D. Miller. Forum: A Multiple-Conclusion Speci cation Logic. Theoretical Computer Science, 165(1):201-232, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... For example, Miller outlines a translation from the calculus into linear logic: processes become LL propositions and calculus reduction becomes LL entailment [Mil92] These ideas are generalized and reformulated as a logical framework in Miller s proposal for the speci cation logic Forum [Mil96, Chi95]. Forum o ers a paradigm for viewing logic programming as concurrent computation, and so it is likely that the logic programming interpretation for CLF will draw heavily from it. However, in Forum proofs cannot be manipulated as rst class objects not even cut elimination is treated, let al..one an ....

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

..... 5.00. 1. INTRODUCTION Linear logic [20] provides a logical characterization of concepts and mechanisms peculiar to concurrency like locality, recursion, non determinism in the de nition of a process [3, 22] and synchronization. Following the paradigm of proofs as computations proposed in [2, 28], provability in fragments of linear logic can be used then as a formal tool to reason about behavioural aspects of concurrent systems (see e.g. 31] In other paradigms for concurrency like the theory of Petri Nets there exist however a number of consolidated algorithmic techniques for the ....

....with nested conjunctive and disjunctive goals) In our setting, a natural way of augmenting the expressivity of the speci cation language is to consider rst order fragments of linear logic. First order formulas can be used, in fact, to color the internal state of processes with structured data [3, 28]. The combination between rst order formulas and linear connectives provides a well founded interpretation of the dynamics in the evolution of the internal state of a process [3, 27, 28] First order quanti cation in goal formulas has several interesting interpretations here: it can be viewed ....

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

..... 30 7 Conclusions and Related Work 30 1 Introduction Linear logic [24] provides a logical characterization of concepts and mechanisms peculiar to concurrency like locality, recursion, non determinism, and synchronization [3, 32, 26] Following the paradigm of proofs as computations proposed in [2, 33], provability in fragments of linear logic can be used then as a formal tool to reason about behavioural aspects of concurrent systems (see e.g. 36] In other paradigms for concurrency like the theory of Petri Nets there exist however a number of consolidated algorithmic techniques for the ....

....with nested conjunctive and disjunctive goals) In our setting, a natural way of augmenting the expressivity of the speci cation language is to consider rst order fragments of linear logic. First order formulas can be used, in fact, to color the internal state of processes with structured data [3, 33]. The combination between rst order formulas and linear connectives provides a well founded interpretation of the dynamics in the evolution of the internal state of a process [3, 32, 33] First order quanti cation in goal formulas has several interesting interpretations here: it can be viewed ....

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

....to study algorithmic techniques for the analysis of a vast range of concurrency models it seems important to nd a uniform framework to reason about their characteristic features. In our approach we will adopt linear logic [Girard, 1987] as a uni ed logical framework for concurrency. As shown in [4, 28, 34, 11, 38], linear logic provides a logical characterization of concepts and mechanisms peculiar of concurrency like locality, recursion, and non determinism in the de nition of a process [4, 28, 34] communication via synchronization and value passing [11, 37] internal state and updates to its current ....

.... logic provides a logical characterization of concepts and mechanisms peculiar of concurrency like locality, recursion, and non determinism in the de nition of a process [4, 28, 34] communication via synchronization and value passing [11, 37] internal state and updates to its current value [4, 38]; and generation of fresh names [12, 37] In this setting provability in fragments of linear logic can be used as a formal tool to reason about behavioral aspects of the concurrent systems. The overall goal of this paper is to investigate the theoretical foundations of algorithmic veri cation ....

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Miller, D. (1996). Forum: A Multiple-Conclusion Speci cation Logic. Theoretical computer science, 165(1), 201{ 232.

....(L ) A; A (C ) 2. 2 The language and its interpretation Historically speaking, the rst proposal for a Linear Logic (LL) programming language was LO (Linear Objects) 3] which is a subset of LinLog [5] A list of other LL programming languages includes (and is not limited to) Forum [71], Lolli [39] Lygon [35] and ACL [58] Our subset of LL consists of LL synchronous connectives plus and ( The applicability of that LL fragment to AI planning is emphasised in [49] We use the following fragment for describing LL programs, where the symbols A and A c represent atomic ....

D. Miller. Forum: A Multiple-Conclusion Speci cation Logic. Theoretical Computer Science, Vol. 165, No. 1, pp. 201-232, 1996.

....systems, based on the general purpose logical speci cation language Forum 1 This micro level method has not been experimented in depth in CaseLP. In particular, the translation between HEMASL and E hhf has to be investigated in depth. 2 See Section 3.3.6 for a description of ProlAg. 87 [61]. E hhf is a multiset based logic language, which combines features of logic programming extensions like Prolog (e.g. goals with implication and universal quanti cation) with the notion of formulas as resources that is the basis of linear logic [33] A E hhf program P is a collection of ....

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

....do not change the current state of an OO system. This is restrictive, but nonetheless interesting, since it allows us to deal with, for example, correctness of queries in distributed OO systems. Our approach is model theoretic, so it is di erent from proof theoretic or typetheoretic approaches [1, 13]. It is closer to other model theoretic approaches, like the algebraic approach [2] However, it extends algebraic methods since, by the threetier architecture of speci cation frameworks, it allows us to introduce pre proved correctness, even at the level of distributed systems of objects. Our ....

D. Miller. Forum: A multiple-conclusion specication logic. TCS, 165(1):201-231, 1996.

....on multiset rewriting. Formally, this logic can be regarded as as an extension of multiset rewriting with universal quanti cation or equivalently as a fragment of Girard s linear logic [Gir87] 1 , and its operational semantics can be given through a goal driven sequent system, in the style of [Mil96] In the rest of this section, we will use the NS protocol presented before to give an informal description of this logic. Term Representation. First of all, we will need a term representation for the entities involved, such as principals, keys, and messages. The notation pr(id,s) will denote a ....

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

....can be regarded as a re nement 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 Specication Logic. TCS, 165(1):201-232, 1996.

....be the use 14 state Alice (bob 1) enc PK Nb) Intruder: keypair bob SK PK, knows (bob 1) Nb, knows (alice 0) Na, knows (trudy ) Nb, knows (trudy ) Na. Fig. 6. Prolog code for Trudy s goal. of a combination of aspects of linear and intuitionistic logic, achieved in languages like Forum [Mil96]. In this setting it could be possible to specify locally the actions of single agents (as in [CDLMS99] whereas the knowledge common to all agents could still be modeled using the intuitionistic component of Forum. Without xing the number of principals in the initial stage of the protocol, ....

D. Miller. Forum: A Multiple-Conclusion Specication 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 dicult they also give rise to new possibilities. Some applications for linear logic programming are illustrated in [19]. 172 Heiko Mantel Christoph Kreitz a000 ; a0100 axiom a000 ; o010 o a000 ; 01 a0010 ; a 0 0100 axiom a0010 ; o 0 010 o a0010 ; 0 01 001 ; 0 01 00 ; 01 ; 0 01 00 ; 01 c 0 lab( 0 ) 0 (A A) ....

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

....whereas denotes a multiset of G formulas. A sequent is provable if all branches of its proof tree terminate with instances of the r axiom. The proof system of Fig. 1 is a specialization of more general uniform proof systems for linear logic like Andreoli s focusing proofs [And92] and Forum [Mil96] The rule bc denotes a backchaining (resolution) step. Note that bc can be executed only if the right hand side of the current LO sequent consists of atomic formulas. Thus, LO clauses behave like multiset rewriting rules. LO clauses having the form H 2 c, where c is a satis able constraint, ....

.... rst order LO programs, i.e. an alternative operational semantics that could be useful to study new applications of linear logic programming (as discussed in [HW98] For this reason, we think it would be important to extend our method to other linear logic languages like Lolli [HM94] and Forum [Mil96] The work presented in this paper can also be a source of further investigations concerning the analysis of programs and the development of new observable semantics. In particular, the semantics could be extended in order to cope with observables like the non ground success set and computed ....

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

....In the paper, and in the framework of the proofs as computations interpretation of linear logic, we present an extension of hereditary Harrop formulas and a corresponding logical system which are the foundations of the logic programming language ehhf . The starting point of this study is Forum [26], a presentation of higher order linear logic in terms of uniform proofs. A subset of its formulas have been isolated and proved to be well suited to encode descriptions of various programming paradigms. Keywords: Extensions of Logic Programming, Linear Logic, Higher Order Logic. 1 Introduction ....

....atomic goal formulas using clauses nondeterministically selected from the program. Besides the logic of Horn Clauses [31] and of hereditary Harrop formulas [28] several fragments of Girard s linear logic [15] have been isolated and proved to be ALPLs. Examples are LO [2] Lolli [18] and Forum [26]. Furthermore, Lygon [17] has been proved to be the largest fragment of linear Preprint submitted to Elsevier Science 3 March 2000 logic enjoying uniform provability, in the sense stated in [30] All these languages adhere to the so called proofs as computations interpretation of linear logic, an ....

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

....simple to handle a nite number of agents, it is not clear yet if the same ideas could be applied to the general case of a multiagent communication protocol. A interesting direction could be the use of a combination of aspects of linear and intuitionistic logic, achieved in languages like Forum [Mil96] In this setting it could be possible to specify locally the actions of single agents (as in [CDL 99] whereas the 13 state Alice Bob (enc PK (c M1 M2) trudy 0) keypair trudy SK PK, announce PKB, pi Nt ( knows key (trudy 0) PK) knows key (trudy 0) PKB) knows key (trudy ....

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

....email:bozzano disi.unige.it y email:martelli disi.unige.it ational 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 [2] and the uniform proofs of Miller [27] 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 ....

..... Gk : The formula F on the left hand side of a sequent indicates that F can be used in a proof an arbitrary number of times. From the left rules of and (see e.g. [2, 27]) this implies that an LO Program can be viewed also as a set of reusable clauses. According to this view, the operational semantics of LO is given via the uniform (goal driven) proof system de ned in Fig. 1. In Fig.1, P is a set of implicational clauses, A denotes a multiset of atomic formulas, ....

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

....admissible, our calculus is in principle a good candidate for such applications like automated deduction, logic programming, etc. The most direct comparison should be done with linear logic, of course, and then with the study of its uniform proofs. In this respect, the reference paper is Miller s [13], where it is de ned the formal system Forum, which is equivalent to linear logic. 60 Alessio Guglielmi The biggest problem in the proof search as computation eld is eliminating as much as possible unnecessary, spurious non determinism in the bottom up search for a proof. Forum succeeds in ....

Dale Miller. Forum: A multiple-conclusion specication logic. Theoretical Computer Science, 165:201-232, 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 de ned 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 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 Specication Logic. Theoretical Computer Science, 165(1):201-232, 1996. 15

....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 sequent calculi that integrate the operational semantics at levels of both sequents and proofs. For instance, the ....

....the connected studies about permutability and reduction of non determinism, often leads to new equivalent sequent calculi that integrate the operational semantics at levels of both sequents and proofs. For instance, the actual presentations of systems like Lolli [74] Lygon [142, 171, 69] or Forum [108] are based on variants sequent calculi from which only uniform proofs can be directly built and for which sequents are re ned in di erent parts. Then the operational meanings are involved at the logical level. Speci cation Logics. We consider at rst two views of logic programming based on LF ....

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

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

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