Proceedings of the American Mathematical Society, 1993
Linear congruences on a free monoid X ∗ {X^{\ast }} coincide with the congruences on X ∗ {X^{\ast... more Linear congruences on a free monoid X ∗ {X^{\ast }} coincide with the congruences on X ∗ {X^{\ast }} induced by nontrivial homomorphisms into the additive group of integers. For a finite X X , we characterize abstractly several classes of linear congruences on X ∗ {X^{\ast }} , in particular, π \pi -linear congruences, called p p -linear and determined by Reis, ξ \xi -linear congruences, introduced by Petrich and Thierrin, and general linear congruences, introduced by the authors. These characterizations include descriptions involving maximality as prefix congruences.
We show that the word problem is decidable for an amalgamated free product of finite inverse semi... more We show that the word problem is decidable for an amalgamated free product of finite inverse semigroups (in the category of inverse semigroups). This is in contrast to a recent result of M. Sapir that shows that the word problem for amalgamated free products of finite semigroups (in the category of semigroups) is in general undecidable.
A finite deterministic automaton A = (Q,, ) is k-com- pressible if there is a word w 2 + such tha... more A finite deterministic automaton A = (Q,, ) is k-com- pressible if there is a word w 2 + such that the image of the state set Q under the natural action of w is reduced by at least k states. In such case w is called a k-compressing word forA. It is known that, for any alphabet and any k 2, there exist words that are k-compressing for each k-compressible automaton with the input alphabet . Such words are called k-collapsing. It has been proved that recognizing 2- collapsing words over a 2-element alphabet may be done in polynomial time, while recognizing 2-collapsing words over an alphabet of size 3 is co-NP-complete. A natural question in this context, whether recog- nizing 3-collapsing words over a 2-element alphabet is easy or hard, has remained open. In this paper we provide results on 3-compressible bi- nary automata, which allow to prove that that the latter problem is co-NP-complete.
A finite deterministic (semi)automaton A = (Q, Σ, δ) is k-compressible if there is some word w ∈ ... more A finite deterministic (semi)automaton A = (Q, Σ, δ) is k-compressible if there is some word w ∈ Σ + such that theimage of its state set Q under the natural action of w is reduced by at least k states. Such word w, if it exists, is calleda k-compressing word for A and A is said to be k-compressed by w. A word is k-collapsing if it is k-compressing foreach k-compressible automaton, and it is k-synchronizing if it is k-compressing for all k-compressible automata withk+1 states. We compute a set W of short words such that each 3-compressible automaton on a two-letter alphabetis 3-compressed at least by a word in W. Then we construct a shortest common superstring of the words in W and,with a further refinement, we obtain a 3-collapsing word of length 53. Moreover, as previously announced, we showthat the shortest 3-synchronizing word is not 3-collapsing, illustrating the new bounds 34 ≤ c(2, 3) ≤ 53 for the length c(2, 3) of the shortest 3-collapsing word on a two-letter alphabet.
In this paper we summarize some results of a work in progress on the computational complexity of ... more In this paper we summarize some results of a work in progress on the computational complexity of the alphabetical satisfiability problem for linear trace equations. In particular we prove that the problem is in P under some conditions on the maximal I-cliques.
A word w over a ÿnite alphabet is said to be n-collapsing if for an arbitrary ÿnite automaton A =... more A word w over a ÿnite alphabet is said to be n-collapsing if for an arbitrary ÿnite automaton A = Q; −•− , the inequality |Q • w| 6 |Q| − n holds provided that |Q • u| 6 |Q| − n for some word u (depending on A). We give an algorithm to test whether a word is 2-collapsing. To this aim we associate to every word w a ÿnite family of ÿnitely generated subgroups in ÿnitely generated free groups and prove that the property of being 2-collapsing re ects in the property that each of these subgroups has index at most 2 in the corresponding free group. We also ÿnd a similar characterization for the closely related class of so-called 2-synchronizing words.
We use the description of the Schützenberger automata for amalgams of finite inverse semigroups g... more We use the description of the Schützenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi in [5] to obtain structural results for such amalgams. Schützenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential information about the automaton. Using this crucial fact, and the Bass-Serre theory, we show that the maximal subgroups of an amalgamated free-product are either isomorphic to certain subgroups of the original semigroups or can be described as fundamental groups of particular finite graphs of groups build from the maximal subgroups of the original semigroups.
Automata, Logic and Semantics International audience Given a word w over a finite alphabet Sigma ... more Automata, Logic and Semantics International audience Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.
International Journal of Foundations of Computer Science, 1996
A new class of languages, called multi-push-down (mpd), that generalize the classical context-fre... more A new class of languages, called multi-push-down (mpd), that generalize the classical context-free (cf, or Chomsky type 2) ones is introduced. These languages preserve some important properties of cf languages: a generalization of the Chomsky-Schützenberger homomorphic characterization theorem, the Parikh theorem and a “pumping lemma” are proved. Multi-push-down languages are an AFL. Their recognizers are automata equipped with a multi-push-down tape. Multi-push-down languages form a hierarchy based on the number of push-down tapes.
We study inverse semigroup amalgams [S 1 , S 2 ; U ], where S 1 and S 2 are finitely presented in... more We study inverse semigroup amalgams [S 1 , S 2 ; U ], where S 1 and S 2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S 1 and S 2. We use a modified version of Bennett's work on the structure of Schützenberger graphs of the R-classes of S 1 * U S 2 to state sufficient conditions for the amalgamated free products S 1 * U S 2 having decidable word problem.
The inverse hull of a left reductive right cancellative semigroup S is represented as a quotient ... more The inverse hull of a left reductive right cancellative semigroup S is represented as a quotient semigroup of the free inverse monoid generated by .S. Necessary and sufficient conditions are established on S in order for its inverse hull to be Eunitary, in which case its P-representation is constructed. The inverse hull of a reversible cancellative semigroup is proved to be an F-inverse semigroup, for which an F-representation is constructed. The class of right cancellative monoids is provided with suitable morphisms; the resulting category is proved to be equivalent to a certain category of inverse monoids via the inverse hull.
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2007
Fuzzy aggregation is the way in which different contributions to the same fuzzy fact are merged t... more Fuzzy aggregation is the way in which different contributions to the same fuzzy fact are merged together to obtain a possibility distribution representative of the acquired knowledge. The choice of the aggregation function is a fundamental step in the definition of inference framework. In most cases aggregation has some monotonicity property and this can lead to saturation problems in complex frameworks, particularly in stateful rational agents. In this paper, we propose an extension to the fuzzy aggregation to handle these cases and apply fuzzy reasoning to complex KBs. We especially focus on Mamdani inference framework, where aggregation is implemented by a triangular conorm.
Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q , Σ, δ , the i... more Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q , Σ, δ , the inequality |δ(Q , w)| ≤ |Q | − n means that under the natural action of the word w the image of the state set Q is reduced by at least n states. A word w is n-collapsing if this inequality holds for any deterministic finite automaton that satisfies such an inequality for at least one word. In this paper we prove that the problem of recognizing n-collapsing words is generally co-NP-complete, while restricted to 2-collapsing words over 2-element alphabet it belongs to P. This is connected with introducing a new approach to collapsing words, which is shown to be much more effective in solving various problems in the area. It leads to interesting connections with combinatorial problems concerning solving systems of permutation conditions on one hand, and coloring trees with distinguished nodes on the other hand.
Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q, Σ, δ , the in... more Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q, Σ, δ , the inequality |δ(Q, w)| ≤ |Q| − n means that under the natural action of the word w the image of the state set Q is reduced by at least n states. The word w is n-collapsing if this inequality holds for any deterministic finite automaton that satisfies such an inequality for at least one word. In this paper we present a new approach to the topic of collapsing words, and announce a few results we have obtained using this new approach. In particular, we present a direct proof of the fact that the language of n-collapsing words is recursive.
K-depth grammars extend context-free grammars allowing k 1 rewriting points for a single non-term... more K-depth grammars extend context-free grammars allowing k 1 rewriting points for a single non-terminal at every step of a derivation. The family of languages generated by k-depth grammars is a proper extension of the family of context-free languages, while retaining many context-free properties, such as closure properties, a version of Chomsky Schu tzenberger theorem, the existence of an accepting device (the multi-pushdown automaton). Here a polynomial-time parsing algorithm for k-depth languages is defined, and its correctness is proved.
The new Associative Language Description (ALD) model, a combination of locally testable and const... more The new Associative Language Description (ALD) model, a combination of locally testable and constituent structure ideas, is proposed, arguing that in practice it equals context-free (CF) grammars in explanatory adequacy, yet it provides a simple description and it excludes mathematical sets based on counting properties, which are rarely (if ever) used in compiler construction or in computational linguistics. The ALD model has been recently proposed as an approach consistent with current views on brain organization. ALD is a "pure", i.e., nonterminal-free definition. The strict inclusion of ALD languages in CF languages is proved, based on a lemma which strengthens the Pumping Lemma for CF languages. Basic nonclosure and undecidability properties are considered and compared with those of CF languages. It is shown that the hardest context-free language is in ALD, that there exists a hierarchy of ALD languages and that each ALD tree language enjoys the noncounting property of parenthesized CF languages. Typical technical languages (Pascal, HTML) can be rather conveniently described by ALD rules.
Discrete Mathematics & Theoretical Computer Science, 2007
International audience The Associative Language Description model (ALD) is a combination of local... more International audience The Associative Language Description model (ALD) is a combination of locally testable and constituent structure ideas. It is consistent with current views on brain organization and can rather conveniently describe typical technical languages such as Pascal or HTML. ALD languages are strictly enclosed in context-free languages but in practice the ALD model equals CF grammars in explanatory adequacy. Various properties of ALD have been investigated, but many theoretical questions are still open. For instance, it is unknown, at the present, whether the ALD family includes the regular languages. Here it is proved that several known classes of regular languages are ALD: threshold locally testable languages, group languages, positive commutative languages and commutative languages on 2-letter alphabets. Moreover, we show that there is an ALD language in each level of (restricted) star height hierarchy. These results seem to show that ALD languages are well-distribut...
Proceedings of the American Mathematical Society, 1993
Linear congruences on a free monoid X ∗ {X^{\ast }} coincide with the congruences on X ∗ {X^{\ast... more Linear congruences on a free monoid X ∗ {X^{\ast }} coincide with the congruences on X ∗ {X^{\ast }} induced by nontrivial homomorphisms into the additive group of integers. For a finite X X , we characterize abstractly several classes of linear congruences on X ∗ {X^{\ast }} , in particular, π \pi -linear congruences, called p p -linear and determined by Reis, ξ \xi -linear congruences, introduced by Petrich and Thierrin, and general linear congruences, introduced by the authors. These characterizations include descriptions involving maximality as prefix congruences.
We show that the word problem is decidable for an amalgamated free product of finite inverse semi... more We show that the word problem is decidable for an amalgamated free product of finite inverse semigroups (in the category of inverse semigroups). This is in contrast to a recent result of M. Sapir that shows that the word problem for amalgamated free products of finite semigroups (in the category of semigroups) is in general undecidable.
A finite deterministic automaton A = (Q,, ) is k-com- pressible if there is a word w 2 + such tha... more A finite deterministic automaton A = (Q,, ) is k-com- pressible if there is a word w 2 + such that the image of the state set Q under the natural action of w is reduced by at least k states. In such case w is called a k-compressing word forA. It is known that, for any alphabet and any k 2, there exist words that are k-compressing for each k-compressible automaton with the input alphabet . Such words are called k-collapsing. It has been proved that recognizing 2- collapsing words over a 2-element alphabet may be done in polynomial time, while recognizing 2-collapsing words over an alphabet of size 3 is co-NP-complete. A natural question in this context, whether recog- nizing 3-collapsing words over a 2-element alphabet is easy or hard, has remained open. In this paper we provide results on 3-compressible bi- nary automata, which allow to prove that that the latter problem is co-NP-complete.
A finite deterministic (semi)automaton A = (Q, Σ, δ) is k-compressible if there is some word w ∈ ... more A finite deterministic (semi)automaton A = (Q, Σ, δ) is k-compressible if there is some word w ∈ Σ + such that theimage of its state set Q under the natural action of w is reduced by at least k states. Such word w, if it exists, is calleda k-compressing word for A and A is said to be k-compressed by w. A word is k-collapsing if it is k-compressing foreach k-compressible automaton, and it is k-synchronizing if it is k-compressing for all k-compressible automata withk+1 states. We compute a set W of short words such that each 3-compressible automaton on a two-letter alphabetis 3-compressed at least by a word in W. Then we construct a shortest common superstring of the words in W and,with a further refinement, we obtain a 3-collapsing word of length 53. Moreover, as previously announced, we showthat the shortest 3-synchronizing word is not 3-collapsing, illustrating the new bounds 34 ≤ c(2, 3) ≤ 53 for the length c(2, 3) of the shortest 3-collapsing word on a two-letter alphabet.
In this paper we summarize some results of a work in progress on the computational complexity of ... more In this paper we summarize some results of a work in progress on the computational complexity of the alphabetical satisfiability problem for linear trace equations. In particular we prove that the problem is in P under some conditions on the maximal I-cliques.
A word w over a ÿnite alphabet is said to be n-collapsing if for an arbitrary ÿnite automaton A =... more A word w over a ÿnite alphabet is said to be n-collapsing if for an arbitrary ÿnite automaton A = Q; −•− , the inequality |Q • w| 6 |Q| − n holds provided that |Q • u| 6 |Q| − n for some word u (depending on A). We give an algorithm to test whether a word is 2-collapsing. To this aim we associate to every word w a ÿnite family of ÿnitely generated subgroups in ÿnitely generated free groups and prove that the property of being 2-collapsing re ects in the property that each of these subgroups has index at most 2 in the corresponding free group. We also ÿnd a similar characterization for the closely related class of so-called 2-synchronizing words.
We use the description of the Schützenberger automata for amalgams of finite inverse semigroups g... more We use the description of the Schützenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi in [5] to obtain structural results for such amalgams. Schützenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential information about the automaton. Using this crucial fact, and the Bass-Serre theory, we show that the maximal subgroups of an amalgamated free-product are either isomorphic to certain subgroups of the original semigroups or can be described as fundamental groups of particular finite graphs of groups build from the maximal subgroups of the original semigroups.
Automata, Logic and Semantics International audience Given a word w over a finite alphabet Sigma ... more Automata, Logic and Semantics International audience Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.
International Journal of Foundations of Computer Science, 1996
A new class of languages, called multi-push-down (mpd), that generalize the classical context-fre... more A new class of languages, called multi-push-down (mpd), that generalize the classical context-free (cf, or Chomsky type 2) ones is introduced. These languages preserve some important properties of cf languages: a generalization of the Chomsky-Schützenberger homomorphic characterization theorem, the Parikh theorem and a “pumping lemma” are proved. Multi-push-down languages are an AFL. Their recognizers are automata equipped with a multi-push-down tape. Multi-push-down languages form a hierarchy based on the number of push-down tapes.
We study inverse semigroup amalgams [S 1 , S 2 ; U ], where S 1 and S 2 are finitely presented in... more We study inverse semigroup amalgams [S 1 , S 2 ; U ], where S 1 and S 2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S 1 and S 2. We use a modified version of Bennett's work on the structure of Schützenberger graphs of the R-classes of S 1 * U S 2 to state sufficient conditions for the amalgamated free products S 1 * U S 2 having decidable word problem.
The inverse hull of a left reductive right cancellative semigroup S is represented as a quotient ... more The inverse hull of a left reductive right cancellative semigroup S is represented as a quotient semigroup of the free inverse monoid generated by .S. Necessary and sufficient conditions are established on S in order for its inverse hull to be Eunitary, in which case its P-representation is constructed. The inverse hull of a reversible cancellative semigroup is proved to be an F-inverse semigroup, for which an F-representation is constructed. The class of right cancellative monoids is provided with suitable morphisms; the resulting category is proved to be equivalent to a certain category of inverse monoids via the inverse hull.
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2007
Fuzzy aggregation is the way in which different contributions to the same fuzzy fact are merged t... more Fuzzy aggregation is the way in which different contributions to the same fuzzy fact are merged together to obtain a possibility distribution representative of the acquired knowledge. The choice of the aggregation function is a fundamental step in the definition of inference framework. In most cases aggregation has some monotonicity property and this can lead to saturation problems in complex frameworks, particularly in stateful rational agents. In this paper, we propose an extension to the fuzzy aggregation to handle these cases and apply fuzzy reasoning to complex KBs. We especially focus on Mamdani inference framework, where aggregation is implemented by a triangular conorm.
Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q , Σ, δ , the i... more Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q , Σ, δ , the inequality |δ(Q , w)| ≤ |Q | − n means that under the natural action of the word w the image of the state set Q is reduced by at least n states. A word w is n-collapsing if this inequality holds for any deterministic finite automaton that satisfies such an inequality for at least one word. In this paper we prove that the problem of recognizing n-collapsing words is generally co-NP-complete, while restricted to 2-collapsing words over 2-element alphabet it belongs to P. This is connected with introducing a new approach to collapsing words, which is shown to be much more effective in solving various problems in the area. It leads to interesting connections with combinatorial problems concerning solving systems of permutation conditions on one hand, and coloring trees with distinguished nodes on the other hand.
Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q, Σ, δ , the in... more Given a word w over a finite alphabet Σ and a finite deterministic automaton A = Q, Σ, δ , the inequality |δ(Q, w)| ≤ |Q| − n means that under the natural action of the word w the image of the state set Q is reduced by at least n states. The word w is n-collapsing if this inequality holds for any deterministic finite automaton that satisfies such an inequality for at least one word. In this paper we present a new approach to the topic of collapsing words, and announce a few results we have obtained using this new approach. In particular, we present a direct proof of the fact that the language of n-collapsing words is recursive.
K-depth grammars extend context-free grammars allowing k 1 rewriting points for a single non-term... more K-depth grammars extend context-free grammars allowing k 1 rewriting points for a single non-terminal at every step of a derivation. The family of languages generated by k-depth grammars is a proper extension of the family of context-free languages, while retaining many context-free properties, such as closure properties, a version of Chomsky Schu tzenberger theorem, the existence of an accepting device (the multi-pushdown automaton). Here a polynomial-time parsing algorithm for k-depth languages is defined, and its correctness is proved.
The new Associative Language Description (ALD) model, a combination of locally testable and const... more The new Associative Language Description (ALD) model, a combination of locally testable and constituent structure ideas, is proposed, arguing that in practice it equals context-free (CF) grammars in explanatory adequacy, yet it provides a simple description and it excludes mathematical sets based on counting properties, which are rarely (if ever) used in compiler construction or in computational linguistics. The ALD model has been recently proposed as an approach consistent with current views on brain organization. ALD is a "pure", i.e., nonterminal-free definition. The strict inclusion of ALD languages in CF languages is proved, based on a lemma which strengthens the Pumping Lemma for CF languages. Basic nonclosure and undecidability properties are considered and compared with those of CF languages. It is shown that the hardest context-free language is in ALD, that there exists a hierarchy of ALD languages and that each ALD tree language enjoys the noncounting property of parenthesized CF languages. Typical technical languages (Pascal, HTML) can be rather conveniently described by ALD rules.
Discrete Mathematics & Theoretical Computer Science, 2007
International audience The Associative Language Description model (ALD) is a combination of local... more International audience The Associative Language Description model (ALD) is a combination of locally testable and constituent structure ideas. It is consistent with current views on brain organization and can rather conveniently describe typical technical languages such as Pascal or HTML. ALD languages are strictly enclosed in context-free languages but in practice the ALD model equals CF grammars in explanatory adequacy. Various properties of ALD have been investigated, but many theoretical questions are still open. For instance, it is unknown, at the present, whether the ALD family includes the regular languages. Here it is proved that several known classes of regular languages are ALD: threshold locally testable languages, group languages, positive commutative languages and commutative languages on 2-letter alphabets. Moreover, we show that there is an ALD language in each level of (restricted) star height hierarchy. These results seem to show that ALD languages are well-distribut...
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Papers by A. Cherubini