D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165(1):201--232, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[Kleene, 1951, Wallen, 1990, Lincoln and Shankar, 1994, Stone, 1999] Thus, unlike systems of structural scope, systems of explicit scope permit general goal directed reasoning. If we adopt Miller s characterization of uniform proof for sequent calculi with multiple conclusions [Miller, 1994, Miller, 1996] then any modal theorem has a uniform proof in a lifted, explicitly scoped inference system. Put another way, explicitly scoped inference assimilates modal proof to classical proof, and we know that uniformity is not really a restriction on classical proof [Harland, 1997, Nadathur, 1998] This ....

.... 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 an extension ....

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Miller, D. (1996). Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165:201--232.

....logic views logical assumptions as consumable resources. Therefore, in linear logic programming languages, a resource can be represented as a formula rather than a term. Several logic programming languages based on linear logic have been proposed: Lygon [7] LO [2] LinLog [1] Lolli [9] Forum [14], etc. In particular, Forum is complete for classical linear logic and its execution can be viewed as a goal directed proof search called uniform proof [15] However, the proof search of Forum programs is highly non deterministic. For example, in the sorting program shown in section 6, execution ....

D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165(1):201--232, 1996.

....which translate machine found proofs back into the usual sequent calculus. 1 Introduction Linear logic [12] has become known as a very expressive formalism for reasoning about action and change. During its rather rapid development linear logic has found applications in logic programming [14, 19], modeling concurrent computation [11] planning [18] and other areas. Its expressiveness, however, results in a high complexity. Propositional linear logic is undecidable. The multiplicative fragment (MLL) is already NP complete [16] The complexity of the multiplicative exponential fragment ....

....that all formulas must be used up in a proof. The # rule requires that all formulas in the context are of type #. Though the connectives of linear logic make proof search more di#cult they also give rise to new possibilities. Some applications for linear logic programming are illustrated in [19]. 171 a000 , a0100 a000 , o010 o a000 , #01 # a0010 , a # 0100 a0010 , o # 010 a0010 , # # 01 #001 , # # 01 #00 , #01 , # # 01 #00 , #01 c #0 lab(# # ) # # (A# A) ....

D. Miller. FORUM: A Multiple-Conclusion Specification Logic. TCS, 165(1):201-232, 1996.

....] where RA , UB , T# and V # correspond to the formulae A, B and the multisets of formulae # and #.Thetwos instances could be swapped:thesubstructures in the par context can be brought inside the times structure independently. We have no combinatorial explosion in the splitting of a times context [9, 11], which depends on the impossibility, in the sequent calculus, of representing the middle structure in the derivation above. In fact, the lazy splitting algorithm of [9] is here represented naturally and simply. System SBV is designed to ensure thesubformula property: all the rule premises are ....

....its atomic form, which is the key to modularity. One possible problem with our calculus is that, since rules apply anywhere deep into structures, proof search can be very non deterministic. Research is in progress in our group to focus proofs not only along lines induced by the logical relations [3, 11], but also based on the depth of structures. Classical logic is also studied. One can easily port additive rules to our calculus, but the question, again, is whether we can get decomposition and a modular cut elimination proof. Recent work, in preparation, by Brunnler and Tiu, shows that ....

Dale Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165:201--232, 1996.

....case can thus be revised and adapted to the non commutative case. 1 Introduction Linear Logic [4] has raised a lot of interest in computer research, especially because of its resource sensitive nature. One line of research, supported by systems such as LO [3] Lambda Prolog [8] Forum [9] or Lolli [7] studies proof construction procedures and their interpretation as computational models, in the Logic Programming tradition. An efficient proof search procedure for Linear Logic, based on a proof normalization result called Focusing , has been described in [2] Focusing is ....

D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165(1):201--232, 1996.

....logic views logical assumptions as consumable resources. Therefore, in linear logic programming languages, a resource can be represented as a formula rather than a term. Several logic programming languages based on linear logic have been proposed: Lygon [7] LO [2] LinLog [1] Lolli [9] Forum [14], etc. In particular, Forum is complete for classical linear logic and its execution can be viewed as a goal directed proof search called uniform proof [15] However, the proof search of Forum programs is highly non deterministic. For example, in the sorting program shown in section 6, execution ....

D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165(1):201--232, 1996.

....particular by Hodas Miller and Harland Pym. Hodas Miller have developed two systems for linear logic programming. CHAPTER 6. A SEQUENT CALCULUS FOR INTUITIONISTIC LINEAR LOGIC 186 The first of these, Lolli ( HM94] is based on a fragment of Intuitionistic Linear Logic. The second, Forum ([Mil96]) is based on full classical linear logic. Harland Pym s system, Lygon ( HP94] is based on a fragment of classical linear logic. This section will briefly describe Lolli (the language most closely related to SILL) and compare it with SILL. 6.7.1 SILL and Lolli In [MNPS91] the idea of a ....

....SILL appears to be too complicated to be practically used as a logic programming language, and its interest is restricted to its theoretical properties of naturally corresponding in a 1 1 with normal natural deductions, and hence giving a semantic rationale to Lolli. 6.7. 2 SILL and Forum Forum ([Mil96]) is another linear logic programming language. It is based on full classical linear logic and exploits the symmetries of linear logic to give a calculus for the whole of linear logic whilst avoiding the use of connectives that have rules which do not fit well with goal directedness. The calculus ....

D. Miller. Forum: A Multiple Conclusion Specification Logic. Theoretical Computer Science, 165(1):201--232, 1996.

....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 most common proof theoretic characterisation of logic programming languages is via goaldirected proofs [17] In ....

....logic significantly changes the overall analysis from that of classical logic. 1 Also, the determination of an appropriate notion of goal directed proof in the presence of presence of multiple conclusioned sequents is not obvious, and there are a number of different approaches to this issue[21, 12, 18, 19], which include restricting attention to a single conclusioned version, requiring don t care nondeterminism in the choice of goal formula (as in Forum[18] or allowing don t know nondeterminism in such choice (as in Lygon[21, 11] It should also be noted that single conclusioned and ....

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Miller D. Forum: A multiple-conclusion specification-logic, Theoretical Computer Science 165:1:201-232, 1996.

....logic views logical assumptions as consumable resources. Therefore, in linear logic programming languages, a resource can be represented as a formula rather than a term. Several logic programming languages based on linear logic have been proposed: Lygon [13] LO [4] LinLog [3] Lolli [18] Forum [33], etc. In particular, Forum is complete for classical linear logic and its execution can be viewed as a goal directed proof search called uniform proof [34] However, the proof search of Forum programs is highly non deterministic. For example, in a list sorting program, execution time rises ....

....can be used for e#cient memory management [29] and for a type system [1] The application to logic programming language is one of the most active topic in linear logic. Several logic programming languages based on linear logic have been proposed: Lygon [13] LO [4] LinLog [3] Lolli [18] Forum [33], HACL [28] etc. We discuss more about linear logic programming languages in the next chapter. 8 A # A (Axiom) # # A, # # # , A # # # #, # # # #, # # (Cut) Logical axiom and Cut rule # # A, # #A # # # ( # L ) #, A # # # # A # , # ( # R ) #, A # # #, A B # ....

D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165(1):201--232, 1996.

....and object oriented paradigms than with the logic one. A MAS Approach to Vehicle Monitoring in Motorway 7 E hhf . The linear logic language E hhf [3] is an executable language for modeling concurrent and resource sensitive systems based on the general purpose specification logical language Forum [10]. E hhf has a multiset based logic combining features of extensions of logic programming languages like Prolog, e.g. goals with implication and universal quantification, with the notion of formulas as resources at the basis of linear logic. A E hhf program is a collection of guarded ....

D. Miller. Forum: A Multiple-Conclusion Specification Logic. Theoretical Computer Science, 165(1), 1996.

....would use cut 4 (P,M ,x.M) Our own notation, chosen initially for the applications in [12] and a forthcoming verification [1] thereof in Coq) made it easier for us to see how to order terms as required for the proof using the r.p.o. theorem. We have adopted a notation used in logic programming [21] for judgments with a privileged or stoup formula rather than Herbelin s G;P Q (with P optional) Howard [18] also has a calculus which allows a bijective correspondence with (normal) natural deduction; but this correspondence no longer works well when disjunction is taken into account. The ....

Miller, D.: "FORUM: a multiple-conclusion specification logic", Theoretical Computer Science, 165 (1996), 201--232.

....facilities [16] and negations and disjunctions in the heads of clauses [17] Secondly, this process can be applied to logics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11] Forum[15], LinLog [1] LO [2] Lolli [12] ACL [13] and LC [21] A popular proof search strategy is the notion of goal directed proof [16] which, roughly speaking, requires that the goal be decomposed before the program, and hence the computation uses the program as a context, and the goal as the ....

....would look like in such systems, and what the results of the previous analysis would be. This is a particularly interesting question given that there has been a significant amount of investigation of notions of goal directed provability for multiple conclusioned systems such as linear logic [1, 15, 19, 21] and classical logic [10, 18] Thus it seems appropriate to investigate the design of logic programming languages via goal directed provability for a multiple conclusioned system for intuitionistic logic, and to compare the results with the single conclusioned case. 2 Preliminaries 2.1 Sequent ....

D. Miller. Forum: A Multiple-Conclusion Specification Logic. Theoretical Computer Science, 165(1):201--232, 1996.

.... written in LP languages tend to be somewhat more concise, and hence more comprehensible to a human, than the same programs written in an imperative language (such as C or Pascal) There have been various LP systems implemented (Prolog [44, 30] LO[3] Lolli[27, 28] LinLog[1] Prolog[35] Forum[34], Lygon[47, 49] etc) But still there is no precise concept of what constitutes a LP language as well as what is the essential difference between a LP system and a theorem prover. One proof theoretic characterisation and a criterion for identifying LP language is formalised in [33] by the ....

....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 ....

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Miller D. Forum: A multiple-conclusion specification-logic, Theoretical Computer Science 165(1): 201-232, 1996.

.... Also, programs written in LP languages tend to be somewhat more concise, and hence more comprehensible to a human, than the same programs written in an imperative language (such as C or Pascal) There have been various LP systems implemented (Prolog [15] Lolli[6] LinLog[1] Prolog[12] Forum[11], Lygon[17] etc) But still there is no precise concept of what constitutes a LP language as well as what is the essential difference between a LP system and a theorem prover. One proof theoretic characterisation and a criterion for identifying LP language is formalised in [10] by the ....

....The dynamics of the proof that results from the permutation analysis clearly has a strong influence on the operational semantics of the resulting LP language. For example, Lygon[17] is based on the search strategy that some permutations of righthand side rules will lead to a proof, whereas Forum[11] is based on the search strategy that any permutation of right hand rules will lead to a proof. Here we give a brief overview of some applications of permutations for the analysis of particular LP strategies. Example 1 (Defining a proof search strategy) Permutation properties of the inference ....

Miller D.: Forum: A multiple-conclusion specification-logic, Theoretical Computer Science 165, 201-232.

....can be regarded as a refinement of classical as well as of intuitionistic logic. It subsumes these logics because both of them can be embedded into linear logic. Mainly, linear logic has become known as a very expressive logic of action and change. It has found applications in logic programming [14,20], planing [19] modeling concurrent computation [11] and other areas. Its expressiveness, however results in a high complexity. Validity is undecidable for propositional linear logic. The multiplicative fragment is already NP complete [16] The complexity of the multiplicative exponential ....

D. Miller. FORUM: A Multiple-Conclusion Specification Logic. TCS, 165(1):201-232, 1996.

....led to the choice of the fragments studied in the literature. To achieve our main goal, i.e. to study concurrent objects in the linear logic setting, we commit ourselves to a particular interpretation of sequents and proofs, i.e. uniform proofs as computations, as proposed by Miller in (Miller 1996) with the language Forum. Forum is a presentation of full higher order linear logic. This choice is guided by previous work on extensions of logic programming, in which the logical connectives are given a precise operational interpretation in terms of search directives, see e.g. Miller 1989a; ....

....linear logic. This choice is guided by previous work on extensions of logic programming, in which the logical connectives are given a precise operational interpretation in terms of search directives, see e.g. Miller 1989a; Miller 1989b; Andreoli and Pareschi 1991; Hodas and Miller 1994; Miller 1996). In a sense, Forum (Miller 1996) can be viewed as an intermediate refinement step to achieve a readable form of linear logic specifications. Our final refinement step is to consider a specific sublanguage, namely E hhf (Delzanno 1997; Delzanno and Martelli 1998) by which we emphasize the view ....

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D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165(1):201--232, 1996.

....and proof search points of view. Let us recall that ILL can be seen as a model of Petri nets [14, 16] and also as a foundation of functional and logic linear programming languages. Moreover recent specification logics are based on fragments of ILL (like Lolli [25] or CLL (like Forum [28]) the choice being motivated by specific operational aims, for instance sequentiality in ILL and concurrency in CLL. Therefore, the study of intuitionistic provability from classical provability in linear logic would have an important impact on proof search and semantics analysis. In section 2, ....

D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165(1):201--232, 1996.

....facilities [16] and negations and disjunctions in the heads of clauses [17] Secondly, this process can be applied to logics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11] Forum[15], LinLog [1] LO [2] Lolli [12] ACL [13] and LC [21] A popular proof search strategy is the notion of goal directed proof [16] which, roughly speaking, requires that the goal be decomposed before the program, and hence the computation uses the program as a context, and the goal as the ....

....would look like in such systems, and what the results of the previous analysis would be. This is a particularly interesting question given that there has been a significant amount of investigation of notions of goal directed provability for multiple conclusioned systems such as linear logic [1, 15, 19, 21] and classical logic [10, 18] Thus it seems appropriate to investigate the design of logic programming languages via goal directed provability for a multiple conclusioned system for intuitionistic logic, and to compare the results with the single conclusioned case. 2 Preliminaries 2.1 Sequent ....

D. Miller. Forum: A Multiple-Conclusion Specification Logic. Theoretical Computer Science, 165(1):201--232, 1996.

....notion of state and with aspects of concurrency. The operational semantics of this class of languages is given via a sequent calculi presentation of the corresponding fragment of linear logic. Special classes of proofs like the focusing proofs of Andreoli [And92] and the uniform proofs of Miller [Mil96] allow us to restrict our attention to cut free, goal driven proof systems that are complete with respect to provability in linear logic. These presentations of linear logic are the natural counterpart of the traditional top down operational semantics of logic programs. In this paper we ....

....viewed as a set of clauses. According to this view, the operational semantics of LO is given via the uniform (goal driven) proof system defined in Fig. 1. The proof system is a specialization of more general uniform proof systems for linear logic like Andreoli s focusing proofs [And92] and Forum [Mil96]. In Fig. 1, P is a set of implicational clauses, A denotes a multiset of atomic formulas, whereas Delta denotes a multiset of G formulas. Intuitively, a sequent is provable if all branches of its proof tree terminate with instances of the r axiom. The rule bc denotes a backchaining step ....

D. Miller. Forum: A Multiple-Conclusion Specification Logic. Theoretical Computer Science, 165(1):201--232, 1996. 22

....consumed hypotheses can not be used again. Therefore, in linear logic, a resource can be represented as a formula rather than represented as a term. There have been several proposals for logic programming language based on linear logic: LO [2] ACL [7] Lolli [6] 3] Lygon [9] 5] and Forum [8]. Lolli a and Lygon b are implemented as interpreter systems (on SML and #Prolog for Lolli, on Prolog for Lygon) But, none of them have been implemented as a compiler system. BinProlog 5.00 can compile linear implications of a#ne logic (a variant of linear logic) but other connectives are ....

D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science. (to appear).

....fgiorgio,martellig disi.unige.it phone: 39 10 3536727 (Martelli) Abstract Girard s linear logic [18] provides powerful means for studying state transformations and resource consumption in computations within a completely logical framework. The starting point of this work is Forum [22, 23], a presentation of higher order linear logic which is an abstract logic programming language[24] i.e. complete with respect to uniform proofs (cut free and goal directed proofs) A subset of Forum s formulas which form the logic programming language E hhf [11, 13] have been isolated and ....

.... logical way allows to overcome various problems related with traditional approaches (think about the primitives assert and retract in Prolog and the various update languages in deductive databases) Various languages have been developed; these include LO [1, 2, 3] ACL [21] Lolli [20] and Forum [22, 23]. Forum, in particular, is a presentation of full linear logic which is complete with respect to uniform proofs, i.e. cut free and goal directed ones. The language E hhf represents an attempt to 1 isolate a subset of Forum s connectives which is both rich enough to model the notions reported ....

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D. Miller. Forum: A Multiple-Conclusion Specification Logic. Theoretical Computer Science, 165(1):201--232, 1996.

....modern and powerful specification languages. Specific features have been separately studied in previous works: object orientation and concurrency in LO [3, 2] and ACL [29] deductive databases in [27] logic programming in [25] communicating processes in [35] imperative programming paradigms in [11, 38], and transition systems in [42] The importance of such works is not limited to the development of new high level languages. In fact, they can help in the analysis of declarative programs properties and in the verification of their correctness. Experimental results in this field have been ....

....to logical characterizations of objects and agents. In our current work [12] we are looking for a uniform representation of all above mentioned aspects within a unique framework. We have developed a formalism, called E hhf [12] based on a subclass of proofs, sequents and formulas of Forum [38]. The interpretation of these restricted formalism is much closer to the real computational aspects present in the new high level languages we are interested in. Forum is a formulation of Higher Order Linear Logic which has the very nice property of providing uniform proofs only. This class of ....

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D. Miller. Forum: A Multiple-Conclusion Specification Logic. Theoretical Computer Science, 165(1):201--232, 1996.

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D. Miller. Forum: A multiple-conclusion specification logic. Theoretical Computer Science, 165:201--232, 1996.

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