Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....universe. Since trees have been useful, e.g. for structuring data in modern symbolic programming languages like Prolog and ML, this gives the more flexible feature trees an interesting potential. Precisely, feature trees model record structures. They form the semantics of record calculi like [AK86], which are used in symbolic programming languages [AKP91b] and in computational linguistics (cf. the book [Car92] In the logical framework for record structures of [BS92] they constitute the interpretation of a completely axiomatizable, and hence decidable, first order theory. partially ....

....by different ones. Thus, symbolic keywords called features denote the possible argument positions of a node. They access uniquely the node s direct subtrees. All constructor symbols can label a node with any features attached to it, in any, though finite, number. Although thoroughly investigated [AK86, Smo92, BS92, AKPS92], also in comparison with first order trees [ST92] feature trees have never been characterized as composable elements in an algebraic structure, i.e. with operations defined on them. Also, up to now, there has been no corresponding notion of automata. This device has generally proven useful for ....

Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....Feature logic is a formalism for describing record structures, which in turn represent objects such as addresses or lexical entries by the values of their attributes. Feature logic has its origin in the three areas of knowledge representation with concept descriptions, frames, or y terms [13, 34, 35, 1], natural language processing, in particular approaches based on unification grammars [26, 24, 45, 43, 39, 42] and constraint (logic) programming [3, 5, 28, 47] An interesting recent application lies in software configuration management, where feature logic is used to denote software versions ....

.... programming [3, 5, 28, 47] An interesting recent application lies in software configuration management, where feature logic is used to denote software versions and to deduce their mutual consistency [50, 51] The first mathematical treatment of record descriptions was the formalisms of y terms [1]. In other approaches, y terms were called feature structures [40] or feature terms [46] In contrast to earlier work, the notion feature structure was mostly used for designating a record structure itself [14, 39, 42] rather than a record description. Logical descriptions of record structures ....

Hassan At-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....lower level analyses. This problem has been cronpounded by a further lack of detail at the lower, syntagmatte constituency sirarran iu SFG. In contrast Also on indefiaite leave froth the Pentnan Project, USC Information Sciences Institute, Marina del Rey, Los Angeles. Supported by BMFT Grant No. 08 B3116 3 to the generation perspective, work oriented towards analysis particularly within current informationbased grammars such as LFG and HPSG has paid extensive attention to the less abstract strata of the finguistic system and have produced highly detailed accomits of sytttagmatic ....

....architecture within which the SFG of English we are working witIt is embedded decotnposes the mapping froin abstract sethantics to starface string as follows. Nearest the surface titere are ealization statements of syntagmatic orgauization, or syntactic form. These are ACTS DE COLING 92, NAN H:S, 23 28 AO0r 1992 9 1 6 PROC. OF COLING 92, NAIWrE , AUG. 23 28, 1992 classified in terms of a grammatical system network that denotes the paradigmatic, functional alternatives offered by syntactic forths. The decisions in the grammatical systems network are motivated by somaattic distinctious that ....

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Hassan Ai't-Kaci. An Algebraic Semantics Approach to the Effective Resolution of Type Equations. Theoretical Computer Science dS, 29351.

.... Feature trees have been introduced as record like data structures in constraint (logic) programming [4] 28] and as models of feature descriptions in computational linguistics [7] 6] The use of record like structures in logic programming languages, in the form of socalled terms [1], was pioneered by the languages LOGIN [2] and LIFE [3] More recently, Oz [17, 26] uses a feature constraint system, the semantics of which is directly based on feature trees. In computational linguistics, feature structures have a long history in the field of unification grammars (as described ....

Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....consists of a typc symbol from a lattice and a set of fcaturc valne pairs. A TFS cau be seen a generalized concept o[ both atomic and complex FSs. Pollard and Sag[18] intro duce sorts into HPSG (Ilead driven Phr;se StrllcLtn e Grammar) and use sorted FSs to describe linguistic objects. A it Kaci[1] proposes an algebraic framewot k using tile C types and [ypes, one of promishg lt maliza tions of TFSs, based on lattice theory. This brmal ization was originally aimed at formalizing and intograting various kinds of knowledgc reprcseut, at. ion frameworks in AI. In this approach, types are ....

....its syntax is called ifil analhen(c( 4ypc. From ;tugmonied t.crms, an augmented Next, augmented ( terms and J types are defined. representing inhibited features and disagreement of feature address values. Then, type symbol lattices are extended to inclade complement type symbols as suggested in [1]. ACRES DE COLING 92, N NTES, 23 28 o( r 1992 3 8 1 l Roc. OF COLING 92, NANTES, AUG. 23 28, 1992 3.1 Typed Feature Structures as Augmented lb Types In order to define complex term structures, a signature is used to specify their vocabulary. It serves as the interface between their syntax and ....

Hassan AYt-Kaci. An algebraic semantics approach to the effective resolution of type equations. Journal of Theoretical Computer Science, 45:293-351, 1986.

....complexity, second order monadic logic. 1 Introduction Feature logic is a formalism to describe objects by the values of their attributes or features. It has its roots in the three areas of knowledge representation, with concept descriptions, frames, or y terms [Brachman Levesque, 1984, At Kaci, 1986, Nebel, 1990, Nebel Smolka, 1990] natural language processing, especially approaches based on unification grammars [Kay, 1979, Kaplan Bresnan, 1982, Shieber et al. 1983, Shieber, 1986, Pollard Sag, 1994, Rounds, 1997] and constraint programming languages with record structures [At Kaci ....

.... 1986, Nebel, 1990, Nebel Smolka, 1990] natural language processing, especially approaches based on unification grammars [Kay, 1979, Kaplan Bresnan, 1982, Shieber et al. 1983, Shieber, 1986, Pollard Sag, 1994, Rounds, 1997] and constraint programming languages with record structures [At Kaci Nasr, 1986, Mukai, 1988, At Kaci Podelski, 1993, Smolka, 1995] Two main approaches to feature logics can be distinguished according to the underlying logical structures. Both approaches rely on quite similar syntactic constructions called feature constraints but differ significantly in their semantics. ....

Hassan At-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

.... and Schmolze, 1985) KRYPTON (Brachman, Gilbert and Levesque, 1985) BACK (Nebel, 1990) LOOM (MacGregor, 1991) CLASSIC (Borgida, Brachman, McGuiness and Resnick, 1989) and KRIS (Baader and Hollunder, 1991) Deep theoretical foundations have been developed for such technologies in recent years (At Kaci, 1986; Nebel, 1990) The availability of these foundations makes it possible to develop general purpose knowledge representation services that are wellprincipled, space and time efficient, and embeddable as sub systems in a wide variety of applications. Since complexity analyses show inference in ....

.... been shown to provide adequate foundations for set theory and arithmetic (Tarski and Givant, 1987) and has been used to model description logics (Brink and Schmidt, 1992) A number of precise characterizations of algebraizable logics have been developed (Blok and Pigozzi, 1989) In the mid 1980s At Kaci (1984, 1986) gave a lattice theoretic model of knowledge base languages with operational semantics through term rewriting that resolved many of the issues of complexity and deduction algorithms for term subsumption knowledge representation systems. This y calculus is particularly interesting because it ....

At-Kaci, H. (1986). An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science 45 293-351.

....are used as addresses of arguments, and the atomic symbols are subject to a subsort ordering. In the earlier literature, various versions of feature type systems have been provided with fixed point semantics and the corresponding operational semantics in the shape of rewriting rules (cf. Ait Kaci [1], Ait Kaci Nasr [2] Emele Zajac [6] Carpenter et al. 3] Here, we present alternatively a model theoretic semantic and a resolution calculus along the lines of Horn logic, as an elaboration of work done by Smolka [15] and Dorre Eisele [4] On the one hand, we expect that this representation ....

....ffl logical connective: r(x[e 1 e 2 ] r(x[e 1 ] r(x[e 1 ] 1 The rank of a set of feature terms is the sum of the ranks of the individual terms. The following example is the feature clause counterpart of the usual Prologdefinition of the append predicate which concatenates two lists (cf. [1]) Example 1 Some feature clauses append2 append1 arity(nil) fg arity(nelist) ffirst; restg arity(append1) ffront; back; resultg arity(append2) arity(append1) fgoal1g list Gamma x 0 [nil] Delta list x 0 nelist rest : x 2 [list] append 0 B B x 0 2 6 6 4 ....

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Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....normal form , are only performed when they are no longer unavoidable. It is shown that the calculus is sound and complete with respect to the so called feature structure interpretations of feature terms. 1 Introduction Both in general knowledge representation (e.g. Ait Kaci Nasr s LOGIN [2] [1]) and in computational linguistics (e.g. Pollard Sag [10] the data structures for representing the terminological knowledge are usually more complex than first order terms. The main differences are the following 1. the argument positions are labeled with feature or attribute names 2. the number ....

....Example 1. Feature term x 0 2 6 6 6 6 6 6 6 6 6 6 4 person sex : female male) mother : x 1 person sex : female father : x 2 person sex : male parent : x 1 x 2 ) roof shape : 3 7 7 7 7 7 7 7 7 7 7 5 This example looks very similar to an example given in Ait Kaci [1] but here it is not meant as a (recursive) definition of a person data type. The symbol person is just a pointer into the sort lattice, not to a piece of code. Definition 3. An expression e is normal if it is one of the following (a 2 S) a :a :x f : x : f : x) f : A set T of feature terms ....

Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....A, logical variables V , and types T . In the following, we refer to a type hierarchy I by a pair hT ; i, such that T Theta T is a decidable partial order, i.e. is reflexive, antisymmetric, and transitive. A typed feature structure (TFS) is essentially either a term or an ffl term [ Ait Kaci 1986 ] i.e. hx; Phii j hx; Thetai such that x 2 V , 2 T , Phi = ff 1 : 1 ; fn : n g, and Theta = f 1 ; n g, where each f i 2 F and i is again a TFS. We will call the equation f : a feature constraint (or an attribute value pair) 3 Phi is ....

....over a certain (downward) continuous function. 5 The first approach is in general closer to an implementation (and thus to our algorithm) in that the function which is involved in the fixpoint construction corresponds more or less to the unification substitution of TFS (see for instance [ Ait Kaci 1986 ] or [ Pollard and Moshier 1990 ] The latter approach is based on the assumption that TFS are only syntactic sugar for first order formulae. If we transform these descriptions into an equivalent set of definite clauses, we can employ techniques that are fairly common in logic programming, viz. ....

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Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....Congruence relation o 1 = o 2 is defined as o 1 v o 2 o 1 w o 2 . We assume that a set of object terms with the subsumption relation and special objects, and , constitutes a lattice without loss of generality because it is easy to construct a lattice from a partially ordered set, as in [Ait Kaci] Meet and join operations of object terms are denoted by # and , respectively. Properties are defined as a set of subsumption constraints of an oid and used with the oid as follows: applejf apple:species v rose; apple:area wH faomori; naganogg, where apple is an object term and the right ....

Hassan Ait-Kaci, "An Algebraic Semantics Approach to the Effective Resolution of Type Equations", Theoretical Computer Science, no.45, 1986.

....this setting, there is no reason to restrict to equations of the form f = id X because an equation of the form f = g also has an equalizer when f; g are monotone, and this again turns out to be the poset of all one point solutions, including the least one point solution if there is one. Ait Kaci [2] gives a fixpoint based approach to solving what he calls type equations for the semantics of a language called KBL. However, this approach seems a bit awkward since infinitary structures are not really needed, and it has been subsumed by some elegant later work of Smolka and Ait Kaci [46] ....

Hassan A it-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....in two essential ways: ffl using a data structure richer than that provided by rst order constructor terms; and, ffl allowing interpretable functional expressions as bona de terms. The rst extension is based on terms which are attributed partially ordered sorts denoting sets of objects [1, 2]. In particular, terms generalize rst order constructor terms in their role as data structures in that they are endowed with a unication operation denoting type intersection. The second extension deals with building into the unication operation a means to reduce functional expressions using ....

....two terms with variables renamed apart; i.e. such that Var( 1 ) Var( 2 ) Let X 1 and X 2 be their respective root variables. Let OE be the normal form of the OSF constraint 1 2 X 1 : X 2 . Proposition 11 Term Unication. The normal form OE is the false constraint if and only if [[ 1 ]] A [ 2 ] A = for all OSF algebras A. Otherwise, OE is the conjunction of equality constraints and of the dissolved version of some term . This term is the GLB of 1 and 2 up to variable renaming; i.e. A = 1 ] A [ 2 ] A . Proposition 12 v LUB of two terms. The term ....

[Article contains additional citation context not shown here]

Hassan At-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293-351 (1986).

....A, logical variables V, and types T . In the following, we refer to a type hierarchy I by a pair hT ; i, such that T Theta T is a decidable partial order, i.e. is reflexive, antisymmetric, and transitive. A typed feature structure (TFS) is essentially either a term or an ffl term [ Ait Kaci, 1986 ] i.e. hx; Phii j hx; Thetai such that x 2 V, 2 T , Phi = ff 1 : 1 ; f n : n g, and Theta = f 1 ; n g, where each f i 2 F and i is again a TFS. We will call the equation f : a feature constraint (or an attribute value pair) 3 Phi is ....

....of a fixpoint over a certain continuous function. 4 The first approach is in general closer to an implementation (and thus to our algorithm) in that the function which is involved in the fixpoint construction corresponds more or less to the unification substitution of TFS (see for instance [ Ait Kaci, 1986 ] or [ Pollard and Moshier, 1990 ] The latter approach is based on the assumption that TFS 4 In both cases, there is, in general, more than one fixpoint, but it seems desirable to choose the greatest one. are only syntactic sugar for first order formulae. If we transform these descriptions ....

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Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....T , which assigns a type to each address, and a tag function which associates a variable with each address. A feature term is called inconsistent iff its denotation is the empty set in all interpretations. Feature terms come with a set theoretic semantics which is described in detail in Ait Kaci [2] 2 . Feature terms can be understood as expressions of an attributive representation language that is basically an instance of feature logic. Features are interpreted as partial functions whereas in languages of the KL ONE family they generalize to roles. It has been shown (cf. 14] that this ....

Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....in two essential ways: ffl using a data structure richer than that provided by first order constructor terms; and, ffl allowing interpretable functional expressions as bona fide terms. The first extension is based on terms which are attributed partially ordered sorts denoting sets of objects [1, 3]. In particular, terms generalize first order constructor terms in their role as data structures in that they are endowed with a unification operation denoting type intersection. This gives an elegant means to incorporate a calculus of multiple inheritance into symbolic programming. Importantly, ....

....two terms with variables renamed apart; i.e. such that Var( 1 ) Var( 2 ) Let X 1 and X 2 be their respective root variables. Let OE be the normal form of the OSF constraint 1 2 X 1 : X 2 . Proposition 3 ( Term Unification) The normal form OE is the false constraint if and only if [[ 1 ]] A [ 2 ] A = for all OSF algebras A. Otherwise, OE is the conjunction of equality constraints and of the dissolved version of some term . This term is the GLB of 1 and 2 up to variable renaming; i.e. A = 1 ] A [ 2 ] A . Proposition 4 (v LUB of two terms) The term ....

[Article contains additional citation context not shown here]

Hassan At-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351 (1986).

....descriptions in logic programming has been advocated and studied [3, 4, 5, 6, 23] Essentially, feature descriptions provide a logical version of records, a data structure found in many programming languages. Feature descriptions have been proposed in various forms with various formalizations [1, 2, 15, 20, 14, 11, 12, 22, 7, 8, 19]. We will follow the logical approach pioneered by [22] which accommodates feature descriptions as standard first order formulae interpreted in first order structures. In this approach, a semantics for feature descriptions can be given by means of a feature theory (i.e. a set of closed feature ....

Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....in two essential ways: ffl using a data structure richer than that provided by first order constructor terms; and, ffl allowing interpretable functional expressions as bona fide terms. The first extension is based on terms which are attributed partially ordered sorts denoting sets of objects [1, 3]. In particular, terms generalize first order constructor terms in their role as data structures in that they are endowed with a unification operation denoting type intersection. This gives an elegant means to incorporate a calculus of multiple inheritance into symbolic programming. Importantly, ....

....with variables renamed apart; i.e. such that Var( 1 ) Var( 2 ) Let X 1 and X 2 be their respective root variables. Let OE be the normal form of the OSF constraint 1 2 X 1 : X 2 . Proposition 3 ( Term Unification) The normal form OE is the false constraint if and only if [[ 1 ]] A [ 2 ] A = for all OSF algebras A. Otherwise, OE is the conjunction of equality constraints and of the dissolved version of some term . This term is the GLB of 1 and 2 up to variable renaming; i.e. A = 1 ] A [ 2 ] A . Proposition 4 (v LUB of two terms) The term ....

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Hassan At-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351 (1986).

....query language. This resulted in logical query languages like LDL [14] and NAIL [13] So called complex objects have recently been studied for use in database systems [7, 8] Much of what has been proposed in those studies is derived from earlier work extending first order terms to terms [1]. The latter notion has had a more direct application in programming language design [4, 2, 6] than in database systems. Still, the functionality and naturalness of deductive queries over terms is a strong motivation for providing a logic programming language using terms with an effective ....

....Similarly, f g is identified with the basic term . Again, this is natural since they are both least elements. However, the empty set is also the least element of coc(Y) and hence we can identify all three: f g = The following is a particular case of a more general result in [1]. Theorem 1 The poset hcoc(Y) i is a lattice. Proof: Greatest lower bounds are constructed as follows. For basic terms p and p 0 , the (possibly disjunctive) term p p 0 is the set of maximal elements of the set of all basic terms u = hDu ; ui such that: ffl Du = Dp [ D p 0 , ffl 8a 2 ....

Hassan At-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351 (1986).

....key of a quite unique and hitherto unheard of generative behavior of programs, which could construct missing information as needed to accommodate success. Finally, the most original part of LIFE is the structure oriented component which consists of a calculus of type structures the calculus [AK84, AK86] and rigorously accounts for some of the (multiple) inheritance convenience typically found in so called object oriented languages. An algebra of term structures adequate for the representation and formalization of frame like objects is given a clear notion of subsumption interpretable as a ....

....encoded into a formula, perspicuously expressed as the term: X : person(name ) id(first ) string; last ) S : string) spouse ) person(name ) id(last ) S) spouse ) X) We shall abstain in this summary from giving a complete formal definition of term syntax. Such may be found elsewhere [AK86, AKN86]. Nevertheless, it is important to distinguish among the three kinds of symbols which participate in a term expression. Thus we assume given a set S of type constructor symbols, a set A of feature function symbols (also called attribute symbols) and a set R of reference tag symbols. In ....

[Article contains additional citation context not shown here]

Hassan At-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986.

....in two essential ways: ffl using a data structure richer than that provided by first order constructor terms; and, ffl allowing interpretable functional expressions as bona fide terms. The first extension is based on terms which are attributed partially ordered sorts denoting sets of objects [1, 2]. In particular, terms generalize first order constructor terms in their role as data structures in that they are endowed with a unification operation denoting type intersection. The second extension deals with building into the unification operation a means to reduce functional expressions ....

Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351 (1986).

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Hassan AYt-Kaci. <<An Algebraic Semantics Approach to the Effective Resolution of Type Equations>>;. Theoretical Computer Sci- ence 45,293-351.

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Hassan AYt-Kaci. An algebraic se- mantics approach to the effective resolution of type equations. Theoretical Com. puler Science $5, pp. 293-351, 1986

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Hassan AYt-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Compuler Science, 45(3):293- 351, 1986.

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Hassan Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science 45, pp. 293--351, 1986

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