J. Harland. On Hereditary Harrop Formulae as a basis for Logic Programming. PhD thesis, University of Edinburgh, UK, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....embedded implication seems to lead to curious paradoxes, as non eliminability of the cut rule or failure of weakening. In the literature the issue has been addressed in essentially three ways: 1. By enforcing a strict distinction between CWA and OWA predicates and applying NF only to the former [10], where the latter would require minimal negation, for example as in [16] 2. By switching to a modal logic, which is able to take into account arbitrary extensions of the program as possible worlds (see the completion construction in [9] for N Prolog and [4] for Hypothetical Datalog) 3. By ....

J. Harland. On Hereditary Harrop Formulae as a Basis for Logic Programming. PhD thesis, Edinburgh, Jan. 1991.

....program in a totally unpredictable way. This makes it in general impossible to talk about the closure of such a program. In the literature the issue has been addressed in essentially three ways: 1. By enforcing a strict distinction between CWA and OWA predicates and applying NF only to the former [8], where the latter would require minimal negation. 2. By switching to a modal logic, which is able to take into account arbitrary extensions of the program as possible worlds (see the completion construction in [7] for N Prolog and [4] for Hypothetical Datalog) 3. By embracing the idea of ....

J. Harland. On Hereditary Harrop Formulae as a Basis for Logic Programming. PhD thesis, Edinburgh, Jan. 1991.

....the rules exist and atom are non standard since some of the involved parameters (the term t and the clause 8 x: G A 0 ) respectively) are subject to extensional universal quantification. Therefore, exist can be viewed as a rule with an infinite number of premisses (Harland has shown in [8] that it is sufficient to consider a finite set of representations) Similarly atom is better seen as a rule with a variable number of premisses depending on the number of matching clauses. Let us now give some examples that better illustrate the distinction among provable, finitely nonprovable ....

....sequents. First, since we defined negative sequents with the aim of formalizing finite non provability, it should not be possible that both the positive and the negative sequents involving the same program and goal are provable. Harland has proved the following result for a similar rule system [8, 9]. Property 3.1 (Consistency of positive and negative sequents) For given program P and goal G, either P = G or P 6 = G is not derivable. This property can be sharpened by considering finite non provability. We have indeed that a positive sequent is finitely non provable iff the corresponding ....

J. Harland. On Hereditary Harrop Formulae as a basis for Logic Programming. PhD thesis, University of Edinburgh, UK, 1991.

....the rules exist and atom are non standard since some of the involved parameters (the term t and the clause 8 x: G A 0 ) respectively) are subject to extensional universal quantification. Therefore, exist can be viewed as a rule with an infinite number of premisses (Harland has shown in [8] that it is sufficient to consider a finite set of representations) Similarly atom is better seen as a rule with a variable number of premisses depending on the number of matching clauses. Let us now give some examples that better illustrate the distinction among provable, finitely non provable ....

....sequents. First, since we defined negative sequents with the aim of formalizing finite non provability, it should not be possible that both the positive and the negative sequents involving the same program and goal are provable. Harland has proved the following result for a similar rule system [8, 9]. Property 3.1 (Consistency of positive and negative sequents) For given program P and goal G, either P = G or P 6 = G is not derivable. This property can be sharpened by considering finite non provability. We have indeed that a positive sequent is finitely non provable if and only if the ....

J. Harland. On Hereditary Harrop Formulae as a basis for Logic Programming. PhD thesis, University of Edinburgh, UK, 1991.

.... of constructive negation, in the sense of [8, 49, 18] We present a version of constructive negation based on the notion of regular splitting, a transformation technique where the failure axiom(s) of a predicate occurring negatively in a program are split into new clauses according to a covering [26, 17] of the underlying signature. Keywords. Logic programming, proof theory, negation as failure, constructive negation, search spaces. 1 Introduction The aim of this paper is to show the fruitfulness and fecundity of the proof theoretic analysis of logic programming developed in [33] both for ....

....program, obtaining in this way a regular system, where it is possible to answer negative open queries. Splitting is a simple transformation technique where the failure axiom(s) of the predicate definition occurring negatively in the source program are split into new clauses according to a covering [17, 30] of the underlying signature and then executed in an opportune inference system. This is similar to the method in [5] and ancestors (see Section 6 for a comparison) We want to stress at this point that our interest lies mainly in showing the versatility and the adaptability of our approach once ....

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J. Harland. On Hereditary Harrop Formulae as a Basis for Logic Programming. PhD Thesis, Edinburgh, 1991.

....program, and hence the program is not used until the goal is reduced to just an atom. The properties of such proofs have been thoroughly investigated for various fragments of first and higher order intuitionistic logic (classes of formulas called Horn clauses [30] and hereditary Harrop formulas [22, 33]) Whilst this analysis is quite thorough, there remains the question of how to lift the concept of goal directed proofs to other logics (for example multiple conclusioned logic, linear, light linear logic, affine, relevant linear logic, etc. Given the evolution of programming languages towards ....

Harland J. On Hereditary Harrop Formulae as a Basis for Logic Programming PhD Thesis, Department of Computer Science, University of Edinburgh, July, 1991.

....N 2 . Intuitively, the two sets of names indicate which predicates are completely defined (den(N) and which are not (ass(N ) The latter set of predicates may be safely assumed (i.e. added to the program) whereas the former set may not. Further discussion on this point is beyond our scope; see [6, 7] for more detail. An operational notion of provability, denoted by o was given in [11] for the above class of formulae when there are no negations present. In order to account for Negation as Failure, we need a more intricate notion of success and failure. We do so by introducing two relations ....

....P 6 f p. The above definitions of s and f may be used to derive a proof system by interpreting the left hand side of the iff as the derived sequent and the right hand side as the previous sequent or sequents. Space prevents us developing this further here; the interested reader may consult [7, 5]. The intuitive reading of P s G and P f G may be given as G succeeds and G fails respectively. In [11, 10] the relationship between o and provability in intuitionistic logic is discussed, and it is shown that for the definition of o given in [10] P o G iff P I G, where I ....

J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. Thesis, University of Edinburgh, forthcoming.

....logic and show the full abstraction theorem. Finally in section 7 we discuss some further work and remaining issues. Due to space limitations, it is not possible to include proofs of all the results in this paper. Complete proofs of all results stated here and further discussion may be found in [2, 4, 5]. 2 Preliminaries Definition 2.1 D iH and G iH formulae are given by the grammar D : A j 8xD j D 1 D 2 j G oe A G : A j 9xG j G 1 G 2 j G 1 G 2 j D oe G We refer to this class of formulae as implicative Horn formulae. 1 D hhf and G hhf formulae are given by the grammar D : A j 8xD j ....

....of a goal into an answer formula makes it clear that computation in this setting is a matter of converting an indefinite constraint into a definite answer. The answer formula is a particularly definite piece of information; in fact, it is not hard to rewrite 8(G i ) as a definite formula (see [2, 3, 4]) Hence we may add answer formulae to the program, and hence memoise [10] successful goals. Memoisation is a well known technique, and may considerably improve performance, as we do not need to re compute previously computed results. Whilst we cannot in general store successful goals themselves, ....

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J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. thesis, University of Edinburgh, July, 1991.

....of some fixed but arbitrarily chosen term. For example, given the following program, p(t) succeeds for every t 2 fa; f(a) f(f(a) g, but p(y) fails. 3 . 2 In fact, the natural logic in which to interpret hereditary Harrop formulae is slightly stronger than intuitionistic logic; see [11, 13] for details. 3 Note, however, that the goal 9xp(x) succeeds 20 p(a) 8x p(f(x) p(x) Hence we may wish for a slightly stronger rule for universal quantification, in that if P u p(y) then P u p(t) for any ground term t. As discussed in [13] this will require some form of structural ....

....stronger than intuitionistic logic; see [11, 13] for details. 3 Note, however, that the goal 9xp(x) succeeds 20 p(a) 8x p(f(x) p(x) Hence we may wish for a slightly stronger rule for universal quantification, in that if P u p(y) then P u p(t) for any ground term t. As discussed in [13], this will require some form of structural induction. This difference between the two rules is similar to the relationship between D and jDj. It is clear that D I D 0 for any D 0 2 jDj, and so (assuming we may write infinitary formulae) D I V jDj, but the converse is not true. We may ....

J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. Thesis, Department of Computer Science, University of Edinburgh, July, 1991.

....p(f(y) both succeed, but p(y) does not succeed. This result cannot be extended to DHHF Gamma formulae either, as for the above program, p(y) does not fail, although :p(a) and :p(f(y) both do. Full discussion of this point is beyond the scope of this paper; further discussion may be found in [9]. The intuitive reading of P s G and P f G may be given as G succeeds and G fails respectively. The validity of this interpretation is shown by the proposition below. Proposition 3.4 Let hP; Gi be a derivation pair where P = hD; Ni. Then 1. P s G ) P 6 f G 2. P f G ) P 6 s G ....

J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. Thesis, Department of Computer Science, University of Edinburgh, July, 1991.

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J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. Thesis, Department of Computer Science, University of Edinburgh, July, 1991.

....use unification. We discuss some approaches to the two problems above, and consider some further properties of the sequent calculi LK for classical logic, and LJ for intuitionistic logic in order to incorporate a Negation as Failure connective. The approach in this paper differs from that of [4], as in this case we are interested in a general NAF connective rather than one restricted to atomic formulae and a particular fragment of intuitionistic logic. The current approach may be considered an update of [5] It should be said that in many ways, proof theory seems a natural place to ....

J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. thesis, Department of Computer Science, University of Edinburgh, July, 1991. Published as Technical Report CST-81-91.

.... point for a given goal is reached, i.e. the value of j such that if T j (I) P 6j=j= G, then T (I) P 6j=j= G. The relationship between the powers of T and the success or failure level of a goal G is given in the lemma below. The proof of this and other results quoted here may be found in [7]. Lemma 5.1 Let P be a program, and let G be a goal. Then for any i 0 1. slevel P (G) i , T i (I ) P j=j= G 2. flevel P (G) i ) T i (I ) P 6j=j= G Note that the converse to 2 is not true. For a counterexample, consider an atom p with slevel P (p) flevel P (p) 1. Then from 1 we ....

J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. thesis, University of Edinburgh, forthcoming.

....existential case. This means that the above results will still hold in the presence of Negation as Failure. However the requisite technicalities are somewhat overwhelming, and we have not included them here for the sake of clarity. More details of this extension (including proofs) may be found in [6]. Note that the results above are expressed purely in terms of o , and so we think of the transformations given above as preserving operational equivalence. Maher [10] has shown that there are many natural notions of equivalence for logic programs, and in our context, another obvious notion of ....

....of :A with the success of A under Negation as Failure, and it seems that this may be a natural setting for investigations of logic programming. Further discussion on this point is beyond our scope; more on the relationship between operational and logical equivalence and I 0 may be found in [7, 6]. 4 Memoisation In the light of the above results, we may think of D normal programs as a normal form for D mod programs. We may imagine the transformation occurring at compile time, and so we may think of the above processes as statically converting the program into a more specific form. ....

J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. thesis, University of Edinburgh, forthcoming.

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J. Harland, On Hereditary Harrop Formulae as a Basis for Logic Programming, Ph.D. thesis, University of Edinburgh, July, 1991.

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J. Harland. On Hereditary Harrop Formulae as a basis for Logic Programming. PhD thesis, University of Edinburgh, UK, 1991.

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J. Harland. On Hereditary Harrop Formulae as a basis for Logic Programming. PhD thesis, University of Edinburgh, UK, 1991.

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