Extended finite automata over groups

Abstract

Some results from Dassow and Mitrana (Internat. J. Comput. Algebra (2000)), Griebach (Theoret. Comput. Sci. 7 (1978) 311) and Ibarra et al. (Theoret. Comput. Sci. 2 (1976) 271) are generalized for finite autómata over arbitrary groups. The closure properties of these autómata are poorer and the accepting power is smaller when abelian groups are considered. We prove that the addition of any abelian group to a finite automaton is less powerful than the addition of the multiplicative group of rational numbers. Thus, each language accepted by a finite automaton over an abelian group is actually a unordered vector language. Characterizations of the context-free and recursively enumerable languages classes are set up in the case of non-abelian groups. We investigate also deterministic finite autómata over groups, especially over abelian groups.

Automata with Counters that Recognize Word Problems of Free Products

“…As noted in [16,Theorem 2], it follows immediately from this lemma that for any monoid M , the family of languages L(M ) is equal to the union L(M 0 ) as M 0 runs through all finitely generated submonoids of M . Combining this with Lemma 5 yields a general result for any monoids K and H: if W (K 0 ) ∈ L(H) for every finitely generated submonoid K 0 of K, then L(K) ⊆ L(H).…”

“…In [16,Theorem 10], it is shown that L(F 2 × F 2 ) is the family of recursively enumerable languages.…”

“…Following [16], we will call such a machine an extended finite automaton with register monoid M (or M -automaton, for short). First let us set up some notation and recall the basic definition of a finite automaton over a monoid.…”

“…One way of increasing the accepting power of finite automata that is currently getting lots of consideration is by adding a counter that behaves like a given group or monoid M ; see, for example, [6,8,10,13,14,16]. Such a machine is called an extended finite automaton with register M or, for short, an M -automaton (see Sec.…”

Unambiguous Constrained Automata

“…A DetCA is an UnCA; moreover, DetCA are effectively equivalent [9] to deterministic extended finite automata over (Z k , +, 0) (defined in [10]). Thus:…”