W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Volume 1: Logical Foundations, pages 449-503, Oxford University Press, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and their model theoretic semantics. A framework F is a full first order logical theory (with identity) with an intended model. The syntax of F is similar to that used in algebraic abstract data types (e.g. 13, 34, 28] However, whilst an algebraic abstract data type is an initial model ([12, 15]) of its specification, the intended model of F is an isoinitial model. Definition 2.1 A model i is an isoinitial model of F iff, for every other model i of F there is a unique isomorphic embedding h : i i. An isomorphic embedding h : i i is a homomorphism with the additional ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Volume 1: Logical Foundations, pages 449-503, Oxford University Press, 1993.

....As a matter of fact, the strategy is incorrect unless the theory is convex. Definition 7. A theory is convex i# for every set # of literals and every finite non empty set P of positive literals, # p#P p i# # p for some p P . By well known results about Horn logic (see, e.g. [10]) one can show that the class of convex theories (properly) includes all Horn theories. The general strategy we propose is motivated by the following result. Proposition 8. Let # be a set of ground literals satisfiable in a convex theory and let # be a set of non positive ground clauses. If ....

Wilfrid Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1, pages 449--503. Oxford University Press, 1993.

....# convex for some signature #. This shows that an important portion of 22 candidate background theories for theory reasoning are in fact convex, which justifies the particular relevance of the results presented in this section. These results and their proofs will use Horn formulas. Following [Hod93a] we call a basic Horn formula a formula of the form where n #) For the rest of this section, we will fix two signatures # 1 , # 2 such that # 1 and two universal theories 2 of respective signature # 1 , # 2 such that satisfiable. Also, we assume that both theories is ....

....# 2 is universal. 31 A.3 # Convex Theories In this subsection we show that every Horn theory, and in particular every universal Horn theory, is # convex for any #. Then, we point to some examples of non Horn # convex theories. We start by defining (universal) Horn theories, again following [Hod93a] Recall that a basic Horn formula is a formula of the form #q #) A Horn formula is a formula of the form Q. # 1 n ) where Q is an arbitrary quantifier prefix, n 0 and each # j is a basic Horn formula. A Horn sentence is Horn formula with no free variables and a universal Horn ....

[Article contains additional citation context not shown here]

Wilfrid Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1, pages 449--503. Oxford University Press, 1993.

....which each term contains at most one negative literal. It is well known that the Horn functions f are those whose set F (f) of false vectors is closed under intersection (see Section 2) they play an important role in artificial intelligence, logical databases, and logic in computer science, cf. [16, 7, 21]. As shown in [28, 6] a Horn extension of a pdBf can be found in polynomial time. In fact, a Horn extension for (T ; F ) exists precisely if the true vectors T are disjoint from the closure of the false vectors F under intersection. However, this characterization shows that the Horn extension ....

W. Hodges, Logical features of Horn clauses, in: Handbook of Logic in Artificial Intelligence and Logic Programming I: Logical Foundations, D.M. Gabbay, C.J. Hogger, and J.A. Robinson (eds), Clarendon Press, Oxford UK, 1993.

....applied in DPLL(T ) whenever the background theory T is convex. De nition 8 (Convex Theory) A theory T is convex i for every set of literals and every nite non empty set P of positive literals, j= T p2P p i j= T p for some p 2 P : By well known results about Horn logic (see, e.g. [8]) one can show that the class of convex theories (properly) includes all Horn theories. The general strategy we propose is motivated by the following result. Proposition 9. Let be a set of ground literals satis able in a convex theory T , and let be a set of non positive ground clauses. If ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Arti cial Intelligence and Logic Programming, volume 1, pages 449-503. Oxford University Press, 1993.

....# 0 . # n 3. If #, # X, then # 4. If # X, then #x# # The set of universal Horn sentences is the set of universal Horn formulas with no free variables. Universal Horn formulas form an important class for a number of reasons (for a survey, the reader may consult [Hod92] For us, the following characterization is useful: Theorem 6.16 (Mal cev) Let # be a first order sentence over some signature. Then # is preserved under taking finite direct products and substructures i# # is equivalent to a universal Horn sentences. With theorem 6.8 from the previous ....

Wilfried Hodges. Logical features of horn clauses. In Handbook of logic in artificial intelligence and logic programming. Oxford University Press, 1992.

....PA, USA, mobile cs.cmu.edu 2 Dipartimento di Scienze dell Informazione, Universita degli studi di Milano, Via Comelico 39 41, Milano, Italy, ornaghi dsi.unimi.it Abstract. We give a proof theoretic analysis of logic programs transformations, viewed as operations on proof trees in the sense of [3, 4, 9, 10]. We present a logic for reasoning about (equivalence preserving) transformations of logic programs. Our main tool is the usage of inference rules; the target program may be obtained as a set of clause introduction proofs with axioms from the source program. The rules are admissible, that is every ....

....languages satisfying the simple requirement of regularity. Our perspective and overall aim is to develop the proof theory of (logic) program transformation. 1 Introduction We give a proof theoretic analysis of logic programs transformations, viewed as operations on proof trees in the sense of [3, 4, 9, 10]. We present a logic for reasoning about (equivalence preserving) transformations of logic programs. Our framework is natural deduction a la Prawitz [12] although a totally analogous presentation can be given in terms of sequent calculi with definitional reflection [14] We are not going to ....

[Article contains additional citation context not shown here]

Hodges, W.: Logical Features of Horn Clauses. In: Gabbay, D.M., Hogger, C.J, Robinson, J.A., (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming. Volume 1: Logical Foundations. Oxford University Press (1993) 449--503

....are convex for some signature . This shows that an important portion of 21 candidate background theories for theory reasoning are in fact convex, which justi es the particular relevance of the results presented in this section. These results and their proofs will use Horn formulas. Following [Hod93a] we call a basic Horn formula a formula of the form :p 1 :p n q where n 0 and each of p 1 ; p n ; q is a positive literal (possibly ) 15 Assumptions. For the rest of this section, we will x two signatures 1 ; 2 such that 1 F = 2 F and two universal ....

....T 1 j= and T 2 j= ut A.3 Convex Theories In this subsection we show that every Horn theory, and in particular every universal Horn theory, is convex for any . Then, we point to some examples of non Horn convex theories. We start by de ning (universal) Horn theories, again following [Hod93a] 21 This substructure exists because 2 contains at least a constant symbol; moreover, it is a model of [ T 2 by Lemma A.3 because [ T 2 is universal. 30 Recall that a basic Horn formula is a formula of the form :p 1 :p n q where n 0 and each of p 1 ; p n ; q is a ....

[Article contains additional citation context not shown here]

Wilfrid Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Articial Intelligence and Logic Programming, volume 1, pages 449-503. Oxford University Press, 1993.

....trading expressiveness for tractability, theories have been restricted to fragments of the full language. A most important such fragment is the class of Horn theories. Fields such as Logic Programming and Deductive Databases are based on their appealing semantical and computational properties, cf. [11]. In fact, deduction of a clause from a propositional Horn theory is possible in linear time [5] while this is co NP complete in general. In this paper, we study the problem of computing the Boolean di erence between two Horn theories 1 and 2 , i.e. 1 n 2 . In general, the resulting ....

W. Hodges. Logical Features of Horn Clauses. In D. Gabbay, C. Hogger, and J. Robinson, editors, Handbook of Logic in Articial Intelligence and Logic Programming, volume I : Logical Foundations, pages 449-503. Clarendon Press, Oxford; New York, 1994.

....the originator of isoinitial semantics: Pierangelo Miglioli (1946 1999) 1 Introduction The intended model of a de nite logic program P is its Herbrand model H . It interprets P under the Closed World Assumption [16] Among the class of all the models of P , H interprets P as an initial theory [9]. A distinguishing feature of an initial theory P is that, in general, it proves (computes) only positive literals in P , so it does not deal with negation. One way to handle negation is to consider program closures (e.g. program completion) In this paper, we introduce isoinitial semantics [2] ....

....; s 1 ) Cdef(sum) The minimum Herbrand model M(P ) is de ned in the usual way. We have the following theorem: Theorem 3. For a (de nite) logic program P , M(P ) is an initial model of P and of Comp(P ) but it is not an isoinitial model of P . Proof. The initiality of M(P ) is well known [9]. M(P ) cannot be an isoinitial model of P , because P cannot be atomically complete (no negated formula is provable from it) We might expect M(P ) to be an isoinitial model of Comp(P ) but this is not necessarily so, as shown by the following example: 4 Here s (n) denotes the iteration of ....

[Article contains additional citation context not shown here]

W. Hodges. Logical features of horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson. editors, Handbook of Logic in Articial Intelligence and Logic Programming, Volume 1:449-503, Oxford University Press, 1993.

....Horn clauses (which, with a finite language, can be a fully instantiated set of first order Horn clauses) things are rather better. Indeed, the problem of building an argument is not only decidable but also may be achieved in time proportional to the number of propositions in the language [16]. The problem of building a rebutting argument is equivalent to building an argument for a proposition, so this is also decidable and takes time proportional to the size of the language. In the worst case undercutting an argument involves attempting to rebut every step in the argument, and so is ....

W. Hodges. Logical features of Horn clauses. In J. van Leeuwen, editor, Logical Foundations, Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 1, pages 449--503. Oxford University Press, 1993.

....Semantics, isoinitial models, negation 1 Introduction The intended model of a standard (Horn clause) logic program P is its Herbrand model H. It interprets P under the Closed World Assumption [11] Considering the class of all the models of P , H interprets P as an initial (Horn) theory [5]. A distinguishing feature of an initial theory P is that, in general, it proves (computes) only positive literals (atoms) in P , so it does not deal with negation. In this paper, we introduce isoinitial models [2] of logic programs. If the completion Comp(P ) of a program P has an isoinitial ....

....a Herbrand Model is an interpretation of the predicates of R P over the domain of H P . The Minimum Herbrand Model M(P ) is defined in the usual way. Theorem 3.2 M(P ) is an initial model of P and of Comp(P ) but it is not an isoinitial model of P . Proof. The initiality of M(P ) is well known [5]. M(P ) cannot be an isoinitial model of P , because no negated formula is provable from P (that is, P is not and cannot be atomically complete) 2 One would expect that M(P ) is an isoinitial model of Comp(P ) However, this is not necessarily true, as shown by the following example: Example 3.2 ....

[Article contains additional citation context not shown here]

W. Hodges. Logical features of horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson. editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Volume 1: Logical Foundations, pages 449--503, Oxford University Press, 1993. Isoinitial Models for Logic Programs: A Preliminary Study 135

....has been adopted. For instance, in logic programming, this relation can be particularly close since LS could also be a logic language (see e.g. 5] Indeed logic programs are often regarded and used as (executable) specifications themselves. After all, a logic program is a Horn clause theory ([4]) and as such it can double as a definition. This would seem to suggest that LS could be just LP , and the notion of correctness is redundant. In this paper, we examine the relation between S and a logic program P , i.e. LP is Horn clause logic. We shall argue that this relation should be firmly ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Vol 1, pages 449-- 503, Oxford University Press, 1993.

....the notion of correctness that has been adopted) For instance, if L S is also a logic language, then this relation could be very close (see e.g. 8] Indeed logic programs are often regarded and used as (executable) specifications themselves. After all, a logic program is a Horn clause theory ([7]) and as such it can double as a definition. This would seem to suggest that L S could be just Horn clause logic, and the notion of correctness is redundant. In this paper, we argue that the relationship between S and P should be set firmly in the context of the underlying problem domain, which ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Vol 1, pages 449--503, Oxford University Press, 1993.

....the notion of correctness that has been adopted) For instance, if L S is also a logic language, then this relation could be very close (see e.g. 7] Indeed logic programs are often regarded and used as (executable) specifications themselves. After all, a logic program is a Horn clause theory ([6]) and as such it can double as a definition. This would seem to suggest that L S could be just Horn clause logic, and the notion of correctness is redundant. In this paper, we argue that the relation between S and P should be set firmly in the context of the underlying problem domain, which we ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Vol 1, pages 449--503, Oxford University Press, 1993.

....model of a set of formulas can be obtained as the parallel propositional circumscription (see [15] of all the propositional variables in the formula. As Makowsky points out, generic models give a semantics to Reiter s Closed World Assumption [23] applied to propositional knowledge bases (see also [13]) The relationship between CWA and circumscription has been explored in [16] 6.4 Data Types and Generic Functions There is a notion of generic function that appears in relation to the concept of parametric polymorphic function in data type specifications [6] Polymorphic functions are ....

Hodges, W. "Logical Features of Horn Clauses". Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 1, 449--503, Oxford University Press, 1993.

....out that if a formula is preserved under direct products then it must be equivalent to a Horn formula which gives a characterization of Horn formulas. In the present section we show that this important characterization carries over to many valued structures and Horn formulas. See for example [H 92, H 93b] for an extended discussion of direct products and related notions. Definition 20 Let (M i ) i2J =hD i ; I i i be a non empty family of many valued structures over the same signature. Then we define the direct product Q i2J M i = hD; Ii as follows: The domain D is the direct ....

....necessary. Our many valued Horn formulas have many desirable properties of classical Horn formulas, while a smaller class of Horn formulas with similar computational characteristics as defined in [EI MS 94] is not sufficient. We are convinced that other properties of classical Horn formulas (see [H 92, EI MS 94] hold for our many valued version as well. We believe that the present work at the same time unifies and justifies [S 86, L M R 93, M R 93, EI MS 94] L = 1 p S [ f 1 pg ff 1 qg; f 1 2 qg; f 1 2 qgg 2 S [ f 1 2 pg ff 1 qg; f 0 pg; f 1 2 p; 1 2 qg, f 1 2 p; 1 2 qg ff ....

W. Hodges. Logical features of Horn clauses. In D. M. Gabbay, C. J. Hogger, and J. A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1: Logical Foundations, pages 449--503. Oxford University Press, 1992.

....the notion of correctness that has been adopted) For instance, if LS is also a logic language, then this relation could be very close (see e.g. 8] Indeed logic programs are often regarded and used as (executable) specifications themselves. After all, a logic program is a Horn clause theory ([7]) and as such it can double as a definition. Address correspondence to Kung Kiu Lau, Department of Computer Science, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom, email: kungkiu cs.man.ac.uk, or Mario Ornaghi, Dipartimento di Scienze dell Informazione, Universita ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Vol 1, pages 449--503, Oxford University Press, 1993.

....if its elements can be represented by ground terms; a reachable model of F is isoinitial iff ground quantifier free formulas are true in it whenever they are true in every model of F . Followingthe tradition of algebraic ADTs [14, 18] initial models have also been proposed for logic programs [5, 6]. We have preferred isoinitial models to properly deal with negation (see also the primal models proposed in [7] In general, a framework may have no isoinitial model. Hence the following adequacy condition: Definition 2.2 A closed framework F is adequate if it has a reachable isoinitial model. ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 1: Logical Foundations, pages 449--503, Oxford University Press, 1993.

....however, on exactly what form a specification should take, what part it should play in synthesis, and what its precise relationship with the specified program should be. In logic programming, the role of specification is all the more unclear because a logic program itself is a Horn clause theory ([10]) and as such can double as a definition or specification. Indeed logic programs are often used as executable specifications, for instance in rapid prototyping. In our work in deductive synthesis of logic programs (see e.g. 16, 17] we maintain a strict distinction between specifications and ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Vol 1, pages 449--503, Oxford University Press, 1993.

....however, on exactly what form a specification should take, what part it should play in synthesis, and what its precise relationship with the specified program should be. In logic programming, the role of specification is all the more unclear because a logic program itself is a Horn clause theory ([11]) and as such can double as a definition or specification. Indeed logic programs are often used as executable specifications, for instance in rapid prototyping. In our work in deductive synthesis of logic programs (see e.g. 17, 18] we maintain a strict distinction between specifications and ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Vol 1, pages 449--503, Oxford University Press, 1993.

....for the pre defined sorts are imported and new axioms are added to define the (new) functions and relations on T . The syntax of a framework F is thus similar to that used in algebraic abstract data types (e.g. 13, 29, 24] However, whilst an algebraic abstract data type is an initial model ([12, 15]) of its specification, the intended model of F is an isoinitial model. Of course, a framework may have no intended (i.e. reachable isoinitial) model. We will only ever use frameworks with such models, i.e. adequate frameworks: Definition 3. Adequate Closed Frameworks) A closed framework F is ....

W. Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming , Volume 1: Logical Foundations, pages 449-503, Oxford University Press, 1993.

....definite goals 35 . If T i is an Sigma i theory in T Sigma , then T i is Sigma stable over Res(L Sigma i ; Sigma) Proof. Since T i is non trivial for being collapse free, it can be shown to have a model A that is free in Mod(T i ) over a basis X of cardinality Card ( Sigma i ) see [Hod93a] for instance) In particular, by Lemma 2.23, A is free over X in the Sigma i variety of T i . By Prop. 2.29 and Prop. 2.27 then, A Sigma is free over X in the Sigma variety of T i . We show below that every formula of Res(L Sigma i ; Sigma) unsatisfiable in A is unsatisfiable in T i ....

Wilfrid Hodges. Logical features of Horn clauses. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1, pages 449--503. Oxford University Press, 1993.

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Wilfred Hodges. Logical features of Horn clauses. In Dov M. Gabbay, C.J. Hogger, and J.A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, pages 449--503. Oxford University Press, 1993.

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