D. J. Pym. On bunched predicate logic. In G. Longo, editor, Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS'99), pages 183--192, Trento, Italy, July 1999. IEEE Computer Society Press.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....some restrictions, new connection based characterizations and methods for both logics. 1 Introduction The logic of bunched implications (BI) provides a logical analysis of the basic notion of resource, that is central in computer science, with well de ned prooftheoretic and semantic foundations [10,12]. Its propositional fragment freely combines multiplicative (or linear) and connectives and additive (or intuitionistic) and connectives [11] and can be seen as a merging of intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL) BI has a Kripke style semantics ....

D. Pym. On bunched predicate logic. In 14h Symposium on Logic in Computer Science, pages 183192, Trento, Italy, July 1999. IEEE Computer Society Press.

....can be viewed as a merging of intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL) that allows to capture interferences between resources. It can be well understood through its Kripke resource models [21] but also proof theoretically through its bunched sequent calculus [34] and the corresponding calculus [27] With its so called sharing interpretation, the BI logic appears as an appropriate logical foundation for logic programming [2] for reasoning about mutable data structures [20] and for problems dealing with interferences between resources. In this context, ....

....relationships in order to determine if and how similar labelled calculi could be de ne for logics with well de ned resource semantics. In section 2, we present BI with its basic proof theory and Kripke resource semantics. The completeness w.r.t. this semantics only holds for BI (BI without ) [34] and consequently we rst deal with this fragment. In section 3, we introduce the labelled tableau calculus TBI with appropriate de nitions of labels, labelled formulas and constraints. We de ne expansion rules that include the generation of two kinds of constraints called assertions and ....

[Article contains additional citation context not shown here]

D. Pym. On Bunched Predicate Logic. In 14h Symposium on Logic in Computer Science, pages 183192, Trento, Italy, July 1999. IEEE Computer Society Press.

....other applications beyond programming languages for an ordered framework such as GSOS with priorities [23] The latter may require a more sophisticated notion of order, such as branching. It is conceivable that this could be mirrored by refining linear ordered context in the sense of bunches in BI [19]. As far as the infrastructure is concerned, note that similarly to [13] in this case study we only needed to induct closed terms, although we reason (typically by inversion) in presence of hypothetical judgments. Inducting HOAS style over open terms is a major challenge [20] in this setting ....

D. J. Pym. On bunched predicate logic. In G. Longo, editor, Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS'99), pages 183--192, Trento, Italy, July 1999. IEEE Computer Society Press.

....rules, we need to determine whether to execute such rules in a depth rst or breadth rst (or iterative deepening) manner. In short, development of an appropriate operational semantics is required. The extension of the mixed mode system to other logics (such as ane logic, light linear logic, BI [37], etc. is also a topic of interest. Another item is the further pursuit of this particular notion of agentoriented programming. For example, it is not clear how the notion of intention ts into our framework (although it seems reasonable that the search strategy used to solve the goals execute ....

D. Pym, On Bunched Predicate Logic, Proceedings of the IEEE Symposium on Logic in Computer Science, Trento, July, 1999.

....space. The lL calculus can be seen to arise in two ways. Firstly, in logical frameworks [9, 18] in which it provides a language that is a suitable basis for a framework capable of properly representing linear and other relevant logics. Secondly, from the logic of bunched implications, BI [15, 19], in which the antecedents of sequents are structured not as lists but as bunches, which have two combining operations, which admits Weakening and Contraction, and , which does not. The lL calculus stands in propositions as types correspondence with a fragment of BI [13, 12] The purpose ....

....f Here ( Gamma; Gamma) is cartesian pairing and [ Gamma; Gamma] is the pairing operation defined by Day s tensor product construction in Set . The resource semantics can be seen to combine Kripke s semantics for intuitionistic logic and Urquhart s semantics for relevant logic [14, 22] see [15, 19]. Suppose we have a category E where the propositions will be interpreted. Then we will index E in two ways for the purposes of interpreting the type theory. First, we index it by a Kripke world structure W . This is to let the functor category [W ; E ] have enough strength to model the ....

[Article contains additional citation context not shown here]

DJ Pym. On Bunched Predicate Logic. In Proc. LICS '99, Trento, Italy. IEEE Computer Society Press, 1999.

....such as linear abd strict logic as well as various other usage logics. An abstract normalization procedure is sketched, which however needs commutative conversions (e.g. the case contraction arrow elimination) already in the purely implicational fragment. In a series of papers Pym et at. see [Pym99] for detailed references and applications) introduce a calculus which aims to couple multiplicative and addititive implication, as well as quanti cation in a novel way. The two kind of operators are distinct by allowing two di erent constructors for contexts which are known in the relevanve logic ....

David J. Pym. On bunched predicate logic. In G. Longo, editor, Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS'99), pages 183-192, Trento, Italy, July 1999. IEEE Computer Society Press.

.... a multiplicative (or linear) and an additive (or intuitionistic) implication in the same logic [12] and its propositional fragment can be viewed as a merging of intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL) with well de ned proof theoretic and semantic foundations [11,13]. The semantics of BI is motivated by modelling the notion of resource. With its underlying sharing interpretation, it has been recently used for logic programming [1] or reasoning about mutable data structures [8] Our aim is to propose useful and e cient proof search procedures for such a mixed ....

.... in labels and moreover design a similar calculus with countermodel generation in BI (with ) starting from Grothendieck sheaftheoretic models [12] 2 The Logic of Bunched Implications (BI) In this section, we remind the essential features of the propositional logic of Bunched Implications (BI) [13], in which additive (intuitionistic) and multiplicative (linear) connectives cohabit. 2.1 Syntax and Proof Theory of BI The propositional language of BI consists of a multiplicative unit I , the multiplicative connectives , the additive units , the additive connectives , a ....

D. Pym. On bunched predicate logic. In 14h Symposium on Logic in Computer Science, pages 183192, Trento, Italy, July 1999. IEEE Computer Society Press.

.... (or linear) and an additive (or intuitionistic) implication cohabit [27] The propositional fragment of BI can be viewed as a merging of intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL) This new logic has been studied from proof theoretic and semantic point of views [33] with main focuses on Kripke resources models [18] and on the corresponding calculus [25] With its sharing interpretation, BI is an appropriate foundation for some computer science applications as for instance logic programming [2] or for reasoning about mutable data structures [17] In this ....

....with other proof search methods, they provide e cient proof search and an easy and direct generation of countermodels. In section 2 we present BI with its basic proof theory and Kripke resource semantics. We recall that the completeness w.r.t. this semantics only holds for BI without the unit [33] and thus we consider from now this fragment, called here BI . In section 3 we introduce the labelled tableau calculus TBI with appropriate de nitions of labels and constraints. We present the expansion rules that include the generation of constraints called assertions and requirements, ....

[Article contains additional citation context not shown here]

D. Pym. On bunched predicate logic. In 14h Symposium on Logic in Computer Science, pages 183192, Trento, Italy, July 1999. IEEE Computer Society Press.

....we need to determine whether to execute such rules in a depth first or breadth first (or iterative deepening) manner. In short, development of an appropriate operational semantics is required. The extension of the mixed mode system to other logics (such as affine logic, light linear logic, BI[31], etc. is also a topic of interest. Another item is the further pursuit of this particular notion of agent oriented programming. For example, it is not clear how the notion of intention fits into our framework (although it seems reasonable that the search strategy used to solve the goals execute ....

D. Pym, On Bunched Predicate Logic, Proceedings of the IEEE Symposium on Logic in Computer Science, Trento, July, 1999.

....Such a presentation retains type theoretic locality at the price of the loss of some global symmetry. Finally, we give an example which goes a little beyond the framework for type theoretic languages that we have introduced. Example 2. 5 (bunched logic) BI, the logic of bunched implications [115, 116, 147], uses sequents with antecedents structured not as lists but as bunches. Bunches have two combining operations, and , Di erent structural properties are speci ed for ; which admits weakening and contraction, and , which admits neither. This richer sequential structure allows additive ....

....in Boolean algebras but such a semantics is not type theoretic: there is no way to distinguish between di erent proofs of the same sequent in such a semantics. Again, we conclude this point with bunched logic. Example 2. 9 (bunched logic) Models of propositional BI, including its proof objects, [116, 149, 147] are characterized by bicartesian doubly closed categories, i.e. categories which enjoy two (symmetric) monoidal closed structures, one of which is cartesian, and which also have co products. A host of examples, relying on Day s tensor product construction [38] is provided by categories of ....

[Article contains additional citation context not shown here]

D.J. Pym. On bunched predicate logic. To appear: Proc. LICS, 1999, IEEE. Available on the web at http://www.dcs.qmw.ac.uk/~pym.

....not needed in [26] because in the intuitionistic semantics the unit of is true (this is because Weakening for is present) We also use the BI symbol instead of for multiplicative conjunction. The quanti ers are standard; we do not include the more unusual multiplicative quanti ers of BI [25]. We adopt typical syntactic sugar: P for P ) false; true for : false) The atomic formulae include an equality relation and the points to relation: E = E 0 Equality j E E 1 ; E n Points to j 3.1 The Classical Semantics The semantics of assertions is given by a forcing ....

....very challenging: structural control is required on the level of individuals rather than just propositions. A more principled logical approach would be most welcome, and seems a good test problem for the multiplicative approaches to dependent types and predication described by Ishtiaq and Pym [14, 25]. The reader might have been bemused by our apparently devious use of dangling pointers in the treatment of frame axioms. The local nature of a speci cation fPgCfQg comes about from the fact that if C tried to alter a cons cell not guaranteed to exist by P then we could contradict the speci ....

Pym, D. J. On bunched predicate logic. In Proceedings, 14th Annual IEEE Symposium on Logic in Computer Science (1999), IEEE Computer Society Press, p. 10pp.

....prefer a less leisurely approach can skip forward directly to the synopsis in Section 2.5. Some of the material in this paper was presented, in preliminary form, at the 1999 TLCA conference [33] Related material on the logic BI of bunched implications, the logical cousin of , may be found in [34, 38, 20]. See in particular Pym s forthcoming monograph for a treatment of some of the foundational issues avoided here [39] 2 Routes to Bunched Typing 2.1 Sharing and Contraction The work reported in this paper arose originally from a failed attempt to reconcile two substructural type systems, ....

D. J. Pym. On bunched predicate logic. In Proceedings, 14th Annual IEEE Symposium on Logic in Computer Science, pages 183-192. IEEE Computer Society Press, 1999.

....computer memory, logic programming, and money. 1 Introduction The purpose of this paper is is to explore, from the point of view of resources and in the context of a broader investigation of resource modelling , algebraic and possible worlds semantics for BI, the logic of bunched implications [38, 41, 37]. Propositional BI, our focus in this paper, freely combines the ; I ; fragment of propositional linear logic and propositional intuitionistic logic via the formulation of contexts not as finite sequences of propositions but rather as finite bunches of propositions. The basic formulation of ....

.... E i 1 2 (i = 1; 2) I ( E Table 1: Natural Deduction System for BI: NBI . Commutative monoid equations for ; and ; a ; Congruence: if , then ( Given this structure, we can define BI, the logic of bunched implications [38, 41, 42], as a natural deduction system, as in Table 1, in which we use , pronounced magic wand , for multiplicative implication and , pronounced star , for multiplicative conjunction, and for their additive counterparts, and for disjunction. The units of , and are denoted I , and ....

[Article contains additional citation context not shown here]

D.J. Pym. On bunched predicate logic. In Proc. LICS'99, pages 183--192. IEEE Computer Society Press, 1999.

....computer memory, logic programming, and money. 1 Introduction The purpose of this paper is is to explore, from the point of view of resources and in the context of a broader investigation of resource modelling , algebraic and possible worlds semantics for BI, the logic of bunched implications [39, 42, 38]. Propositional BI, our focus in this paper, freely combines the ; I ; fragment of propositional linear logic and propositional intuitionistic logic via the formulation of contexts not as finite sequences of propositions but rather as finite bunches of propositions. The basic formulation of ....

.... E ; I ( E ; I ; E E i 1 2 (i = 1; 2) I ( E Table 1: Natural Deduction System for BI: NBI Given this structure, we can define BI, the logic of bunched implications [39, 42, 43], as a natural deduction system, as in Table 1, in which we use , pronounced magic wand , for multiplicative implication and , pronounced star , for multiplicative conjunction, and for their additive counterparts, and for disjunction. The units of , and are denoted I , and ....

[Article contains additional citation context not shown here]

D.J. Pym. On bunched predicate logic. In Proc. LICS'99, pages 183--192. IEEE Computer Society Press, 1999.

....a range of strategies, from the lazy to the eager, for solving sets of constraint equations. We indicate how to apply our methods systematically to large family of relevant systems. 1 Introduction Proof search in logics, such as linear logic (LL) 7] or the logic of bunched implications (BI) [13, 14, 15], which have multiplicative connectives requires a mechanism by which the distribution of formul, sometimes viewed as resources, between different branches of a proof may be calculated. Such mechanisms are usually specified by intricate rules of inference which are used to keep track of the ....

....logic. In x3, we extend the results of the previous section to include the additive and exponential connectives of linear logic, i.e. to propositional linear logic, PLL. This is quite straightforward. In x4, we extend the approach of x2 to the (propositional) logic of bunched implications, BI [13, 14, 15, 16]. This may be understood as a different extension of MLL in which the additive connectives, including a semantically adequate implication, are combined freely with the multiplicatives via the notion of bunches [17, 13, 14, 15] The results we obtain for BI are correspondingly more subtle. In x5, ....

[Article contains additional citation context not shown here]

D. Pym, On Bunched Predicate Logic, Proceedings of the 14th IEEE Symposium on Logic in Computer Science, 183-192, Trento, Italy, July,

....a predicate bunched logic, in which the variables in the antecedent have a bunched structure too, then we can form two kinds of quanti ers, a linear one 8 new and an intuitionistic one 8. The Kripke style semantics for predicate BI can be given by extending the above ideas, and are discussed in [23, 32]. Although presheaf DCCs are adequate for such a semantics of predicate BI, they do not yield a good interpretation of proofs. For this, we must move to an indexed or bred setting in which the predicate BI judgement (X) is interpreted by interpreting the propositional judgement over the ....

.... Here ( is cartesian pairing and [ is the pairing operation de ned by Day s tensor product construction in Set M op . The resource semantics can be seen to combine Kripke s semantics for intuitionistic logic and Urquhart s semantics for relevant logic [18, 37] Further details are in [23, 32]. Suppose we have a category E where the propositions will be interpreted. Then we will index E in two ways for the purposes of interpreting the type theory. First, we index it by a Kripke world structure W . This is to let the functor category [W ; E ] have enough strength to model the f ; ....

[Article contains additional citation context not shown here]

DJ Pym. On bunched predicate logic. In LICS 1999. IEEE, 1999.

....a range of strategies, from the lazy to the eager, for solving sets of constraint equations. We indicate how to apply our methods systematically to large family of relevant systems. 1 Introduction Proof search in logics, such as linear logic (LL) 2] or the logic of bunched implications (BI) [8, 10, 11], which have multiplicative connectives requires a mechanism by which the distribution of formul, sometimes viewed as resources, between different branches of a proof may be calculated. Such mechanisms are usually specified by intricate rules of inference which are used to keep track of the ....

....we extend the results of the previous section to include the additive and exponential connectives of linear logic, i.e. to propositional linear logic, PLL. This is quite straightforward. In x4, we extend the approach of x2 to the (propositional fragment of) the logic of bunched implications, BI [10]. This may be seen as a different extension of MLL, in that the additive connectives are introduced in a more intricate manner. Accordingly the results are correspondingly more subtle in this case. In x5, we discuss the manner in which the equations generated by the inference rules may be ....

[Article contains additional citation context not shown here]

D. Pym, On Bunched Predicate Logic, Proceedings of the 14th IEEE Symposium on Logic in Computer Science, 183-192, Trento, Italy, July, 1999. IEEE Computer Society, 1999.

.... is its implication adjoint. A Boolean variant of BI is obtained by adding the law of reductio ad absurdum to the intuitionistic system. The purpose of this paper is to explore in some detail the possible worlds semantics of propositional BI sketched in [1] A similar analysis for predicate BI [2], building on the present analysis, is treated in [3] Our first reason for doing this is simple: the possible worlds semantics shows how the logic has many interesting and naturally occurring models. Furthermore, although they can sometimes provide semantics for proofs, many of these models make ....

....all interpretations [ in BI algebras (BI algebras obviously give complete models of the proof system just given. This way of formulating the system is unsophisticated proof theoretically, but it is adequate for capturing provability. BI can also be presented as a system of natural deduction [1, 2, 3] or, indeed, as a sequent calculus [2, 3] where the two conjunctions are mimicked by two context combining operations. Normalization and cut elimination results for these calculi are established in [3] We will give several models in which the additives are treated classically. So we define ....

[Article contains additional citation context not shown here]

D. J. Pym. On bunched predicate logic. In Proceedings, 14th Annual IEEE Symposium on Logic in Computer Science, pages 183--192. IEEE Computer Society Press, 1999.

....be used to freely mix together a substructural implication Gamma and a conventional additive (intuitionistic or classical) implication . The purpose of this paper is to explore in some detail the possible worlds semantics of propositional BI sketched in [1] A similar analysis for predicate BI [2], building on the present analysis, is treated in [3] Our first reason for doing this is simple: the possible worlds semantics shows how the logic has many interesting and naturally occurring models. Furthermore, although they can sometimes provide semantics for proofs, many of these models make ....

....n j= p and m where m j= q, we have that n Delta m j= p q and, because of the existential formula characterizing (p Gamma 0) 0, this is enough to give us the judgement. The unprovability of the syntactic judgement is easy to establish via the cut elimination theorem for BI s sequent calculus [2, 3]. Also, at the end of x 5.3, we give an explicit counter model. A more conceptual, partial explanation of this incompleteness can be seen by considering where a standard completeness argument breaks down. In this (which is essentially a Yoneda lemma argument) we use the propositions of BI to ....

[Article contains additional citation context not shown here]

D. J. Pym. On bunched predicate logic. In Proceedings, 14th Annual IEEE Symposium on Logic in Computer Science, pages 183--192. IEEE Computer Society Press, 1999.

....a predicate bunched logic, in which the variables in the antecedent have a bunched structure too, then we can form two kinds of quantifiers, a linear one 8 new and an intuitionistic one 8. The Kripke style semantics for predicate BI can be given by extending the above ideas, and are discussed in [20, 24]. Although presheaf DCCs are adequate for such a semantics of predicate BI, they do not yield a good interpretation of proofs. For this, we must move to an indexed or fibred setting in which the predicate BI judgement (X ) Gamma OE is interpreted by interpreting the propositional judgement ....

....is cartesian pairing and [ Gamma; Gamma] is the pairing operation defined by Day s tensor product construction in Set M op . The resource semantics can be seen to combine Kripke s semantics for intuitionistic logic and Urquhart s semantics for relevant logic [15, 33] Further details are in [20, 24]. Suppose we have a category E where the propositions will be interpreted. Then we will index E in two ways for the purposes of interpreting the type theory. First, we index it by a Kripke world structure W . This is to let the functor category [W ; E ] have enough strength to model the f ; ....

[Article contains additional citation context not shown here]

DJ Pym. On Bunched Predicate Logic. To appear: Proc. of LICS, 1999, IEEE.

....Such a presentation retains type theoretic locality at the price of the loss of some global symmetry. Finally, we give an example which goes a little beyond the framework for type theoretic languages that we have introduced. Example 2. 5 (bunched logic) BI, the logic of bunched implications [115, 116, 147], uses sequents with antecedents structured not as lists but as bunches. Bunches have two combining operations, and , Different structural properties are specified for ; which admits weakening and contraction, and , which admits neither. This richer sequential structure allows ....

....in Boolean algebras but such a semantics is not type theoretic: there is no way to distinguish between different proofs of the same sequent in such a semantics. Again, we conclude this point with bunched logic. Example 2. 9 (bunched logic) Models of propositional BI, including its proof objects, [116, 149, 147] are characterized by bicartesian doubly closed categories, i.e. categories which enjoy two (symmetric) monoidal closed structures, one of which is cartesian, and which also have co products. A host of examples, relying on Day s tensor product construction [38] is provided by categories of ....

[Article contains additional citation context not shown here]

D.J. Pym. On bunched predicate logic. To appear: Proc. LICS, 1999, IEEE. Available on the web at http://www.dcs.qmw.ac.uk/~pym.

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D. J. Pym. On bunched predicate logic. In G. Longo, editor, Proceedings of the 14th Annual Symposium on Logic in Computer Science (LICS'99), pages 183--192, Trento, Italy, July 1999. IEEE Computer Society Press.

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D. Pym. On bunched predicate logic. In Proc. 14th Symposium on Logic in Computer Science, 183--192, Trento, Italy, July 1999. IEEE Computer Society Press.

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