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21
Dominance Constraints in Context Unification
, 1998
"... Tree descriptions based on dominance constraints are popular in several areas of computational linguistics including syntax, semantics, and discourse. Tree descriptions in the language of context unification have attracted some interest in unification and rewriting theory. Recently, dominance constr ..."
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Cited by 14 (10 self)
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Tree descriptions based on dominance constraints are popular in several areas of computational linguistics including syntax, semantics, and discourse. Tree descriptions in the language of context unification have attracted some interest in unification and rewriting theory. Recently, dominance constraints and context unification have both been used in different underspecified approaches to the semantics of scope, parallelism, and their interaction. This raises the question whether both description languages are related. In this paper, we show for a first time that dominance constraints can be expressed in context unification. We also prove that dominance constraints extended with parallelism constraints are equal in expressive power to context unification.
Linear SecondOrder Unification and Context Unification with TreeRegular Constraints
 Proc. of the 11th Int. Conference on Rewriting Techniques and Applications (RTA’2000), volume 1833 of LNCS
, 2000
"... Linear SecondOrder Unification and Context Unification are closely related problems. However, their equivalence was never formally proved. Context unification is a restriction of linear secondorder unification. Here we prove that linear secondorder unification can be reduced to context unificatio ..."
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Cited by 12 (3 self)
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Linear SecondOrder Unification and Context Unification are closely related problems. However, their equivalence was never formally proved. Context unification is a restriction of linear secondorder unification. Here we prove that linear secondorder unification can be reduced to context unification with treeregular constraints. Decidability of context unification is still an open question. We comment on the possibility that linear secondorder unification is decidable, if context unification is, and how to get rid of the treeregular constraints. This is done by reducing rankbound treeregular constraints to wordregular constraints.
Calculating ChurchRosser Proofs in Kleene Algebra
 Relational Methods in Computer Science, 6th International Conference, volume 2561 of LNCS
, 2002
"... We prove ChurchRosser theorems for nonsymmetric transitive relations, quasiorderings and equations in Kleene algebra. Proofs are simple, rigorous and general, using solely algebraic properties of the regular operations. They are fixed pointbased, inductionfree and often amenable to automata. The ..."
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Cited by 9 (4 self)
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We prove ChurchRosser theorems for nonsymmetric transitive relations, quasiorderings and equations in Kleene algebra. Proofs are simple, rigorous and general, using solely algebraic properties of the regular operations. They are fixed pointbased, inductionfree and often amenable to automata. They are mere calculations as opposed to deduction and in particular suited to automation. In the ChurchRosser proofs for the calculus, the term and algebra part are cleanly separated. In all our considerations, Kleene algebra is an excellent means of abstraction.
Context unification and traversal equations
 In: Proc. of the 12th International Conference on Rewriting Techniques and Applications (RTA’01
, 2001
"... Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing firstorder variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are secon ..."
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Cited by 8 (7 self)
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Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing firstorder variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are secondorder variables that are restricted to be instantiated by linear terms (a linear term is a λexpression λx1 ···λxn.t where every xi occurs exactly once in t). In this paper, we prove that, if the so called rankbound conjecture is true, then the context unification problem is decidable. This is done reducing context unification to solvability of traversal equations (a kind of word unification modulo certain permutations) and then, reducing traversal equations to word equations with regular constraints. 1
On Unification Problems in Restricted SecondOrder Languages
 In Annual Conf. of the European Ass. of Computer Science Logic (CSL98
, 1998
"... We review known results and improve known boundaries between the decidable and the undecidable cases of secondorder unification with various restrictions on secondorder variables. As a key tool we prove an undecidability result that provides a partial solution to an open problem about simultaneous ..."
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Cited by 6 (3 self)
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We review known results and improve known boundaries between the decidable and the undecidable cases of secondorder unification with various restrictions on secondorder variables. As a key tool we prove an undecidability result that provides a partial solution to an open problem about simultaneous rigid Eunification.
Towards Specifying with Inclusions
, 1997
"... In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalizatio ..."
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Cited by 5 (2 self)
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In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalization of the conventional algebraic specification by means of equations. The main aim of this generalization is to facilitate the incremental refinement of specifications. Furthermore, inclusional specifications admit a natural visual syntax which can also be used to visualize the reasoning process. We show that reasoning with inclusions is well captured by birewriting, a rewriting technique introduced by Levy and Agust'i [15]. However, there are still key problems to be solved in order to have executable inclusional specifications, necessary for rapid prototyping purposes. The article mainly points to the potentialities and difficulties of specifying with inclusions.
Deriving Focused Calculi For Transitive Relations
 Rewriting Techniques and Applications, 12th International Conference, volume 2051 of LNCS
, 2001
"... We propose a new method for deriving focused ordered resolution calculi, exemplified by chaining calculi for transitive relations. Previously, inference rules were postulated and a posteriori verified in semantic completeness proofs. We derive them from the theory axioms. Completeness of our calculi ..."
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Cited by 4 (4 self)
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We propose a new method for deriving focused ordered resolution calculi, exemplified by chaining calculi for transitive relations. Previously, inference rules were postulated and a posteriori verified in semantic completeness proofs. We derive them from the theory axioms. Completeness of our calculi then follows from correctness of this synthesis. Our method clearly separates deductive and procedural aspects: relating ordered chaining to KnuthBendix completion for transitive relations provides the semantic background that drives the synthesis towards its goal. This yields a more restrictive and transparent chaining calculus. The method also supports the development of approximate focused calculi and a modular approach to theory hierarchies.
Query Answering by Means of Diagram Transformation
, 1998
"... In previous work we presented a diagrammatic syntax for logic programming which clearly `resembles' the semantics of predicates as relations, i.e. sets of tuples in the Universe of Discourse. This paper shows diagrams as an alternative formal notation for pure logic programming which not only emphas ..."
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Cited by 3 (1 self)
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In previous work we presented a diagrammatic syntax for logic programming which clearly `resembles' the semantics of predicates as relations, i.e. sets of tuples in the Universe of Discourse. This paper shows diagrams as an alternative formal notation for pure logic programming which not only emphasizes some structural features of logical statements, but could also be useful to conduct visual inferences and to communicate them. This paper describes the current state of our research on a visual inference system for answering visually posed queries by means of diagram transformations. Although the transformations are shown by example we point to their correctness and formal character.
KnuthBendix Completion for NonSymmetric Transitive Relations
 Second International Workshop on RuleBased Programming (RULE2001), volume 59 of Electronic Notes in Theoretical Computer Science
, 2001
"... We extend the KnuthBendix completion procedure from equational rewriting to rewriting with nonsymmetric transitive relations and quasiorderings. The main dierences are the following: Specication of the general nonground case seems beyond rstorder logic. It is within rstorder logic when ter ..."
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Cited by 3 (3 self)
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We extend the KnuthBendix completion procedure from equational rewriting to rewriting with nonsymmetric transitive relations and quasiorderings. The main dierences are the following: Specication of the general nonground case seems beyond rstorder logic. It is within rstorder logic when terms are linear or functions nonmonotonic. The procedure requires criticalpair computations and need not terminate even in the ground case. Simplication is not don't care nondeterministic, but search based. Applications include ordered resolution and ordered chaining calculi, development of rulebased declarative procedures and algorithms, program and reachability analysis (in rewriting logic) and propagation of inequality constraints. Finally the paper also contains a brief introduction to nonsymmetric rewriting. 1
From Queries to Answers in Visual Logic Programming
 In 13th Annual IEEE Symposium on Visual Languages. IEEE Computer
, 1997
"... In VL'96 we presented a visual declarative programming language based on two main graphical constructs: directed acyclic graphs representing predicate application and graphical set inclusion representing logical implication. We showed that with these simple visual constructs we can cover most of the ..."
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Cited by 3 (1 self)
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In VL'96 we presented a visual declarative programming language based on two main graphical constructs: directed acyclic graphs representing predicate application and graphical set inclusion representing logical implication. We showed that with these simple visual constructs we can cover most of the representational demands of computational logic allowing a blend of functional and relational styles of programming. In this paper we explore the advantages of using directly our visual syntax for solving queries, by giving a way to visually ask questions about a visual program by means of query diagrams, and by defining visual inferences which operate on those diagrams. The result is an operational semantics for declarative programming which is intended to be visual, intuitive and formal. Visual because the inference rules display graphically the transformation of query diagrams into answer diagrams. Intuitive because it is intimately linked with the visual syntax of the declarative langua...