Harland, J. (1994). A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):69--88.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....goal directed proof is possible in a logic if and only if any theorem has a uniform proof. In systems of structural scope, this is not possible, and we must instead restrict our to inference in specific logical fragments, as described for the intuitionistic case in, e.g. Miller et al. 1991, Harland, 1994, Harland et al. 2000] By contrast, systems of explicit scope can be lifted by a suitable analogue to the HerbrandSkolem Godel theorem for classical logic so that any pair of unrelated inferences can be interchanged [Kleene, 1951, Wallen, 1990, Lincoln and Shankar, 1994, Stone, 1999] Thus, ....

.... case of structural control of inference that has attracted particular interest is linear logic, where linear disjunction must be understood to specify synchronization between concurrent processes rather than proof by case analysis; see, e.g. Andreoli, 1992, Hodas and Miller, 1994, Pym and Harland, 1994, Miller, 1996, Kobayashi et al. 1999] The investigation of fragments of linear logic remains essential, as linear logic has no analogue of an explicitly scoped proof system, and so unlike intuitionistic logic and modal logic must be understood as a refinement of classical logic rather than ....

Harland, J. (1994). A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):69--88.

....included in this introduction is the link between permutation free sequent calculi and logic programming. One view of logic programming is that it is about backwards proof search (as in proof enumeration) in constructive logics. This view is laid out by Miller et al. in [MNPS91] see also [Har94]) We describe goal directed proof search in the Horn formula and hereditary Harrop formula fragments of first order intuitionistic logic, as given in [MNPS91] We also present calculi for goal directed proof search in these fragments. 1.4.1 Uniform Proofs and Abstract Logic Programming ....

....the development of the view of logic programming as the backwards search for a proof of a formula in a constructive logic. The hereditary Harrop formula fragment of intuitionistic logic can be seen as the maximal fragment of intuitionistic logic for which goal directed proof search is complete ([Har94]) Logic programming is not just about what is provable, but about how something is proved proof enumeration, not just theorem proving. If one holds the view that the proofs that should be enumerated are normal natural deductions, then one would like a suitable system for enumerating these ....

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J. Harland. A Proof-Theoretic Analysis of Goal-Directed Provability. Journal of Logic and Computation, 4(1):69--88, 1994.

....If it is not provable, the answer is no . Such a programming system needs an efficient method of searching for a proof. Prolog is a traditional logic programming language based on Horn clauses, a fragment of classical first order logic for which there are efficient implementation techniques [1]. Prolog s expressiveness is restricted because programmers have only Horn clauses with which to represent their problem. It is also restricted by the underlying logic (classical first order) Classical logic contains three structural rules that determine its expressive power. In particular, two ....

Harland, J., "A proof-theoretic analysis of goal-directed provability," J. Logic Computat., Vol. 4 No.1, 1994, pp. 69--88

....which a goal is decomposed before the 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 [5] and hereditary Harrop formulas [10] 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 logic, light linear logic, affine logic, relevant linear logic, etc. ....

....the basis of a given problem to be solved. Still there is no clear technical statement of why, and what kind of restrictions on some classes of formulas are necessary, in order to use classes of formulas in logic programs. There have been various proof theoretic techniques, approaches and analyses [1, 2, 3, 5, 6, 9, 13] used to design and analyse LP languages. Many of the existing approaches and analyses which lead to some of the answers are all rather sophisticated and involve complex manipulations of proofs. Many are restricted to particular logic or classes of formulae. Almost all are designed for analysis on ....

Harland J.: A proof-theoretic Analysis of Goal-directed Provability, Journal of Logic and Computation, Vol.4 No 1 (1994) 69-88.

....of the LL inference rules in the corresponding sequent calculus. Example are given by the proof search strategies defined in (Galmiche and Perrier 1994a; Lincoln and Shankar 1994; Tammet 1994) and by the classes of proofs defined in (Andreoli 92; Galmiche and Perrier 1994b; Hodas 1994; Pym and Harland 1994), that are complete w.r.t. the provability. Let us analyse how we can naturally design and justify such strategies and proofs for a given logical fragment. For this purpose, we recall a general, two steps, method applied to LL and to other fragments in (Galmiche and Perrier 1994a; Galmiche and ....

J. Harland. A Proof-theoretic Analysis of Goal-directed Provability. Journal of Logic and Computation, 4(1):69--88, 1994.

.... ; G can be used to represent the state of an idealized logic programming interpreter where is the signature, or current set of non logical constants, is the current program and G is the current goal. These ideas have been discussed, for intuitionistic, classical and linear logics, in [67, 69, 107, 109, 113, 142, 152]. Languages which adopt this point of view include Prolog [113] Lolli [74] Lygon [69, 142, 171] Forum [108] and Elf [132, 133] This study of searching for normal proofs, together with the connected studies about permutability and reduction of non determinism, often leads to new equivalent ....

J. Harland. A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):69-88, 1994.

....with restrictions on de nite formulae (i.e. resources) goal formulae and queries de nitions. They mainly take as conceptual departure a speci c proof form (focusing or uniform proof) that is complete) assuming a goal directed proof search strategy as characteristic feature of logic programming [8]. In [2] the main point is to treat Horn clauses with multiple atoms in the head, in [10] it is to use resolution in a large LL fragment and in [12] considering some computational problems, it is to make a suitable logic programming language in a fragment of intuitionistic linear logic. For ....

.... works devoted to the de nition of such classes, for logic programming, mainly based on uniform proofs and the class of hereditary Harrop formulae of intuitionistic logic [16] Let us mention that they assume that the characteristic feature of logic programming is the goal directed proof search [8]. The notion of uniform proof has been adapted to adequate fragments of linear logic [12] for which a suitable proof search method, for instance resolution proof [9, 10] can be de ned. Here, we will analyze the general concept of canonical proof in linear logic, dedicated to frameworks based on ....

J. Harland. A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):6988, 1994.

....(dedicated here to program synthesis) but also applicable to applications based on proof search as computation paradigm. 4 Proof search in ILL Recent works have been devoted to proof search methods and to their applications to various topics of computer science as logic (concurrent) programming [2, 7, 10] or concurrency [10, 13] These various results are based on different fragments of (intuitionistic or not) LL, different proof search approaches (top down or bottom up proof construction, goal oriented proof search) 4, 5, 14] and lead to proposals for designing automated proof search procedures. ....

J. Harland. A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):69--88, 1994.

....G can be used to represent the state of an idealized logic programming interpreter where Sigma is the signature, or current set of non logical constants, Delta is the current program and G is the current goal. These ideas have been discussed, for intuitionistic, classical and linear logics, in [67, 69, 107, 109, 113, 142, 152]. Languages which adopt this point of view include Prolog [113] Lolli [74] Lygon [69, 142, 171] Forum [108] and Elf [132, 133] This study of searching for normal proofs, together with the connected studies about permutability and reduction of non determinism, often leads to new equivalent ....

J. Harland. A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):69--88, 1994.

....are fffij convertible. In [MNPS91] it is suggested that an abstract logic programming language should be characterised by the completeness of uniform proofs; there it is shown that this is the case for first order hereditary Harrop sequents. For first order intuitionistic logic it is shown in [Har94] that hereditary Harrop formulae are essentially the largest fragment for which uniform proofs are complete. The works [MNPS91] Har94] are merely concerned with completeness of derivability, i.e. if there is a proof then there is a (simple) uniform proof. However, to integrate logic and ....

....of uniform proofs; there it is shown that this is the case for first order hereditary Harrop sequents. For first order intuitionistic logic it is shown in [Har94] that hereditary Harrop formulae are essentially the largest fragment for which uniform proofs are complete. The works [MNPS91] [Har94] are merely concerned with completeness of derivability, i.e. if there is a proof then there is a (simple) uniform proof. However, to integrate logic and functional programming we are also concerned with the structural theory of proofs, i.e. one wants to capture all the different proofs under some ....

J. Harland. A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):69--88, 1994.

....and their computation may be identified as searching for proofs: given a program P and a goal G we attempt to satisfy G by searching for a proof of P G using the inference rules of a given logic. There have been various proof theoretic approaches to the design of logic programming languages [1, 2, 5, 7, 12, 16, 21] and a corresponding variety of languages implemented (Prolog [24] Lolli[12] LinLog[1] Prolog[20] Forum[18] Lygon[26] among others) However, despite many similarities in such analyses, the issue of a criterion for the identification of logic programming languages remains problematic. The ....

Harland J. A Proof-theoretic Analysis of Goal-directed Provability, Journal of Logic and Computation 4:1:69-88, 1994.

....terms, this represents a cut; when searching for a proof of P G, we first determine a program P 0 such that P V P 0 and P 0 G. The key technical question is then to find the appropriate rules for . This procedure is reminiscent of interpolation results, in particular those of [7], in which it is shown that for any uniform proof of P G, there is a program P 0 such that P V P 0 and P 0 G, where the latter proof is considerably simpler than the original one. In particular, P 0 can be constructed from G by replacing indefinite formul (such as p q) with ....

J. Harland. A Proof-Theoretic Analysis of Goal-Directed Provability. Journal of Logic and Computation 4:1:69-88, January, 1994.

....calculus for intuitionistic logic is LJ, which is single conclusioned. It is known that hereditary Harrop formulae are a logic programming language in intuitionistic logic, using goal directed proof search in LJ. Further, there is evidence that this class of formulae is, in some sense, maximal [9] (at least for the first order case) Thus it would seem that the identification of logic programming languages in intuitionistic logic is a solved problem. However, it is less widely known that there are multiple conclusioned sequent calculi for intuitionistic logic [22] Whilst these are not as ....

J. Harland. A Proof-Theoretic Analysis of Goal-Directed Provability. Journal of Logic and Computation, 4(1):69--88, January 1994.

....At the same time, in general, we need complete and tractable classes of proofs with a view to automating and optimising the proof search. The relationship between the logical fragment and the proof strategy is fundamental. There have been various proof theoretic techniques, approaches and analyses [1, 2, 13, 23, 28, 32, 34, 38, 47] used to design and analyse LP languages. Many of the existing approaches and analyses which lead to some of the answers are all rather sophisticated and involve complex manipulations of proofs. Many are restricted to particular logic or classes of formulae. Almost all are designed for analysis on ....

Harland J. A proof-theoretic Analysis of Goal-directed Provability, Journal of Logic and Computation, Vol.4 No 1, 1994, pp.69-88.

....of a formula F with F Gammaffi and thereafter treat as a distinguished atom. 2.2 Uniformity We exploit our analysis of the permutability properties of the linear connectives to obtain our notion of uniform proof for linear logic. This notion differs from that of intuitionistic logic [19, 9] rather delicately. Here we must permit certain occurrences of left rules, other than Gammaffi L, nearer to the endsequent than certain occurrences of right rules but still retain the ability to obtain a computationally acceptable, goaldirected notion of resolution proof , i.e. one in which we ....

J.A. Harland. A Proof-Theoretic Analysis of Goal-directed Provability. Accepted for publication in J. Log. Comp.. To appear.

....sequent is the introduction rule for the principle connective of the succedent. In this way, the sequence of steps in the search for a uniform proof is determined by the structure of the succedent, i.e. the goal. This particular notion of goal directed provability has been studied in various ways, [9, 8, 3], and has also lead to similar investigations in linear logic [4] Whilst the connections between intuitionistic logic and computation are various and well known, from the point of view of proof search, there seems to be no inherent reason why we cannot use classical logic to find a logic ....

....such as natural deduction, Hilbert type systems or tableau systems, or model theoretic means. 2 Notions of Classical Goal Directedness A proof theoretic characterization of goal directed provability is the notion of a uniform proof, which has been studied extensively in the intuitionistic case [8, 9, 3]. Intuitively, a uniform proof requires that the right rules be used closer to the root of the proof tree than the left rules. Definition 2.1 (Uniform intuitionistic proofs) An intuitionistic proof Xi is uniform if for every sequent Gamma Gamma F in Xi in which F is non atomic formula, the ....

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J. Harland, A Proof-Theoretic Analysis of Goal-Directed Provability, accepted for publication in the Journal of Logic and Computation. To appear.

....terms, this represents a cut; when searching for a proof of P G, we first determine a program P 0 such that P V P 0 and P 0 G. The key technical question is then to find the appropriate rules for . This procedure is reminiscent of interpolation results, and in particular those of [7], in which it is shown that for any uniform proof of P G, there is a program P 0 such that P V P 0 and P 0 G, where the latter proof is considerably simpler than the original one. In particular, P 0 can be constructed from G by replacing indefinite formulae (such as p q) with ....

J. Harland. A Proof-Theoretic Analysis of Goal-Directed Provability. Journal of Logic and Computation 4:1:69-88, January, 1994.

....calculus for intuitionistic logic is LJ, which is single conclusioned. It is known that hereditary Harrop formulae are a logic programming language in intuitionistic logic, using goal directed proof search in LJ. Further, there is evidence that this class of formulae is, in some sense, maximal [9] (at least for the first order case) Thus it would seem that the identification of logic programming languages in intuitionistic logic is a solved problem. However, it is less widely known that there are multiple conclusioned sequent calculi for intuitionistic logic [22] Whilst these are not as ....

J. Harland. A Proof-Theoretic Analysis of Goal-Directed Provability. Journal of Logic and Computation, 4(1):69--88, January 1994.

.... particular, this notion of proof has lead to the study of the class of formulae known as hereditary Harrop formulae, which may be used as the basis of both first order and higher order logic programming languages [15] There is some evidence that this class of formulae is, in some sense, maximal [7] (at least for the first order case) Thus it would seem that the identification of logic programming languages in intuitionistic logic is a solved problem. However, it is less widely known that there are multiple conclusioned sequent calculi for intuitionistic logic [26] Whilst these are not as ....

J. Harland. A Proof-Theoretic Analysis of Goal-Directed Provability. Journal of Logic and Computation, 4(1):69--88, January 1994.

....: A j 8xD j D D j G oe A G : A j 9xG j 8xG j G G j G G j D oe G where A is an atomic formula. This class of formulae is known as hereditary Harrop formulae. There is some evidence to suggest that this is the largest class of formulae for which (intuitionistic) uniform proofs are complete [8]. Note that this class of formulae properly includes Horn clauses, and that classical and intuitionistic logic coincide on Horn clauses. For more details about uniform proofs see [22] The notion of a goal directed proof in linear logic is significantly more complicated than in intuitionistic ....

James Harland. A proof-theoretic analysis of goal-directed provability. Journal of Logic and Computation, 4(1):69--88, 1994.

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