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14
Formal languages and groups as memory
, 2006
"... Abstract. We present an exposition of the theory of Mautomata and Gautomata, or finite automata augmented with a multiplyonly register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a grouptheoretic ..."
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Abstract. We present an exposition of the theory of Mautomata and Gautomata, or finite automata augmented with a multiplyonly register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a grouptheoretic interpretation and proof of a key theorem of Chomsky and Schützenberger from formal language theory. 1.
Partially commutative inverse monoids
 PROCEEDINGS OF THE 31TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS 2006), BRATISLAVE (SLOVAKIA), NUMBER 4162 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algo ..."
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Cited by 5 (5 self)
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Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NPcompleteness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.
Small overlap monoids II: Automatic structures and normal forms
 J. Algebra
"... Abstract. We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms ..."
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Cited by 5 (3 self)
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Abstract. We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C(4) monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy analogues of Kleene’s theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automatatheoretic results which may be of independent interest. 1.
On groups and counter automata
 Internat. J. Algebra Comput
"... Abstract. We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind ncounter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gi ..."
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Abstract. We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind ncounter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the MullerSchupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognised by a Gautomaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata. 1.
RATIONAL SUBSETS IN HNNEXTENSIONS AND AMALGAMATED PRODUCTS
"... Several transfer results for rational subsets and finitely generated subgroups of HNNextensions G = 〈H, t; t −1 at = ϕ(a)(a ∈ A) 〉 and amalgamated free products G = H ∗A J such that the associated subgroup A is finite. These transfer results allow to transfer decidability properties or structural pr ..."
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Cited by 4 (3 self)
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Several transfer results for rational subsets and finitely generated subgroups of HNNextensions G = 〈H, t; t −1 at = ϕ(a)(a ∈ A) 〉 and amalgamated free products G = H ∗A J such that the associated subgroup A is finite. These transfer results allow to transfer decidability properties or structural properties from the subgroup H (resp. the subgroups H and J) to the group G. 1.
Tilings and Submonoids of Metabelian Groups
, 903
"... Abstract. In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product Z ≀ (Z × Z). We also show that subsemimodule membership is undecidable for finite rank free (Z × Z)modules. The proof involves an encodi ..."
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Cited by 3 (3 self)
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Abstract. In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product Z ≀ (Z × Z). We also show that subsemimodule membership is undecidable for finite rank free (Z × Z)modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for twodimensional lamplighter groups. 1
On the decidability of semigroup freeness
, 2008
"... This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been clos ..."
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This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over threebythree integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is threefold: (i) to present general results concerning freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic. 1
Bottomup rewriting for words and terms
, 2009
"... For the whole class of linear term rewriting systems, we define bottomup rewriting which is a restriction of the usual notion of rewriting. We show that bottomup rewriting effectively inversepreserves recognizability and analyze the complexity of the underlying construction. The BottomUp class (B ..."
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For the whole class of linear term rewriting systems, we define bottomup rewriting which is a restriction of the usual notion of rewriting. We show that bottomup rewriting effectively inversepreserves recognizability and analyze the complexity of the underlying construction. The BottomUp class (BU) is, by definition, the set of linear systems for which every derivation can be replaced by a bottomup derivation. Membership to BU turns out to be undecidable, we are thus lead to define more restricted classes: the classes SBU(k), k ∈ N of Strongly BottomUp(k) systems for which we show that membership is decidable. We define the class of SBU(k). We give a polynomial sufficient condition for a system to be in SBU. The class SBU contains (strictly) several classes of systems which were already known to inverse preserve recognizability: the inverse leftbasic semiThue systems (viewed as unary term rewriting systems), the linear growing term rewriting systems, the inverse LinearFinitePathOrdering systems. Strongly BottomUp systems by SBU = S k∈N