D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....satisfies several requirements, but not a single and general requirement that justifies the whole of Prolog. At the beginning, Miller and Nadathur present a logic programming language called Prolog, which features higher order Horn clauses, terms, and equivalence [40, 45] Then, Miller et al. [42, 41] introduce the connectives 8 G and )G , and keep the name Prolog. In several articles Miller 1 Similarly, a subscripted D means that a connective can be used as a constructor for definite clauses. For instance, Horn clauses are constructed with the implication connective (i.e. D ) 2 ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

.... clauses, terms, and equivalence [40, 46] Then, Miller formalizes module importation as logical implication in goals, G [35] and module abstraction as universal quantification in goals, 8 G [33] Miller et al. observe that these extensions form a well behaved fragment of intuitionistic logic [43, 42]. This fragment is called hereditary Harrop formulas, and the extended language is still called Prolog. Table 1.1 sums up the introduction of the basic concepts of Prolog in its creators writings. To ease the way of a beginner, it is tempting to define fragments of Prolog by merely dropping some ....

.... Typed Prolog def = Prolog simple types defines a strongly typed variant of Prolog as proposed by Lakshman and Reddy [27] CLP( def = Prolog terms simple types = fffi 3 1986 1987 1988 1989 1990 1991 1992 1993 Higher order LP [40] 41, 46] 47] Modules [37] 35] 36] G [37] [41, 43] [42] 8G [43] 33] 42] Decidable higher order unification [34] 38] Abstract syntax [32] Unification and quantification [39] TABLE 1.1. A bibliography map defines an instance of the scheme CLP [9] for the domain of the simply typed terms endowed with the equivalence relation = fffi [57] ....

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D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, NY, 1987. Revised version in [42].

....are combined in trees to form proofs, and one calls theorem the conclusion of the root of some proof. The success set of a program P is the set of all theorems whose antecedent is the program (i.e. theorems T such that P a T ) Among the possible proofs of a theorem, so called uniform proofs [MNS87] are preferred for their operational reading: only apply clause rules (i.e. clause , clause , 8 clause ) when no other rule applies. So, uniform proofs are almost completely directed by the goals, except for the selection of clauses. Example 3 (Sequent proof) The following is a proof of ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98-105, 1987. 37

.... in semantic interpretation may also be examined in terms of a formalization of semantic interpretation rules in a suitable higherorder logic, as has been done for inference rules of various logics by Felty and Miller [13, 14] using as rule metalanguage the higher order hereditary Harrop formulas [29] implemented in #Prolog [36, 37] Rather than repeating here the precise definitions found elsewhere [13, 29, 36] the language will be introduced by showing how the typing rules of the previous section would be expressed [13] The #Prolog program in Figure 2 consists of certain declarations ....

.... rules in a suitable higherorder logic, as has been done for inference rules of various logics by Felty and Miller [13, 14] using as rule metalanguage the higher order hereditary Harrop formulas [29] implemented in #Prolog [36, 37] Rather than repeating here the precise definitions found elsewhere [13, 29, 36], the language will be introduced by showing how the typing rules of the previous section would be expressed [13] The #Prolog program in Figure 2 consists of certain declarations followed by two clauses. The declarations specify expr and ty as the atomic types of (objectlevel) expressions and ....

D. A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In Proceedings of the Second Symposium on Logic in Computer Science, Ithaca, New York, 1987. Cornell University, IEEE.

....scheme of Jaffar and Lassez [JL87] gives a general approach for incorporating constraints in a logic programming language. This has been implemented in the language CLP(IR) JMSY92] which contains a simplex algorithm for solving constraints over the real numbers. In Prolog (see [Mil91] and [MNS87] functional and logic programming are unified using a typed calculus. Logic programs are sets of Harrop formulae which may contain terms. In fact, each formula is a term of propositional type. Constraint functional programming was also introduced by Darlington et al. in [DGP92] FALCON ....

Dale Miller, Gopalan Nadatur, and Andre Scedrov. Hereditary Harrop Formulas and Uniform Proofs Systems. In Proceedings of the Logic In Computer Science, 1987.

....program under the Negation as Failure rule, in that A fails iff G fails, and so :A succeeds iff :G succeeds. In this paper we show how a form of the completion based on this idea may be given, not just for Horn clauses, but for a larger class of formulae known as hereditary Harrop formulae [11, 10], which properly includes Horn clauses. An advantage of this approach is that the formulae added to the program by the completion process may be considered as clauses in this framework, and so the completion of a hereditary Harrop formula program may be considered as a hereditary Harrop formula ....

....the requisite definitions for hereditary Harrop formulae, and discuss our fundamental approach to the problem. Section 3 deals with the definition of our completion, and section 4 with the properties of our completion. 2 Preliminaries The class of hereditary Harrop formulae was introduced in [10], and the basic properties of this class of formulae were developed in [11] Below we give the definition of programs and goals extended to the case when negated atoms are allowed in goals. Definition 2.1 D and G formulae are given by D : A j D 1 D 2 j 8xD j G oe A G : A j :A j G 1 G 2 j G ....

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D.A. Miller, G. Nadathur and A. Scedrov, Hereditary Harrop Formulas and Uniform Proof Systems, Proceedings of the Symposium on Logic in Computer Science 98-105, Ithaca, June, 1987.

....terms manipulated through high order unification. Let us stress from the start that terms are mostly used as data structures and that Prolog is not a mixture of orthogonal functional and logic programming languages but a natural extension of Horn clause logic based on the concept of uniform proof [9]. 2 However we will not use in this case any of the quantification related features of Prolog so that the code will run with minor modifications on any Prolog system having the call N primitive introduced by [11] primitive operations maps a single element of A to an object of the monad ....

D. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

....formalism is computationally complete [9, 16] many attempts are made to extend it in order to gain more flexibility or expressiveness. One of these attempts is Prolog [10] Among the extensions to Horn clauses, it has the rare quality of preserving the connection with logic in a formal sense [11]. One of the features of Prolog that is really important in this paper is that contextual information can be conveyed through terms, as usual, as well as through program clauses, which is new. Prolog programs can be temporarily augmented by assumptions forming a context. Pereira [14] and Pareschi ....

....Grammars) that uses the features of Prolog at the rule level and at the attribute level. This last section completes the diagram. 2 Prolog and DCG 2. 1 Prolog Miller proposes an extension of both the term language and the formula language of Prolog that still has nice proof theoretic properties [10, 11]. First, the extended term language is the language of the simply typed terms. Simple types are generated by the following grammar 2 : T : U j ( C i T i ) j ( T T ) Where U (resp. C i ) are identifiers of type variables (resp. of type constructors of arity i) There is always ....

[Article contains additional citation context not shown here]

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

....focus attention on a restricted class of programs, such as the locally stratified programs [14] or some similar weak form of stratification ( 13] for example) We have no such restriction; programs such as p ae :p are dealt with by our semantics. 2 Worlds and Accessibility It has been shown in [12] how first order hereditary Harrop formulae, a fragment of first order intuitionistic logic, may be considered as a logic programming language, and that they form a generalisation of Horn clauses. An operational notion of consequence was presented, so that if P is a program and G is a goal, then P ....

D.A. Miller, G. Nadathur and A. Scedrov, Hereditary Harrop Formulas and Uniform Proof Systems, Proceedings of the Second Annual Symposium on Logic in Computer Science, 98-105, Ithaca, June, 1987.

....G hhf formulae satisfy D : A j 8xD j D 1 D 2 j G oe A G : A j 9xG j 8xG j G 1 G 2 j G 1 G 2 j D oe G In all cases, we refer to D formulae as definite formulae, and to G formulae as goal formulae. Also, a program is a set of closed definite formulae, and a goal is a closed goal formula. In [14] and [13] it is shown how an operational notion of proof may be given for all the above classes of programs and goals. Such proofs are known as uniform proofs, and may be defined as follows. We assume the existence of a finite set of constant and function symbols, and a countable set of ....

D.A. Miller, G. Nadathur and A. Scedrov, Hereditary Harrop Formulas and Uniform Proof Systems, Proceedings of the Second Annual Symposium on Logic in Computer Science, 98-105, Ithaca, June, 1987.

....between logic programming and logic 10 . Standard Prolog texts (e.g. 25] insist on model theory, but this setting does not extend well to non Horn formulas. Miller proposes to define a logic programming language as a fragment of a predicate logic that enjoys a uniform sequent proof property [28]. 1.2.2.1 What is in a logic programming language A logic programming language is defined by its legal programs (clauses) and queries (goals) D and G. This defines by induction the legal sequents (P Q, where P 2 D and Q 2 G) and the legal deduction rules and axioms. A sequent proof of a ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

....we might expect this step to be displayed as True True BY T True True In a sense, this is no longer a correct step, since there is no reason to expect T to act appropriately when its argument goal is instantiated. One could fix this by using a logic programming language such as Prolog [14, 6] as the tactic programming language, for then successful executions would be preserved under instantiation. We prefer not to do this for two reasons. First, there is a considerable body of practical evidence that ML is a good language for tactic programming. Secondly, we want to use tactics to ....

D. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In Proceedings of the Second Annual Symposium on Logic in Computer Science, pages 98--105. IEEE, 1987.

....[25] The formalism of Horn programs is computationally complete [1, 37] but one has often tried to augment it to gain more flexibility and expressivity. One of these attempts is Prolog [30] It has the quality, rare among extensions to Horn formulas, to preserve a formal connection to logic [32, 31]. Indeed, a kind of goal directed proofs (proofs that can be used as an operational semantics) is complete for the formulas of Prolog. The following equation sketchs the definition of Prolog: Prolog def = Prolog terms simple types = fffij 8G )G The components of this formula will ....

.... as the same problem modulo fffij equivalence (with j) RR n2390 4 Catherine BELLEANN EE, Pascal BRISSET et Olivier RIDOUX 2 Prolog Miller and Nadathur proposed in 1986 a generalisation of the terms and formulas of Prolog which still has interesting logical and computational properties [30, 32, 31]. It encompasses Prolog as in 2 type append (list T) list T) list T) o . append [ L L . append( L, L ) append [A L1] L2 [A L3] append L1 L2 L3 . append( A j L1] L2, A j L3] append( L1, L2, L3 ) but also makes possible programs with a really new ....

[Article contains additional citation context not shown here]

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

....nouveaut es de Prolog comme les types, la fi r eduction et la suspension de l unification. Mots cl e : Impl ementation, Programmation logique, Prolog, Gestion de m emoire, Mali The Architecture of an Implementation of Prolog: Prolog Mali 3 1 Introduction The logic programming language Prolog [30, 29, 31, 17, 15, 28, 16, 32] improves on standard Prolog because it features more powerful operations on terms and programs while still giving them a logical semantics. A keyword common to all these features is scoping. Terms introduce scoping at the term level, explicit quantifications (universal and existential) introduce ....

....issues are a good guide for implementing logic programming systems. Speed was always our second concern. We assume a knowledge of Prolog and Prolog, their semantics, and their basic algorithms: logical variable, search stack, unification, unification [20] deduction rules, and uniform proofs [34, 32]. We adopt an architectural presentation: in section 2, we present the kernel subsystem that is in charge of the elementary representation problems, in section 3, we present a software layer which is both a specialization and an extension of the kernel, finally, in section 4, we present the ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

....played by closures in the models of the polymorphic calculus and the first order cc calculus. I have recently realized that computation in cc languages can be viewed as deduction in a fragment of intuitionistic logic, though the setting is somewhat different from that in Dale Miller s work [MNS87]. There is a connection here with higher order (untyped) intuitionistic logics that should be examined carefully. Jointly with Spiro Michaylov, a prototype implementation is underway in Prolog based on extending the Krivine machine presented in [ACCL90] However, not much of the power of Prolog ....

Dale Miller, Gopalan Nandathur, and Andre Scedrov. Hereditary Harrop Formulas and Uniform Proof Systems. In Symposium on Logic in Computer Science, pages 98 --- 105. IEEE Computer Society, IEEE Computer Society Press, June 1987.

....of) the term described above. eq(x, x) sample term(t) eq(t, ff(aa, t) PI n878 8 Solange COUPET GRIMAL et Olivier RIDOUX 3. 2 Prolog Miller proposes an extension of both the term language and the formula language of Prolog that still has nice proof theoretic properties [37, 39, 38, 5]. 3.2.1 Syntax First, the extended term language is the language of the simply typed terms. Terms can be considered as functions, and simple types describe from what and to what domain they are defined. Simple types are generated by the following grammar 5 : T : U j ( K i T i ) j ( ....

....prenex variables in types. This means that they obey the theory of simple types as presented by Church [7] augmented with a generic polymorphism capability as introduced by Milner [40] see ML for another example of generic polymorphism) The semantics of Prolog is usually based on proof theory [39, 38] rather than on model theory as for Prolog [34] The main result is that a certain kind of goal directed proofs, called uniform proofs, is complete with respect to intuitionistic provability for these formulas. In other words, every time a hereditary Harrop G formula is a consequence of a ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987. Irisa ADVANCED LOGIC PROGRAMMING LANGUAGES & COMPUTATIONAL LINGUISTICS 29

....our extension of oe calculus with new set oriented rules. ffl Finally, we discuss some related works. We conclude in section 5 and we present an extension of the sterile box example in appendix. 2 Language Definition We present here an extension of the logic programming languages proposed in [18, 20] by introducing set constraints. Our paradigm slightly differs from usual Constraint Logic Programming language in that sense that here constraints are not based on a built in set of predicates related to the constraint domains but on user s predicate definitions. The built in constraint language ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary harrop formulas and uniform proof systems. In D. Gries, editor, 2nd. Symp. Logic in Computer Science, pages 98--105, Ithaca, New york, USA, 1987.

....proving and proof manipulation environments. Its range of applications therefore include the range of applications of such environments (as indicated above) The basic idea behind Elf is to unify logic definition (in the style of LF) with logic pro Elf 2 gramming (in the style of Prolog, see [22, 21, 24]) It achieves this unification by giving types an operational interpretation, much the same way that Prolog gives certain formulas (Horn clauses) an operational interpretation. Here are some of the salient characteristics of this unified approach to logic definition and metaprogramming. First of ....

Dale Miller, Gopalan Nadathur, and Andre Scedrov. Hereditary Harrop formulas and uniform proof systems. In Second Annual Symposium on Logic in Computer Science, pages 98--105, IEEE, June 1987.

....avoid the word logic in logic programming being only a catchword, one needs to define precisely what is the relation between logic programming and logic 9 . Miller proposes to define a logic programming language as a fragment of a predicate logic that enjoys a uniform sequent proof property [53, 52]. 1.2.1 What is in a logic programming language A logic programming language is defined by its legal programs (clauses) and queries (goals) D and G. This defines by induction the legal sequents (P Q, where P 2 D and Q 2 G) and the legal deduction rules and axioms. A sequent proof of a ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

....of a Closure Based Compilation Method for . the following syntax rules: G : A j G G j G G j D oe G C : A j G oe A j 8xC D : C j C D: In the rules above, A represents an atomic formula. The C formulas defined here are a subclass of first order hereditary Harrop formulas [Miller et al. 87] In the programming language to be considered, G formulas will function as queries and collections of C formulas will constitute programs. For this reason, we refer to a G formula as a goal or query, to a C formula as a clause, to a D formula as a program clause, and to a collection of clauses ....

Dale Miller, Gopalan Nadathur, and Andre Scedrov. Hereditary Harrop formulas and uniform proof systems. In David Gries, editor, Symposium on Logic in Computer Science, pages 98--105, Ithaca, NY, June 1987.

....into functions operating on several continuations. The compilation scheme is sometimes an adaptation of the standard Prolog scheme, but at other times it has to handle new features such as types, fi reduction and delayed unification. 1 Introduction The logic programming language Prolog [33, 31, 34, 18, 16, 30, 17, 35] improves on standard Prolog because it features more powerful operations on terms and programs while still giving them a logical semantics. A keyword common to all these features is scoping. Terms introduce scoping at the term level, explicit quantifications (universal and existential) introduce ....

....concern. We assume a knowledge of Prolog and its semantics and basic algorithms: logical variable, search stack, unification, and deduction rule. In section 2, we give a light introduction to the semantics and basic algorithms of Prolog: unification [21] deduction rules, and uniform proofs [37, 35]. We adopt an architectural presentation: in section 3, we present the kernel subsystem that is in charge of the elementary representation problems, in section 4, we present a software layer which is both a specialization and an extension of the kernel, finally, in section 5, we present the ....

D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, New York, USA, 1987.

....Order No. 5404, monitored by the Office of Naval Research under the same contract, and for Scedrov from NSF grants DMS85 01522 and CCR 87 05596 and from the University of Pennsylvania Natural Sciences Association Young Faculty Award. 9. References A preliminary version of this paper appeared as [21]. Theorem 3 of that paper is incorrect. It is corrected by the material in Sections 5 and 6 of the current paper. 1] K. Apt and M. H. van Emden, Contributions to the Theory of Logic Programming, Journal of the ACM 29 (1982) 841 862. 2] A. Church, A Formulation of the Simple Theory of Types, ....

D. Miller, G. Nadathur, and A. Scedrov, Hereditary Harrop Formulas and Uniform Proofs Systems, Proceedings of the Second Annual Symposium on Logic in Computer Science, Ithaca, June 1987, 98 --- 105.

.... Formulas We are interested in the G and D formulas defined by the following syntax rules in which we assume A represents atomic formulas: G : A j G G j G G j 9xG j D oe G j 8xG D : A j G oe A j D D j 8xD: The D formulas defined here are called (first order) hereditary Harrop formulas [17]. These formulas define a logic programming language in the following sense: a G formula can be thought of as a query or goal, a finite set of closed D formulas constitutes a program, and the process of answering a query consists of constructing an intuitionistic proof of the existential closure ....

....leads to the conclusion that no proof exists for the original formula. Although the scheme outlined above functions correctly, it involves keeping track of a potentially 2 It follows from this that a Herbrand like theorem does not hold for hereditary Harrop formulas, contrary to the claim in [17]. A deeper analysis reveals that the source of the problem is that, in contrast to the classical case, certain propositional inference rules in this case the oe L and oe R rules cannot be permuted in our intuitionistic sequent calculus. This observation, coincidentally using the same ....

Dale Miller, Gopalan Nadathur, and Andre Scedrov. Hereditary Harrop formulas and uniform proof systems. In David Gries, editor, Symposium on Logic in Computer Science, pages 98--105, Ithaca, NY, June 1987.

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D.A. Miller, G. Nadathur, and A. Scedrov. Hereditary Harrop formulas and uniform proof systems. In D. Gries, editor, 2nd Symp. Logic in Computer Science, pages 98--105, Ithaca, NY, 1987. Revised version in [MNPS91].

No context found.

Dale Miller, Gopalan Nadathur, and Andre Scedrov. Hereditary Harrop formulas and uniform proof systems. In Second Annual Symposium on Logic in Computer Science, pages 98--105. IEEE, June 1987.

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