G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, 1990. Typeset with Xy-pic  Home/Search   Document Details and Download   Summary   ACM   TOC   Related Articles   Check

....of proof search is a current topic of active research [15] and we can expect that advances there will find applications to this particular domain of computer science. The proof search paradigm comes equipped with notions of abstract data types [22] and higher order predicate abstractions [28], all features that we shall draw upon in this paper. Given that these abstraction mechanism all result from aspects of logic, there is no problem in understanding the interaction between abstractions and other aspects of the logic (for example, with multiset rewriting) The problems surrounding ....

G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777--814, October 1990.

.... The extension in one direction is obtained by including higher order features in the form of quanti cation over function and some occurrences of predicate variables and the replacement of rst order terms by simply typed lambda terms within Horn clauses, resulting in higher order Horn clauses [31]. Along the other direction, Horn clause logic is enhanced by permitting universal quanti ers and restricted uses of implications, resulting in a rst order version of hereditary Harrop formulas [23, 28] The combination of these two logics produces a logic that is simply typed. This typing ....

....The di erences are the use of a curried notation for atoms and the presence of typing. We adopt this kind of a syntax in the discussions below. From a logical perspective, the idea of answering a query can be explained by using provability in classical logic and this aspect is investigated in [31]. Operationally, this results in a recipe for solving a closed query from a program P that is based on the structure of the query: 1) Solve G 1 G 2 by solving both G 1 and G 2 . 2) Solve G 1 G 2 by solving one of G 1 and G 2 . 3) Solve 9xG by solving [t=x]G for some closed positive term ....

[Article contains additional citation context not shown here]

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777{ 814, October 1990.

....some higher order extension thereof in the case of higher order logic programming languages [9, 2] and so on. This rigidity can be unduly constraining. It locks the user into formulating every problem in terms of the same representation (Horn clauses, or higher order hereditary Hattop clauses [10], etc. and the same inference method, even when those are not the proper tools to use. For instance, how does one go about proving De Morgan s laws in Prolog7 How does one derive ( x) P(x) from the assumption (V z) P(z)7 Moreover, how does one write a schema that does this for any given z and P7 ....

G. Nadathur and D. Miller. Higher-order Horn Clauses. Journal of the ACM, 37(4):777 814, 1990.

....Originally the specifications were taken to be first order Horn theories, with a restricted proof search, resolution, as the computational engine. However, the demand for more expressive power and efficiency has led language designers to consider logical extensions to the original Horn Clause core [26, 48, 51, 50, 46, 28]) and to add control features and constructs drawn from other language paradigms (e.g. types [55, 32, 49] partial evaluation [31] and constraints [10, 29, 30] to name just a few) The effect has been to expand the boundaries of the subject and of the very notion of declarative content of a ....

....programming languages [28, 45, 9] and which give rise to useful operations that still agree with some notion of conventional semantics in the limiting case. 7. 2 Directions for future work This foundation opens the way to a categorical approach to modeling other extensions of logic programming [28, 51, 50, 55, 49], and connecting well known techniques for modeling functional evaluation, non local control, side effects (e.g. monads, see [62, 42, 47, 68, 69] to the declarative and operational semantics of logic programs. Our operational semantics ignores control issues (such as the selection rule) since ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. Journal of the ACM, 37(4):777-814, 1990.

....by the type theories of Martin L of. The Coq system [CH85,DFH 91] also features a similar constructive type theory with impredicative type quanti cation and abstraction. Isabelle [Pau94] is a metalogical framework that uses higher order Horn clauses to represent inference rules. Prolog [NM90] is a similar metalogical framework based on the hereditary Harrop fragment of higherorder logic. There is a general impression that higher order logics are not easily automated. We get a di erent picture if we view higher order logic as a framework for embedding small and useful sublogics that ....

....in the context of a theorem proving environment. It is also possible to embed programming vernaculars such as logic programming [Sym98] and imperative programming. Higher order logic programming with the use of higherorder syntactic encoding has emerged as an important medium for metaprogramming [NM90] A number of useful decision procedures from computer algebra, operations research, and engineering, can also be usefully incorporated into a theorem proving environment. Drawing inspiration from the work of Boyer and Moore [BM81] a metaprogramming capability can be used to re ectively develop ....

G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-814, 1990.

....some of its components. For instance, 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 ....

G. Nadathur and D.A. Miller. Higher-order Horn clauses. JACM, 37(4):777--814, 1990.

....show that, in terms of expressive power, they can capture the full second level of the polynomial hierarchy. 7. 6 Other Logics There are also successful approaches to generalize logic programming languages by using intuitionistic, linear, or higher order logics (e.g. Hodas and Miller 1994; Nadathur and Miller 1990]) These extensions seem somewhat orthogonal to our treatment of negation in the context of arbitrary propositional formulas. 22 Delta S. Brass and J. Dix and T. C. Przymusinski 8. CONCLUSION We introduced the class of super programs as a subclass of the class of all nonmonotonic knowledge ....

Nadathur, G. and Miller, D. 1990. Higher-order horn clauses. Journal of the Association for Computing Machinery (JACM) 37, 4, 777--814.

....in compilation. Novel monad structures are described for lazy function lists, clause unfoldings and a monad morphism based embedding of an Prolog in Prolog is given. In the examples that follow we will not limit ourself to Prolog but refer most of the time to its more powerful superset Prolog [7, 8, 14, 13] (in its Prolog Mali incarnation [5, 2] which has been used to run all the programming examples) The paper is organized as follows: we start by describing some monads for elementary data structures the (lists, lazy function lists etc. and monad morphisms. After that, we describe how to ....

G. Nadathur and D. Miller. Higher-order Horn clauses. JACM, 37(4):777-- 814, 1990.

....but also in logic programming since one proposal for negation [Barbuti et al. 1990] called explicit negation, relies on complement problems which are used to compute the clauses representing the negative counter part p of a predicate p. Therefore this approach can be used for lambda prolog [Nadathur and Miller, 1990] or ELF [Pfenning, 1989] if higher order complement problems are as manageable as rst order ones, which we prove in section 5. 2. Undecidability results 2.1. Higher order disunification is not semi decidable In this section we prove that higher order disunication is not semi decidable even when ....

Nadathur and Miller, 1990 Nadathur, G. and Miller, D. (1990). Higher-order horn clauses. Journal of the ACM, 37(4):777814.

....rules are also allowed in rule heads. The authors show that they can capture the full second level of the polynomial hierarchy in terms of expressive power. There are also successful approaches to generalize logic programming languages by using intuitionistic, linear, or higher order logics (e.g. [40, 52]) These extensions seem orthogonal to our treatment of negation in the context of arbitrary propositional formulas. 8.3 Why is it useful We already illustrated in the introduction the need for expressing disjunctive information and thus for extending normal logic programs to disjunctive ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. Journal of the Association for Computing Machinery (JACM), 37(4):777--814, 1990.

....well connected. A number of authors have attempted reconciliations of the two by embedding term rewriting in an extended unification mechanism for logic programming (e.g. 3, 16] and others have attempted reconciliation by treating both as formal systems embedded in the lambda calculus (e.g. [12, 15, 27]) Although it is natural to hope that we might reconstruct term rewriting and narrowing directly in logic programming (or in some resolution based system) this turns out to be not so simple to do, and recognizing why helps explain some of the lack of connectedness. 1.3 Organization of the Paper ....

G. Nadathur, D. Miller, "Higher Order Horn Clauses", J. ACM 37:4, 777--814, October 1990.

.... Forum and higher order LL suggests to consider Forum as a proof theoretical foundation for Logic programming languages defined over particular classes of formulas, as in the case of First Order Logic for Horn Clauses (Prolog [44] and Intuitionistic Logic for hereditary Harrop formulas (Prolog [40]) In our work [12] we have tried to isolate a particular 2 Operations over multisets will be denoted by: for the union and n for the difference. subset of Forum enjoying the good properties that the subset of higher order hereditary Harrop formulas does [39] We called the considered class of ....

G. Nadathur and D. Miller. Higher-Order Horn Clauses. Journal of the ACM, 37(4):777--814, 1990.

....than procedures. In order to formulate a type preservation theorem for an infinite class of type systems it is necessary to define the concept of a type system in general. Such a definition amounts to the construction of a logical framework a formalism for expressing (type) inference rules [4, 15, 3, 10]. Here we use a simple framework consisting of a notion of inference rules with built in notions of free variable and substitution using as the only quantifier. It differs from LF [3] and [10] in that it does not directly support ff fi equivalence of lambda expressions and in particular does not ....

....the construction of a logical framework a formalism for expressing (type) inference rules [4, 15, 3, 10] Here we use a simple framework consisting of a notion of inference rules with built in notions of free variable and substitution using as the only quantifier. It differs from LF [3] and [10] in that it does not directly support ff fi equivalence of lambda expressions and in particular does not provide higher order unification or matching. In the framework used here fi conversion must be handled explicitly. However, the framework used here does support implicit substitution. We call ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. JACM, 37(4):777--814, 1990.

....expressivity of higher order logics provides for direct representation of knowledge that is otherwise difficult to express, so the development of reasoning systems that manipulate fragments of higher order languages continues. Some (relatively) efficient deduction algorithms for higher order logic [NM90] have been successfully developed. This has led to the implementation of higher order languages like prolog [NM90] and HOL [GM93] which are based on deduction mechanisms in certain restricted forms of higher order logic. These languages have been successfully used for formal reasoning in many ....

....express, so the development of reasoning systems that manipulate fragments of higher order languages continues. Some (relatively) efficient deduction algorithms for higher order logic [NM90] have been successfully developed. This has led to the implementation of higher order languages like prolog [NM90] and HOL [GM93] which are based on deduction mechanisms in certain restricted forms of higher order logic. These languages have been successfully used for formal reasoning in many different areas, including hardware design and verification, reasoning about security, proofs about real time ....

G. Nadathur and D. Miller. Higher-order horn clauses. Journal of the ACM, 37(4):777--814, 1990.

....simple syntactic check but a more complex check of fij conversion. Second order unification is not in general decidable. In Prolog, full higher order unification is required and this issue is addressed by implementing a depth first version of the unification search procedure described in [8] See [10, 9]. In this paper, the second order unification problems that result from programs we present are all rather simple: it is easy to see, for example, that all such problems are decidable. In the AUGMENT search operation, clauses get added to the program dynamically. Note that as a result, clauses ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. April 1988. To appear in the Journal of the ACM.

.... terms. Also the equality of terms is not a simple syntactic check but a more complex check of fij conversion. Unification on terms is not in general decidable. In Prolog, this issue is addressed by implementing a depth first version of the unification search procedure described in [15] See [19, 17]. In this paper, the unification problems that result from programs we present are all decidable and rather simple. In the AUGMENT search operation, clauses get added to the program dynamically. Note that as a result, clauses may in fact contain logic variables. The GENERIC operation must be ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. Journal of the ACM, 37(4):777 -- 814, October 1990.

....exist when unifiers do exist. Prolog addresses these issues by implementing a depth first version of the unification search procedure described by Huet [24] It was shown by Miller et al. 30] that such unification is sufficient for determining substitutions, and by Nadathur and Miller [31, 33], that this unification procedure can be smoothly integrated into the usual backtracking mechanism of logic programming languages. The higher order unification problems we shall encounter in this paper are all rather simple. In fact all such problems are decidable. The presence of logic variables ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. Journal of the ACM, 37(4):777--814, October 1990.

.... or provided a general mechanism for integrating constraint systems into the programming language (e.g. via the L(D) construction) Because our work provides automatically a higher order version of Prolog, it is related to other proposals for higher order logic programming, e.g. CKW89] [NM90]. In particular, the connection with [NM90] seems to be close (though that study is proof theoretic in nature and a notion of models has not been elucidated) and bears further investigation. It should be noted, however, that first order cc languages do not require higher order unification to be ....

.... integrating constraint systems into the programming language (e.g. via the L(D) construction) Because our work provides automatically a higher order version of Prolog, it is related to other proposals for higher order logic programming, e.g. CKW89] NM90] In particular, the connection with [NM90] seems to be close (though that study is proof theoretic in nature and a notion of models has not been elucidated) and bears further investigation. It should be noted, however, that first order cc languages do not require higher order unification to be implemented in fact any underlying ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. Journal of the Association for Computing Machinery, 37(4):777--814, October 1990.

....of its corresponding formula in some suitable sequent calculus. Conversely, each formula can be interpreted operationally as a search instruction in the interpreter. We do not introduce the inference rules of the sequent calculus here but rather refer the interested reader to Nadathur and Miller [NM90, NM95] for a complete discussion. The main differences between proof systems for first order and higher order logics are the presence of conversion rules and a richer syntax of formulas. In what follows, we discuss the higher order unification problem and from this develop an implementable ....

G. Nadathur and D. Miller. Higher-order horn clauses. Journal of the A.C.M., 37(4):777--814, 1990.

....could be multiple maximally general unifiers. We would like to note that our use of a nondeterministic rewriting process should be surprising. In this respect, set unification is similar to other unification problems where an equality theory is involved, e.g. the unification of typed lambda terms [Hue75, NM89]. The unification procedure in these papers is also given in terms of a nondeterminstic rewriting process. From the above proposition, we see that in solving a set equation, other objects such as existential variables or the set predicate can appear in a disjunct on the rhs of the equivalence. ....

Nadathur, G. and Miller, D.: Higher-Order Horn Clauses, JACM, vol. 31, pp. 777-814, 1989.

....a restricted kind of higher order operators, syntactically reducible to first order operators, but more expressive and cleaner than PROLOG s use of extralogical builtins like functor, and metacall as higher order substitutes. Higher order unification of the kind studied with Prolog [NM90] however, is orthogonal to the extensions in RELFUN, which for simplicity and efficiency lives without expressions (thus avoiding problems with variables [Bac78] and semantic extensions of Robinson unification. Constructor variables can be used to abstract from, or force equality of, the ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. JACM, 37(4):777--814, October 1990.

....HLcc has no explicit recursion, and the underlying calculus is typed, so strongly terminating. We define two terms s ff and t ff to be equivalent, and write s ff = t ff if they can be shown equivalent using (ff; fi; j) rules. For background on such a treatment of terms, the reader may refer to [NM90]. As shown there, under such conversion rules a term M has a unique normal form; we shall denote it by ae(M ) A sequent is of the form G D where G; D are multisets of formulas (terms of type o) The inference figures for HLL are the expected ones for a higher order logic in the style of ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. Journal of the Association for Computing Machinery, 37(4):777--814, October 1990.

....in formal calculi such as the Calculus of Constructions [ Coquand and Huet, 1985 ] as well as in programming systems such as Edinburgh lego, L. Paulson s isabelle [ Paulson, 1989; Paulson, 1990 ] Andrews s TPS [ Andrews et al. 1984 ] and Miller s Prolog [ Nadathur and Miller, 1988; Nadathur and Miller, 1990; Miller, 1993 ] 3.7 Truth definitions revisited It is rewarding to relate finite order logic to the issue of truth definitions mentioned in Section 2.6 above. The proof of Theorem 2.6.2 can be easily adapted to higher order formulas: 42 i.e. forall( reduces to 8 by a type ....

....[1968] independently proved it for full finite order logic. These proofs use a model theoretic method of partial valuation due to Sch utte [ 1960a ] 64 This is less true for certain fragments of second order logic, in which the complexity of substituted formulas can be effectively controlled [Nadathur and Miller, 1990; Miller et al. 1991] 30 Daniel Leivant does not: the sentence : oe 1 ) see Section 2.2) has a model, but no countable model. 4. A logic L has the Upward Skolem L owenheim Property if every countable set of formulas that has an infinite countable model has arbitrarily large models. First ....

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G. Nadathur and D. Miller. Higher-order horn clauses. Journal of the ACM, 37:777--814, 1990.

....of higher order predicates. Prolog supports first order Horn clauses with only limited higher order features. #Prolog is a higher order logic programming language that extends Prolog by incorporating higher order unification and # terms [30] Because of its support of higher order Horn clauses [31], #Prolog makes an excellent logic programming language 3 for representing logic program schemata [26] In this paper, we present a set of logic program schemata and show how to use them to promote a structured approach to teaching logic programming. Our approach to teaching recursive ....

G. Nadathur and D. Miller. Higher-Order Horn Clauses. Journal of the ACM, 37: 777-814, 1990.

....rules are also allowed in rule heads. The authors show that they can capture the full second level of the polynomial hierarchy in terms of expressive power. There are also successful approaches to generalize logic programming languages by using intuitionistic, linear, or higher order logics (e.g. [40, 52]) These extensions seem orthogonal to our treatment of negation in the context of arbitrary propositional formulas. 8.3 Why is it useful We already illustrated in the introduction the need for expressing disjunctive information and thus for extending normal logic programs to disjunctive ....

Gopalan Nadathur and Dale Miller. Higher-order horn clauses. Journal of the Association for Computing Machinery (JACM), 37(4):777--814, 1990.

....For example, an individual term in predicate calculus may not appear as an atomic formula or be applied to other terms. The work on type systems for logic programs follows two main approaches. One adopts the thesis that the semantics of typed logic programs should be based upon a typed logic [13, 15, 21, 22, 28]. Most of the proposals are designed mainly for predicate calculus like languages and cannot accommodate the flexible syntax of HiLog. The other approach is meta theoretic in the sense that types are essentially constraints over type free logic programs [20, 33, 17, 16] A logic program may have a ....

G. Nadathur and D. Miller. Higher-order horn clauses. Journal of ACM, 37(4):777--814, October 1990.

....presentation of higher order linear logic in terms of goaldriven proofs. E hhf extends the previous proposals with aspects derived by the general purpose logic defined by Forum. E hhf is a multiset based logic which combines features peculiar of extensions of logic programming languages like Prolog[18], e.g. goals with implication and universal quantification, with the notion of formulas as resources at the basis of linear logic. Furthermore, E hhf is defined in a higher order setting, thus facilitating the development of applications based on meta programming. A specification written in E hhf ....

G. Nadathur and D. Miller. Higher-Order Horn Clauses. Journal of the ACM, 37(4):777--814, 1990.

....are identifiers beginning with a lowercase letter, and terminals are such identifiers surrounded by [ and ] A higher order DCG [13] is similar in structure to a first order DCG except that typed terms take the place of firstorder terms. It can be converted into a higher order Horn clause program [16] in a manner similar to the first order case: by adding two extra arguments to each nonterminal symbol, for the input list and remainder list respectively. Procedure SYNTH(G) 1. Let G be an unambiguous CFG having n rules, with start symbol start, and let L(G) be the language generated by G. The ....

G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, pages 777--814, 1990.

....called gentzen for an intuitionistic sequent calculus. Felty and Miller [FM88] have written a Prolog procedure for proof search in the intuitionistic sequent calculus that maintains the eigenvariable conditions through the use of higher order unification and hereditary Harrop program clauses [MNPS91, NM90]. The procedure we describe has a significant efficiency advantage over the approaches of Beeson, and Felty and Miller, since it postpones commitments on the order of introduction of certain quantifiers until the terminal nodes of the search tree are reached. Wallen [Wal90] uses Godel s ....

G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, 1990.

....of higher order predicates. Prolog supports first order Horn clauses with only limited higher order features. #Prolog is a higher order logic programming language that extends Prolog by incorporating higher order unification and # terms [14] Because of its support of higher order Horn clauses [15], #Prolog makes an excellent logic programming language for representing logic program schemata and programming techniques. In this paper, we present a set of logic program schemata and programming techniques. We begin by highlighting those features of #Prolog which are useful in defining logic ....

G. Nadathur and D. Miller. Higher-Order Horn Clauses. Journal of the ACM, 37: 777-814, 1990.

....this as the only legitimate answer to the posed query. We have relied thus far on an intuitive understanding of what it means to solve a goal. This understanding can be made logically precise by equating it with the notion of provability in classical logic, an aspect that is explored at length in [35]. At an operational level, this sanctions a recipe for solving a closed goal from a program P that is based on the structure of the goal: 1. Solve G 1 G 2 by solving both G 1 and G 2 . 2. Solve G 1 G 2 by solving one of G 1 and G 2 . 3. Solve 9xG by solving G[x : t] for some closed ....

....as a composite of a set of goal formulas and a disagreement set. Progress through this state space may be made by simpli cation steps applied to either the goal set or the disagreement set. In any given case, these steps must be relativized to a particular program P. The notion of a P derivation [35] that generalizes SLD derivations described in [6] for rst order Horn clause logic makes this idea precise. De nition 3.1. Let P be a program and let G and be symbols for sets of goal formulas and substitutions, respectively. Further, let D be a symbol for a disagreement set or the special ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-814, October 1990.

.... variable [Ble79,Fel00,Dow93] The logic programming language #Prolog allows some uses of higher order quantification (used for higher order programming) but such uses are restricted so that computing necessary predicate substitutions can be done using (essentially) higher order (pre) unification [NM90]. This can only be done, however, for rather serious restrictions on the use of higher order predicate substitutions (such restrictions do, however, also have a natural operational meaning within the logic programming setting) G[y x] #,D[t x] G[t x] #,D[y x] Fig. 1. ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777--814, October 1990.

.... in one direction is obtained by including higher order features in the form of quantification over function and some occurrences of predicate variables and the replacement of first order terms by simply typed lambda terms within Horn clauses, resulting in higher order Horn clauses [21]. Along the other direction, Horn clause logic is enhanced by permitting universal quantifiers and restricted uses of implications, resulting in a first order version of hereditary Harrop formulas [12, 17] The combination of these two logics produces a logic that is simply typed. This typing ....

....and this procedure is described in [8] This procedure can be factored into the repeated application of certain simple steps and can as such be amalgamated into the abstract interpreter described in Section 4. Such an amalgamation has been carried out for a higher order version of Horn clauses in [21] and has provided the basis for an extended WAM for implementing the higher order additions [19] In a general sense, the new problems that arise are that conversion has to be performed, that sets of terms that have to be unified have to be represented explicitly, and that the unification steps ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, October 1990.

.... in one direction is obtained by including higher order features in the form of quantification over function and some occurrences of predicate variables and the replacement of first order terms by simply typed lambda terms within Horn clauses, resulting in higher order Horn clauses [31]. Along the other direction, Horn clause logic is enhanced by permitting universal quantifiers and restricted uses of implications, resulting in a first order version of hereditary Harrop formulas [23, 28] The combination of these two logics produces a logic that is simply typed. This typing ....

....The differences are the use of a curried notation for atoms and the presence of typing. We adopt this kind of a syntax in the discussions below. From a logical perspective, the idea of answering a query can be explained by using provability in classical logic and this aspect is investigated in [31]. Operationally, this results in a recipe for solving a closed query from a program P that is based on the structure of the query: 1) Solve G 1 G 2 by solving both G 1 and G 2 . 2) Solve G 1 G 2 by solving one of G 1 and G 2 . 3) Solve 9x G by solving [t=x]G for some closed positive term ....

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Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, October 1990.

....easy later to take what is learned from these logics and move them into the linear logic framework. Furthermore, we shall focus almost entirely on first order logics, allowing quantification of predicates occasionally in examples, For more on higher order quantification in logic programming, see [NM90, MNPS91]. 2.1 Types and signatures Let S be a fixed, finite set of primitive types (also called sorts) We assume that the symbol o is always a member of S. Following Church [Chu40] o is the type for propositions. The set of types is the smallest set of expressions that contains the primitive types and ....

....for every G 2 G, Sigma; P G if and only if the sequent Sigma : P Gamma G has a uniform proof. Since these two definitions are restricted to I proofs, we shall refer to them as the singleconclusion version of uniform proofs and abstract logic programming. For more on these definitions, see [MNPS91, NM90]. We shall later introduce a multiple conclusion version of these definitions. The first (abstract) logic programming language we consider is first order Horn clauses: these are weak enough that they do not separate classical from intuitionistic provability. 3.3 The syntax of first order Horn ....

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Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777--814, October 1990.

....over predicate symbols. Such quantification is intended here to be purely syntactic: the type i o denotes the set of closed, simply typed terms of type i o and not some abstract domain of functions. A similar treatment of higher order type quantification for Horn clauses can be found in [NM90]. The conjunctive translation. It is trivial to dualize the disjunctive translation completely. That is, it is possible to map the logical combinators into the dual logical connectives. hhP Qii = hhP ii hhQii hhP j Qii = hhP ii Omega hhQii hh(x)P ii = 9xhhP ii hh P ii = hhP ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777 -- 814, October 1990.

....Prolog [20] has, over the last five years, been a testbed for experimenting with several different extensions to a language such as Prolog. The initial impetus in developing this language was provided by a desire to understand the nature and role of higher order notions within logic programming [12, 19, 21]. This goal was subsequently expanded to include devices for other forms of abstraction such as lexical scoping, modules and abstract data types [9, 10, 13] The purpose of this paper is to discuss another important but somewhat less exposed component of the language, namely its type system. The ....

Nadathur, G. and Miller, D. Higher-order Horn clauses. J. ACM, Vol. 37, No. 4, October 1990, 777 -- 814.

....to be considered, then we can, in fact, restrict our attention to C proofs in which the antecedent and succedent of each sequent occurrence is a subset of D 1 and G 1 [ f g, respectively. The rest of the argument in the proof of Theorem 1 would then carry over to this case as well. In [22] and [24], it is shown that this restriction on substitutions within C proofs in fact preserves the set of provable sequents. That is, if P is a set of higher order Horn clauses and G is a formula in G 2 such that P Gamma G has a C proof, then this sequent has a C proof in which the substitution terms ....

....in the proof, the arguments in the proof of Theorem 1 permit the removal of the unnecessary instance, P b. These observations are made precise in the following theorem. Theorem 2. hohc is an abstract logic programming language. Proof. Again, only an outline is provided. Details can be found in [22, 24], and the proof in Section 6 is also similar in spirit. Let P be a finite set of higher order Horn clauses and let G be a formula in G 2 . The theorem is trivial in one direction, since a uniform proof of P Gamma G is also a C proof. For the other direction, assume that P Gamma G has a ....

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G. Nadathur and D. Miller, Higher-Order Horn Clauses, Journal of the ACM (submitted) .

....the additional restriction that if a quantified variable is of type then does not contain the primitive type o. Thus, predicate quantification is not permitted in this logic. There are various ways to allow forms of predicate quantification in this setting: one approach is described in [22, 25] and another is described in [16] The kinds of meta programs that we discuss here do not require any forms of predicate quantification. A sequent calculus is used to define intuitionistic provability over these formulas. A sequent is a triple, written Sigma ; Gamma Gamma B, where Sigma is a ....

....An untyped version of terms and G and D formulas arises by simply deleting the typing information from inference rules in Figure 3. In that system, essentially existential variables can occur in predicate positions. Logic programming languages with such possibilities have been analyzed elsewhere [22, 25]. Here we shall assume that the two inference rules that permit the inference of an atomic D formula and of an atomic G formula are modified as in Figure 4. There the proviso (x) is that Q contains 8h and that n 0. This restriction ensures that the resulting language is first order in the sense ....

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G. Nadathur and D. Miller (1990). Higher-Order Horn Clauses, Journal of the ACM 37 (4), 777 -- 814.

....for every G 2 G, Sigma; P G if and only if the sequent Sigma : P Gamma G has a uniform proof. Since these two definitions are restricted to I proofs, we shall refer to them as the singleconclusion version of uniform proofs and abstract logic programming. For more on these definitions, see [MNPS91, NM90]. We shall later introduce a multiple conclusion version of these definitions. The first (abstract) logic programming language we consider is first order Horn clauses: these are weak enough that they do not separate classical from intuitionistic provability. 4.3 The syntax of first order Horn ....

....formula is an implication, say G oe D, then we need to need to do two things: we must prove G and continue using D to do backchaining. Combining the inference rules in Figure 4 and 5 now yields a complete proof system for fohc with respect to classical logic. A proof of this fact can be found in [NM90] (see also the exercises below) When read bottom up, these inference rules provide a complete set of reduction steps for finding a proof. Consider a proof of the sequent Sigma : P Gamma G using these inference rules. It is easy to see that every sequent that appears in such a proof with either ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777--814, October 1990.

....carefully defined in the higher order context because of the presence of abstractions in terms. However, there is a simple, and standard, way of doing this using conversion. We assume such a definition here; the reader unfamiliar with this formalization of substitution may look, for example, at [21]. As in the first order context, the idea of programming can be thought of as asking if a proof exists for a goal formula from a set of program clauses. Now, a property very similar to that presented in Theorem 5 holds in the higher order context as well and this is once again useful in designing ....

....described in [7] This procedure can be factored into the repeated application of certain simple steps, and this permits its amalgamation into a notion of derivation akin to the one described in Section 5. Such an amalgamation is described explicitly for a higher order version of Horn clauses in [21], and a similar process can be used in the case of hereditary Harrop formulas. The one difference is that substitutions that are suggested for the purpose of unification must respect the constraints imposed by labels on symbols. As in the first order case, this can be ensured by incorporating ....

[Article contains additional citation context not shown here]

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, October 1990.

....hereditary Harrop formulas in (Miller et al. 1991) Alternately, we can use a slightly higher order variant of the logic over just true; oe; and 8 to define part of the meaning of disjunctions and existential quantifiers. In particular, consider the three higher order Horn clauses (see Nadathur and Miller, 1990, for a treatment of such clauses) 8P8Q[P oe (P Q) 8P8Q[Q oe (P Q) 8B8T [ B T ) oe 9B] Here, and 9 are treated as non logical symbols that have the types (as in Church s Simple Theory of Types (Church, 1940) o o o and (i o) o, respectively, where o is the type of propositions and ....

Nadathur, G., and Miller, D. (1990). Higher-order Horn Clauses. Journal of the ACM, 37 (4), 777 -- 814.

.... is obtained by including higher order features in the form of quantification over function and some occurrences of predicate variables and the replacement of first order terms by simply typed lambda terms within Horn clauses, thereby producing the logic of higher order Horn clauses [22]. Along the other direction, Horn clause logic is enhanced by permitting universal quantifiers and restricted uses of implications, resulting in a first order version of the logic of hereditary Harrop formulas [13, 17] The combination of these two logics produces a simply typed version of the ....

....that can be used to find unifiers for these terms whenever they exist [9] This procedure can be factored into the repeated application of certain simple steps and can be amalgamated as such into the abstract interpreter described in Section 4. A similar amalgamation has been carried out in [22] relative to a higher order version of the Horn clause language and has been used in [20] to describe a WAM based implementation scheme for this language. At a level of detail, the main new implementation concerns in the context of this language are (a) devising a good representation for lambda ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, October 1990.

....goal set may be solved by applying the substitution fhX; ig to it. A P derivation of a goal G is intended to show that G succeeds in the context of a program P. The following lemma is useful in proving that P derivations are true to this intent. A proof of this lemma may be found in [Nad87] or [NM90]. The property of the P derivability relation that it states should be plausible at an intuitive level, given Lemma 11 and the success failure semantics for goals. Lemma 13 Let hG 2 ; D 2 ; 2 ; V 2 i be P derivable from hG 1 ; D 1 ; 1 ; V 1 i, and let D 2 6= F. Further let 2 U(D 2 ) be a ....

....showing that if the goals in the last tuple of a P derivation sequence succeed and the corresponding disagreement set has a unifier, then that sequence can be extended to a successfully terminated sequence. This is the content of the following lemma, a proof of which may be found in [Nad87] and [NM90]. The lemma should, in any case, be intuitively acceptable, given the success failure semantics for goals and Lemmas 11 and 12. Lemma 15 Let hG 1 ; D 1 ; 1 ; V 1 i be a tuple that is not a terminated P derivation sequence and for which F(G 1 ) F(D 1 ) V 1 . In addition, let there be a ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777 -- 814, October 1990.

....paradigm has been formulated using the technical notion of uniform proof [18, 17] To retain completeness of uniform proofs, restrictions on logical formulas need to maintained. For example, completeness of uniform proofs can be achieved in classical logic by restricting to Horn clauses [19]; in intuitionistic logic by restricting to hereditary Harrop formulas [18] and in linear logic by choosing the proper logical connectives [1, 17] There are numerous examples of specifying computations within these logics and with using meta theoretic properties of those logics to infer ....

Gopalan Nadathur and Dale Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, October 1990.

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G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, 1990. Typeset with Xy-pic

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G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-- 814, 1990. Typeset with Xy-pic

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Gopalan Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777-814, October 1990. 18

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G. Nadathur and D. Miller. Higher-order horn clauses. Journal of the Association for Computing Machienery, 37(4):777 -- 814, 1990.

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G. Nadathur and D. Miller. Higher-order Horn clauses. Journal of the ACM, 37(4):777{ 814, 1990.

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