA’s are unique up to isomorphism. We are mainly concerned with the representation of the transition function of DFA’s of size n over an alphabet of k symbols , so we disregard the set of final states and we consider only a quadruple (Q,Σ, δ, q0) called the structure of an automaton and referred as DFA∅. For each of our representations, there will be 2 DFA’s. We denote by ICDFA∅ the structure of an ICDFA. We consider that any integer variable has always a nonnegative value (if not otherwise stated). Let [n]0 = {0, 1, . . . , n} and [n] = {1, . . . , n}. 3 Representations towards a normal form The method used to represent a DFA has a significative role in the amount of computer work needed to manipulate that information, and can give an important insight about this set of objects, both in its characterisation and enumeration. Let us disregard the set of final states of a DFA. A naive representation of a DFA∅ can be obtained by the enumeration of its states and for each state a list of its transitions for each symbol. For the DFA∅ in Fig.1 we have: [[A (a : A, b : B)], [B (a : A, b : E)], [C (a : B, b : E)], [D (a : D, b : C)], [E (a : A, b : E)]]. (1)