Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992. 13  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the sequent calculus is very non deterministic which makes proof search rather ineffective. A complete strategy that cuts down the search space considerably is to alternate phases of invertible steps and noninvertible steps. This strategy of focusing proofs has been formulated by Andreoli [And92] for linear logic and adapted to intuitionistic linear logic [Pfe99] We distinguish four classes of rules. 1. Invertible right rules: oeR, 8R and R. These introduce new hypotheses or parameters, or split the conclusion (the goal) respectively. The axiom R can be added to this group since ....

....distinctions between invertible and non invertible rules, the distinctions between left and right rules, and focusing. This means our design is likely to be applicable to a rich variety of systems for which focusing system can be designed (including classical logic [Hua94] classical linear logic [And92] intuitionistic linear logic, modal logics, temporal logics, and others) Related Work. Lamport gives general guidelines on how to structure rigorous proofs in [Lam93] Comparable to our linear proof syntax with frames given in Sect. 2 is the Block Calculus by Dahn and Wolf [DW94] based on ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992. 13

....on tabled higher logic programming. Similar to the development of uniform proofs for a fragment of intuitionistic logic, we can design a higher order logic programming language based on linear logic. Different proposals for linear logic programming languages exist such as Lolli [19] or LinLog [1] . Similar to the logical DRAFT December 6, 2001 18 : 29 8. EVALUATION 27 framework LF on which the language Elf is based, Cervesato and Pfenning [5] developed the linear logical framework and a linear logic programming language LLF which is based on additive product, additive unit and linear ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

....the sequent calculus is very non deterministic which makes proof search rather ine ective. A complete strategy that cuts down the search space considerably is to alternate phases of invertible steps and noninvertible steps. This strategy of focusing proofs has been formulated by Andreoli [And92] for linear logic and adapted to intuitionistic linear logic [Pfe99] We distinguish four classes of rules. 1. Invertible right rules: R, 8R and R. These introduce new hypotheses or parameters, or split the conclusion (the goal ) respectively. The axiom R can be added to this group since ....

....distinctions between invertible and non invertible rules, the distinctions between left and right rules, and focusing. This means our design is likely to be applicable to a rich variety of systems for which focusing system can be designed (including classical logic [Hua94] classical linear logic [And92] intuitionistic linear logic, modal logics, temporal logics, and others) Related Work. Lamport gives general guidelines on how to structure rigorous proofs in [Lam93] Comparable to our linear proof syntax with frames given in Sect. 2 is the Block Calculus by Dahn and Wolf [DW94] based on ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297-347, 1992. 13

....certain permuted form. That is, one may permute the order of application of inferences in a proof to achieve a kind of normal form. Permutability properties are used to give control over the shape of linear logic proofs, thus enabling a more direct computational reading, see Lincoln [28] Andreoli [5], Hodas and Miller [20] and Lincoln and Scedrov [24] The expressiveness of linear logic proof search as a computational paradigm is brought to light in a body of research over the past several years. Previous work of Miller and others [29] had developed several examples in data bases, theorem ....

....languages was inadequate. In each of these cases, a logic programming language based on linear logic proof search provided correct specifications. In a number of papers Andreoli and Pareschi develop a declarative treatment of object communication and concurrent object oriented computation [3, 4, 5]. As part of this work, Andreoli and Pareschi develop a concurrent specification language, LO [4] The work of Miller and Hodas [20] describes a linear logic programming language that refines the logics behind both Prolog and Prolog. A detailed analysis of this linear logic programming language ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3), 1992.

....eld is eliminating as much as possible unnecessary, spurious non determinism in the bottom up search for a proof. Forum succeeds in eliminating all the non determinism due to permutability of inference rules in linear logic, and employs a clever technique, called focusing and due to Andreoli [4], to keep the search for a proof nailed to certain choices done during a computation. Unfortunately, being intimately tied to the sequent calculus, Forum su ers from the very same context partitioning problem of the tensor rule. Therefore, it will be interesting for me to see whether my system can ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297-347, 1992.

....The other logic programming languages we have mentioned can, of course, capture the expressiveness of full logic by introducing non logical constants and programs to describe their meaning. Felty in [Fel93] uses a meta logical presentation to specify full logic at the object level. Andreoli [And92] provides a compilation like translation of linear logic into LinLog (of which LO is a subset) Forum has a more immediate relationship to all of linear logic since no non logical symbols need to be used to provide complete coverage of linear logic. Of course, to achieve this complete coverage, ....

..... A p is the head of such a clause. If p = 0 then we say that this clause has an empty head. The formulas of LinLog [And92] are essentially clauses in which p 0 and the formula G 1 ; Gm do not contain Gammaffi and ) and where has only atomic scope. 6.2 Proof system for Forum If we add to the Lolli language in the preceding section the connective . ....

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Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

.... we follow Fitting [4, 5] and Boolos [2] Our approach to multimodal logics is similar to the presentation in Halpern and Moses [6] Tense logics are implemented via a calculus developed by Zimmermann [14] which is based on Rescher and Urquhart [10] In the case of linear logic we follow Andreoli [1] and Tammet [13] Decision Procedures: All decision procedures for monotonic systems, which are implemented so far, are based on sequent calculi. We took the standard forms of these systems and combined them with several methods that improve the runtime complexity considerably. For each logic ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

....the cut elimination property is used simply to allow one to concentrate on 4 cut free proofs without loss of generality. Certain permutability properties, which yield optimized presentations of cut free proofs play a central role in the proof search paradigm, see Lincoln [76] Andreoli [11]. 2.1 Proof Reduction Paradigm The earliest work in this direction was Lafont s investigation of a functional programming language implementation in which garbage collection was replaced by explicit duplication operations based on linear logic [63] Another possible application in functional ....

....theories. The remarkable expressiveness of this computational paradigm is brought to light in a body of research accumulated over the past three years. In a number of papers Andreoli and Pareschi develop a declarative treatment of object communication and concurrent object oriented computation [8, 9, 11, 10]. A treatment of linear logic programming is given by Hodas and Miller [59] A role of linear logic in a declarative semantics of SLDNF resolution is considered by Cerrito [32] An approach that spans both the proof reduction and the proof search paradigms is proposed by Girard [51] Recent ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3), 1992.

....[11] The foundation of logic programming languages by means of proof theory has been investigated by Miller et al. in the seminal work [16] where the concept of uniform proof is defined. In the case of logic programming languages based on linear logic, the concept has been further explored in [14, 1, 2, 8]. The use of a clean proof theory as a foundational 2 Paola Bruscoli and Alessio Guglielmi tool has been proved very effective for studying semantics, analysis, transformation and verification issues. Uniform proofs are a logic programming view of the process of making proofs, in the sense that ....

....left is decided to focus on, and left rules may be applied. Since it can be shown that the order at the right of the sequents is not important, we have that the system shown produces uniform proofs only. The reader may find many relationships with the work of Andreoli on LinLog and focusing proofs [1]. It holds the following [15] Theorem Being G 2 , then G has a proof in linear logic iff ; Gamma G has a proof in Forum. This is enough to conclude that Forum is able to express the whole of linear logic, since the missing connectives are directly translatable into the available ones without ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

....of logic programming languages by means of proof theory has been investigated by Miller et al. in the seminal work [23] where the concept of uniform proof 2 Alessio Guglielmi is defined. In the case of logic programming languages based on linear logic, the concept has been further explored in [22, 1, 3, 17]. The use of a clean proof theory as a foundational tool has been proved very effective for studying semantics, analysis, transformation and verification issues. Following these guidelines we define the language LM. It is made up of the linear implication connective ffi Gamma (written reversed, as ....

....programming language 17 a non tree structure in the evolution of concurrent processes. In particular, here processes have complete view of other processes and very different forms of interaction are possible, not only communication through variable sharing. SMR is completely realizable in LinLog [1] (which is a superset of LM) and in particular in the asynchronous fragment of linear logic. We plan to fit SMR in a categorical construction we are presently pursuing in order to give a very general semantical characterization to multiset rewriting languages, following the approaches of [7, 8, ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

....C The other logic programming languages we have mentioned can, of course, capture the expressiveness of full logic by introducing non logical constants and programs to describe their meaning. Felty in [7] uses a meta logical presentation to specify full logic at the object level. Andreoli [1] provides a compilationlike translation of linear logic into LinLog (of which LO is a subset) Forum has a more immediate relationship to all of linear logic since no non logical symbols need to be used to provide complete coverage of linear logic. Of course, to achieve this complete coverage, ....

.... Delta . A p is the head of such a clause. If p = 0 then we say that this clause has an empty head. The formulas of LinLog [1] are essentially clauses in which p 0 and the formula G 1 ; Gm do not contain Gammaffi and ) and where has only atomic scope. 3 Proof Search In this section we consider the abstract character of cut free proofs over the connectives of Forum. Let L 1 be the set of all formulas over ....

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Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

....predicate logic (Horn clauses, generalized Horn clauses) a number of novel approaches have been proposed during recent years. The following are some examples. LP in the one sided sequent calculus of classical linear predicate logic (LinLog formulae and focusing proofs) Andreoli and Pareschi [And92] proposed a language of logic programming in classical linear logic suitable for actor programming. LP in the sequent calculus of intuitionistic higher order predicate logic (hereditary Harrop formulae and uniform proofs) Miller et al. MNPS91] promoted programming with hereditary Harrop ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

....The M proof produced by algorithm 2 is use valid. Proof: Algorithm 2 does not introduce Use rules and all of the lemmas used yield use valid proofs. Xi 6 Comparison with Other Work There are a number of logic programming languages based on linear logic. Some of these, like LinLog [1] are based on proof theoretic analyses, as Lygon is, but, to the authors knowledge, have not been implemented and hence do not involve the problems of lazy splitting discussed in this paper. Others, like ACL [18, 25, 17] and LO [4, 5, 3] use linear logic as motivation and a design guide for ....

....N (b ( a Phi b) are logically equivalent however the first is not a valid goal formula and does not have a uniform proof even though the second does have a uniform proof. Secondly, and perhaps more importantly, our presentation allows us to simply use the permutability properties explored in [6, 1] to reduce the nondeterminism associated with selecting the formula to be reduced. In the 2 Actually ACL includes Omega but only in contexts which prevent lazy splitting from being needed. Lolli encoding selecting the next formula to be reduced is done by the resolution rule. Reducing the ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3), 1992.

....the similar result for Lolli, but this time accounting for the . and connectives. However, Andreoli has a result in [And92] that he calls focused proofs and he shows that focused proofs are complete for linear logic. It is possible to translated between focused proofs and the proofs in F . Completeness for F then follows from completeness for focused proofs. 7.2 Further readings The material in this section is taken ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

....left hand rules will always have an instance of one of the decide rules as its final inference, and these apply only when the right hand side contains only atomic formulas. The proof that this system is complete for Forum follows directly from Andreoli s work on proof search in full linear logic [1]. While Forum contains only half the operators of linear logic, it can nevertheless be considered to be complete for full linear logic since the remaining operators can be encoded using the following equivalences: B j B Gammaffi 0 j Gammaffi 1 j Gammaffi B j (B ) Gammaffi ....

..... C ) Notice that the operators handled directly in Forum are exactly those that Andreoli classified as asynchronous, while those that are handled by encoding are the synchronous operators [1]. 4 Backchaining in Forum The completeness of uniform proofs is the essential requirement for classifying a logic as an abstract logic programming language. However, in rich logics which admit complexly structured program clauses, when the goals are all atomic it may still not be apparent how to ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297-347, 1992.

....by translations to and from the system CLV introduced in the next section. 4. 2 CLV : A Two Sided Formulation of Classical Linear Logic with Four Zones In this section we present a formulation of a sequent calculus for linear logic, called CLV very close to Andreoli s dyadic system Sigma 2 [And92] and similar to Girard s LU [Gir93] This formulation was specifically devised to allow a simple proof of cut elimination. A sequent has the form Psi; Gamma = Delta; Theta which may be interpreted as Psi; Gamma = Delta; Theta in ordinary linear sequent calculus. Thus the outer zones in ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):197--347, 1992.

....linear sequent calculus. Roorda [Roo91] gives a different proof of cut elimination by generalizing the cut rule to multiple occurrences of modal formulas. The main challenge is to isolate the non linear reasoning and the associated structural rules. Our solution is close to Andreoli s Sigma 2 [And92] and Girard s LU [Gir93] in that we divide a sequent into linear and non linear zones, and that we have several forms of cut. The structural aspects of non linear reasoning are treated in the manner of the earlier sections. This leaves a version of dereliction as the only structural rule, and it ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):197--347, 1992.

.... of this paper are closest to what may be broadly called logic programming ramifications, in which computation is expressed by cut free proof search in certain linear logic theories [16] Recent topics in this direction include a treatment of object style inheritance and linear logic programming [1, 2, 4], a treatment of concurrent constraint programming [14] and a closely related treatment of Milner s calculus [12] The expressiveness of cut free linear logic proof search as a computational paradigm is also indicated by the complexity and undecidability of provability in fragments of linear ....

....For the record: Theorem 2.1 If a sequent is provable in mall1, then it is provable in mall1 without using the Cut rule. We end this section by considering some technical properties of mall1, in particular so called permutability properties, some of which have been discussed in several texts [2, 11]. The sign of a formula is defined using the transformations below, where is any binary connective except Gammaffi and where Q is either of the two quantifiers: A GammaffiB] A] Gamma Gammaffi[B] A B] A] B] QX:A] QX: A] Sigma; A] Sigma] ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3), 1992.

....more controlled than in CLL, a system with explicit weakening, contraction, and dereliction for modal formulas. This is achieved by dividing a sequent into linear and non linear zones whose constituents are treated differently in the sequent rules. Examples of such calculi are Andreoli s Sigma 2 [And92], Girard s LU [Gir93] and Hodas Miller s L [HM94, Hod94] We take a two sided version of classical linear logic quite close to Sigma 2 with three rules of Cut as in LU and endow the resulting calculus LV with proof terms. We then prove admissibility of the cut rules in LV by three nested ....

.... AOB j (A Omega B ) j 1 A Phi B j (A NB ) 0 j 9x: A j (8x: A ) 3 LV: Another Sequent Calculus for Linear Logic In this section we present a formulation of a sequent calculus for linear logic, called LV very close to Andreoli s dyadic system Sigma 2 [And92] and similar to Girard s LU [Gir93] It may also be considered a complete classical analogue of Hodas Miller s L [HM94, Hod94] a formulation of a fragment of intuitionistic linear logic. LV will be amenable to a structural proof of cut elimination following ideas from an analysis of ....

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Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):197--347, 1992.

....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 properties of computations. We only mention a few of these examples here. Intuitionistic logic has been used to specify both the dynamic and static ....

Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297--347, 1992.

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