H. 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

....every record has a field f . Next we note that the familiar term notation can still be used in CFT if a little syntactic sugar is provided. For instance, the equational constraint X = point(Y; Z) employing the binary constructor point translates into the conjunction X: point Xf1; 2g X[1]Y X[2]Z: Note that constructors and features are dual in the sense that features are argument selectors for constructors. CFT can also express constructors that identify their arguments by keywords rather than by position. For instance, the equation P = point(xval: X; yval: Y; color: Z) can be taken ....

....= f(y; z) Feature descriptions have a long and winded history. One root are the unification grammar formalisms FUG [14] and LFG [13] developed for applications in computational linguistics (see [8] for a more recent paper in this area) Another, independent root is Ait Kaci s term calculus [1, 2], which is the basis of several constraint programming languages [3, 4, 5] Smolka [20] gives a unified logical view of most earlier feature formalisms and studies an expressive feature constraint logic. Feature trees appeared only recently with the work on FT [7, 6] To our knowledge the notion ....

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

....but has a different signature and constraint language [12] 8 , which results in slightly different expressive power. It can be viewed as the restriction of domains over which LOGIN [7] and LIFE [9] compute when all sorts are disjoint. The close relationship between the constraints and terms [5] is emphasized by a syntactic sugaring of the constraints. Example 2.4. Let Sigma = f=g [ S [ F where S is a set of unary predicate symbols (sorts) and F is a set of binary predicate symbols (features) Let D be the set of (finite or infinite) trees where: each node of each tree is labelled by a ....

H. Ait-Kaci, An Algebraic Semantics Approach to the Effective Resolution of Type Equations, Theoretical Computer Science 45, 293--351, 1986.

.... a kind of is a relation) For example, apple v fruit parallelogram v quadrangle Given partial order between the basic object terms, it is extended between complex object terms as usual: apple[color = green] v apple: As the construction of a lattice from a partially ordered set, like that in [1], is well known, we can assume that a set of object terms with top and bot constitutes a lattice, without losing generality. The meet and join operations of o 1 and o 2 are denoted by o 1 # o 2 and o 1 o 2 , respectively. 3.2 Subsumption Constraints A property of an object is represented as a ....

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

....2 In [21, Sec. 3.1] the term open specification is used in a different way, in a discussion contrasting ALE to Troll [12] We see this terminological overloading as a non issue, provided authors are careful to indicate which definitions are in force in their writings. 3 In his work A it Kaci [1], 2] has used a crown construction to complete hierarchies which may not have all glb s. However, this technique always associates a unique completion with a specification. Thus, we regard it as a means of extracting a total specification from an incomplete specification, rather than one of open ....

H. Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoret. Comput. Sci., 45:293--351, 1986.

....reported here has been done while the author was with IBM Deutschland. The author s article [23] is a more recent work on feature logics. 1 1 Introduction This paper presents a set description logic that generalizes and integrates formalisms that have been developed for knowledge representation [1, 2] and computational linguistics [13, 21, 11, 9] The logic comes with an open world model theoretic semantics, where admissible worlds can be required to satisfy a classification scheme postulated by means of a sort lattice. The logic supports the typically partial description of objects using ....

....The logic is based on the notion of a signature, which specifies which interpretations are admissible. A signature postulates a set of feature symbols and a lattice of sort symbols, where is the least and is the greatest sort symbol. Figure 1 shows an example of a sort lattice taken form [2]. An interpretation of a signature assigns to every sort symbol a set, where the denotation of is called the universe of the interpretation, and to every feature symbol a partial unary function from the universe to the universe. The denotations of the sort symbols must satisfy the following ....

[Article contains additional citation context not shown here]

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

.... descriptions are used as the main data structure of so called unification grammars, which are currently a popular family of declarative formalisms for processing natural language (Shieber (1986) More recently, feature descriptions have been proposed as a constraint system for logic programming (Ait Kaci Nasr (1986), Ait Kaci et al. 1987) Ait Kaci Podelski (1991) Ait Kaci et al. 1992) Smolka Treinen (1992) They provide for a partial description of abstract objects by means of functional attributes called features. As an example consider the feature description (in matrix notation) x : 9y 2 6 6 6 ....

....age : y husband : painter age : y 3 7 7 7 7 7 5 ; which may be read as saying that x is a woman whose father is an engineer, whose husband is a painter and whose father and husband are both of the same age. Feature description have been proposed in various forms with various formalizations (Ait Kaci (1986), Kasper Rounds (1986) Rounds Kasper (1986) Johnson (1988) y This work was supported by a research grant, ITW 9002 0, from the German Bundesministerium fur Forschung und Technologie to the DFKI project DISCO 0747 7171 94 050421 35 08.00 0 c fl 1994 Academic Press Limited 422 R. ....

Ait-Kaci, H. (1986). An algebraic semantics approach to the effective resolution of type equations.

....set inclusion. A subtype is considered to be more informative than its supertype (we assume a subtype relation definition based on type specifications similar to the given in [12] Let T be a set of types with a subtype ordering relation : An extensional type semantics is an order homomorphism [3]: h : T ; PV; In particular: h( V; h( fg and for all T; S in T S T ) h(S) h(T ) These considerations give us a general hierarchy for placement of type specifications. The guiding intuition behind the hierarchy is that type specifications are partial descriptions of ....

....was one of the first who introduced a syntactic calculus of recordlike structures based on a type subsumption ordering forming a lattice structure. Structural properties of types were taken into account. Solving of systems of type equations by iterated rewriting of type symbols became possible [3]. A.Ohori continued investigation of structural type systems (including labeled records, labeled disjoint unions, finite sets and recursion) as a proper generalization of the relational data model. Tractable type inference has been emphasized [15] R.J.Peters introduced an object model with meet, ....

Ait-Kaci H. An algebraic semantic approach to the effective resolution of types equations. Theoretical computer science, 45, 1986, 293 - 351

....in untyped systems. In automatic deduction systems, this has been found to provide a significant run time gain due to the fact that useless branches in the search space can be efficiently detected and pruned before the creation of expensive structural copies or binding frames (Walther 1985, 1988; Ait Kaci and Nasr 1986). In a constraint based linguistic theory such as HPSG, parsing and generation reduces to solving constraints. We allow constraints to be expressed by a feature logic analogue of definite clauses. The benefit of this approach is that it admits a natural and effective method paralleling ....

....list. 6 Implementation The typed system described here has been implemented in both Lisp (Franz 1990) and Prolog. Emele and Zajac (personal communication) report that Franz s (1990) grammar has been ported, with a 100 fold speedup, to their TFS system (1990) which was originally based on Ait Kaci (1984, 1986). We anticipate that a number of the optimizations employed in TFS will carry over to the system described here. In Franz s system, compilation is first carried out on the type scheme and constraints to detect errors and compute minimal satisfiers. A serious processing bottleneck can be traced to ....

Ait-Kaci, H. (1986). An algebraic semantics approach to the effective resolution of type equations.

....used to represent sets of objects by attribute value combinations, and the semantic analysis of natural language [25, 27, 49] In programming languages, attribute value combinations are used in record structures. A t Kaci was the first to study such structures mathematically, calling them # terms [2]. The resulting # term calculus is the formal foundation of the programming languages LOGIN [3] and LIFE [4] which are similar to PROLOG, but use feature unification [51] instead of syntactic unification. In contrast to several description logics, attributes in # terms are functional : they can ....

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

....represents higher order and object oriented concepts directly, both syntactically and semantically. This work builds upon our previous papers, 58, 55, 60] which in turn borrowed several important ideas from Maier s O logic [73] that, in its turn, was inspired by Ait Kaci s work on terms [7, 6]) In [58, 60] we described a logic that adequately covered the structural aspect of complex objects but was short of capturing methods, types, and inheritance. The earlier version of F logic reported in [55] 2 A PERSPECTIVE ON OBJECT ORIENTED VS. DECLARATIVE PROGRAMMING 2 was a step towards a ....

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

....of a clean exposition. The paper does not present the full syntax of TM, but instead presents significant examples to illustrate the main features of the model. The language TM is not unique in that it is based on a formal theory; other notable examples are IQL [AbKa89] Iris [LyVi87] LIFE [Ait K86], Logres [CCCT90] Machiavelli [OhBB89,Ohor90] and F logic [KiLa89] However, we know of no other language that is formally capable of dealing with arbitrary set expressions and (their) powertypes (or set types) in the context of subtyping and multiple inheritance. In fact, TM is a syntactically ....

H. Ait-Kaci, "An Algebraic Semantics Approach to the Effective Resolution of Type Equations," Theoretical Computer Science 45 (1986), 293--351.

....m 2 ) returns a most general unifier for m 1 and m 2 in A under N . Otherwise, Unify(A; N; m 2 ; m 2 ) fails. We shall not spell out the details of the algorithm here, nor shall we prove the above theorem, for there are several similar algorithms and proofs in the literature, e.g. algorithms by Ait Kaci(1986), R emy(1989) and Aponte(1992) The following outline of an algorithm is based on the Commentary of Standard ML (Milner and Tofte, 1991) which we refer to as the Commentary in what follows. Unify first builds the smallest equivalence relation j on names(A) satisfying that (a) m 1 j m 2 and (b) ....

Ait--Kaci, H. 1986. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45 (3): pp. 293--351.

....of constructors. 3.2 Term Tuple Representation with Type Hierarchy [3;4] Data representation in tuple representation has been done as research not only in the database area, but also in knowledge representation language and feature structures in natural language processing. Ait Kaci [3] proposed terms, which is considered as precise formalism and is summarized as follows: 1) Types are represented as tuples called terms. The construction is aimed at overcoming the weak points of first order terms, already mentioned. Consider the following example (a term for person ....

....as student per, employee per, and grad student student, is used to inherit attributes of upper types. in the example of (1) shows top of this lattice. The join operation in the lattice, corresponding to unification in first order terms, results in inheritance of attributes. Furthermore, [3] continues the formulation as follows: 3) Algorithm of join operation unification between terms is provided. 4) terms are extended by introducing set notation called ffl terms, which is used only as abbreviation of a set of terms. 5) A new programming language called KBL is defined as ....

H. Ait-Kaci, "An Algebraic Semantics Approach to the Effective Resolution of Type Equations", Theor. Comput. Sci , vol.45, pp.293-351, 1986.

.... 2) c) The disjoint sets of (b) after performing union(1; 7) followed by union(4; 1) d) The disjoint sets of (c) after performing union(4; 5) ometry problems [31, 45, 46] testing equivalence of finite state machines [3, 26] string algorithms [4, 29] logic programming and theorem proving [6, 7, 28, 63], and several combinatorial problems such as solving dynamic edge and vertex connectivity problems [66] computing least common ancestors in trees [2] solving off line minimum problems [20, 27] finding dominators in graphs [53] and checking flow graph reducibility [52] The focus of the ....

H. Ait-Kaci, "An algebraic semantics approach to the effective resolution of type equations", Theoret. Comput. Sci. 45 (1986).

....constructs of these two paradigms. The LIFE language [3] subscribes to the logic programming, functional programming, and object oriented programming paradigms. The language design of LIFE is most similar to the philosophy of our approach. The central component of LIFE is a calculus of terms [1, 2], an extension of first order terms (Herbrand terms and terms) which provides structural convenience for representing record like data structures. The calculus defines a clear notion of subsumption interpretable as a subtype ordering, together with an efficient unification operation ....

....logic part of OLI for free. Third, objects are inherently typed. It is interesting to investigate if typing can be introduced into the logic part of OLI , resulting in an inheritance scheme and generic mechanism for predicates. For this, we need to generalize the notion of o terms. The calculus [1, 2] would be a good starting point. Fourth, more work is needed for an efficient implementation of the OLI language. A programming environment similar to that of SMALLTALK should also be investigated. Acknowledgement This research was supported in part by a UK HK Joint Research Scheme grant ....

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

....of the results reported in [29, 30] The LIFE language [31] subscribes to the logic programming, functional programming, and object oriented programming paradigms. The language design of LIFE is most similar to the philosophy of our approach. The central component of LIFE is a calculus of terms [32, 33], an extension of first order terms (Herbrand terms and terms) which provides structural convenience for representing record like data structures. The calculus defines a clear notion of subsumption interpretable as a subtype ordering, together with an efficient unification operation ....

....to investigate if typing can be introduced into the logic part of OLI , resulting in an inheritance scheme and generic mechanism for predicates. These issues are very useful in reusing and maintaining logical predicates. For this, we need to generalize the notion of o terms. The calculus [32, 33] would be a good starting point. Fourth, a rudimentary implementation of the OLI language has been constructed [36] but more work is needed for an efficient implementation of the OLI language. A programming environment similar to that of Smalltalk should also be investigated. Fifth, a CASE tool is ....

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

.... a person who employs himself . To cope with such problems, the infinite structure should be embedded in an oid or a name. Thus, we do not take a predicate based notation as an id term in Flogic [8] but tuple based representation with tag such as X person[employ = X ] which is similar to term [3]. The third criterion concerns representation of extrinsic properties and the relation between oids. The representation of partial information should be flexible. Given an oid o, a label l, and a value v, an extrinsic property is represented as a constraint between o:l (l value of o) and v. The ....

....set grouping by combining Hoare ordering and property inheritance. 3) Set constructors are introduced into subsumption constraints and make it possible to represent complex objects. However, as intrinsic properties with sets seem to cause some semantic problems, we take an ffl term approach [3]. The fifth criterion concerns the globality and locality of data and knowledge in a database, which also relate to the construction of very large knowledge bases. To realize the coexistence of inconsistent knowledge or localization of properties or methods, we introduce a module concept as in ....

[Article contains additional citation context not shown here]

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

.... 1987] Johnson, 1988] Johnson, 1990] Dawar and Vijay Shanker, 1990] Carpenter, 1992] The UNO model shares many computational characteristics with the programming language LIFE [Ait Kaci and Richard Meyer and Peter Van Roy, 1993] because the efficiently computable calculus that underlies LIFE [Ait Kaci, 1986] is extended in the UNO model to handle negation and generalized quantifiers. Some of the linguistic theories that the UNO model encompasses and (or) extends include insights of the Montague semantics of natural language [Montague, 1973] Dowty et al. 1981] the Boolean algebra mathematical model ....

Ait-Kaci, H. (1986). An algebraic semantics approach to the effective resolution of type equations. Journal of Theoretical Computer Science, 45:293--251.

.... data structures in constraint (logic) programming (Ait Kaci et al. 1994, Smolka and Treinen 1994) and as models of feature descriptions in computational linguistics (Backofen 1994b, Backofen 1994a) The use of record like structures in logic programming languages, in the form of so called terms (Ait Kaci 1986), was pioneered by the languages LOGIN (Ait Kaci and Nasr 1986) and LIFE (Ait Kaci and Podelski 1991) More recently, Oz (Henz et al. 1995, Smolka 1995) uses a feature constraint system, the semantics of which is directly Specifying Syntactic Structures P. Blackburn and M. de Rijke, eds. ....

.... et al. 1994, Smolka and Treinen 1994) and as models of feature descriptions in computational linguistics (Backofen 1994b, Backofen 1994a) The use of record like structures in logic programming languages, in the form of so called terms (Ait Kaci 1986) was pioneered by the languages LOGIN (Ait Kaci and Nasr 1986) and LIFE (Ait Kaci and Podelski 1991) More recently, Oz (Henz et al. 1995, Smolka 1995) uses a feature constraint system, the semantics of which is directly Specifying Syntactic Structures P. Blackburn and M. de Rijke, eds. Copyright c fl 1996, CSLI Publications. 2 Ralf Treinen Gamma ....

Ait-Kaci, Hassan. 1986. An Algebraic Semantics Approach to the Effective Resolution of Type Equations. Theoretical Computer Science 45:293--351.

....with either of the constraint programming frameworks ALPS, CCP or KAP mentioned at the beginning of this section. The indepence property means among other things that negative constraints can essentially be handled through entailment simplification. One origin of FT is Ait Kaci s term calculus [1], which is at the heart of the programming language LOGIN [3] and further extended in the language LIFE [5] with functions over feature structures thanks to a generalization of the concept of residuation of Le Fun [4] Other precursors of FT are the feature descriptions found in so called ....

....constraints; for instance, the constructor constraint x : A(y; z) can be expressed equivalently as Ax xf1; 2g x1y x2z, if we assume that A is a sort and the numbers 1; 2 are features. Second, it seems attractive to extend FT such that it can accommodate a sort lattice as used in [1, 3, 4, 5, 23]. One possibility to do this is to assume a partial order on sorts and replace sort constraints Ax with quasi sort constraints [A]x whose declarative semantics is given as [A]x j BA Bx: Given the assumption that the sort ordering has greatest lower bounds if lower bounds exist, it seems that ....

H. Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer Science, 45:293--351, 1986. 21

....Vol. 16, No. 4, July, 1994, Pages 1 40. Functions as Passive Constraints in LIFE Delta 3 [At Kaci and Nasr 1989] This simple idea turns out invaluable in practice. Here are a few benefits. Such nondeclarative heresies as the is predicate in Prolog and the freeze metapredicate in some of its extensions [Colmerauer 1982b; Naish 1986] are not needed. Functional computations are determinate and do not incur the overhead of the search strategy needed by logic programming. Higher order functions are easy to return or pass as arguments since functional variables can be bound to partially applied functions. Functions can ....

A it-Kaci, H. 1986. An algebraic semantics approach to the effective resolution of type equations. Theor. Comput. Sci. 45, 293--351.

....which is not a semi lattice fw 1 ; w k g. This is precisely the effect that the following construction achieves. To our knowledge, this construction is not conventional. It is related to what is known as completion by ideals [8] and detailed in the particular following form in [3] and [4]. In what follows, we make the assumption that the poset is finite. 1 First, we need some definitions. The restricted powerset of a poset S; is the set 2 (S) of non empty finite subsets of pairwise incomparable elements of S. Such subsets are called cochains or, more figuratively, ....

Ait-Kaci, H. An algebraic-semantic approach to the effective resolution of type equations. Theoretical Computer Science, 45 (1986) pp. 185--215.

....key of a quite unique and heretofore 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 [2, 4] 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 ....

....expressed as the term: X : person(name ) id(first ) string ; last ) S : string) spouse ) person(name ) id(last ) S ) spouse ) X ) Since it is beyond the informal scope of this paper, we shall abstain from giving a complete formal definition of term syntax. Such may be found elsewhere [4, 3]. Nevertheless, it is important to distinguish among the three kinds of symbols which participate in a term expression. Thus we assume given a signature Sigma of type constructor symbols, a set A of access function symbols (also called attribute symbols) and a set R of reference tag symbols. ....

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Ait-Kaci, H., "An Algebraic Semantics Approach to the Effective Resolution of Type Equations." Journal of Theoretical Computer Science 45, pp. 293--351. 1986.

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H. Ait-Kaci, An Algebraic Semantics Approach to the Effective Resolution of Type Equations, Theoretical Computer Science 45, 293--351, 1986.

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H. Ait-Kaci. An algebraic semantics approach to the effective resolution of type equations. Theoretical Computer 11 In addition, it can be shown ([?]) that, in general, the problem of marker integration is undecidable. 12 in fact, a kind of classifying topos -- for the reader familiar with such things Science, 45:293--351, 1986.

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