Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In this section we summarize some of the applications presented in [2] 4.1 Maude as a Meta Logical Translator A logical framework is a logic in which many other logics can be represented. From the perspective of general logics [9] the quest for a logical framework has been defined in [7] as the search within the space of logics for a logic F such that many other logics L can be represented in F by means of mappings L Gamma F that have particularly nice properties such as being conservative translations. In this line, Mart i Oliet and Meseguer [8] have recently proposed ....

N. Mart'i-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....AL (or abstract AL) Both the core part and the general part have their category theoretic versions. For that of the core part we refer to works of Michael Makkai, cf. e.g. 8] 7] while for the category theoretic version of the general part we refer to works on institutions theory, cf. e.g. [9]. In the present issue we do not discuss the category theoretic approach (or version) and we note that the above references to it are extremely incomplete. Let us return to the non category theoretic approach. The core part consists of the theories of cylindric algebras, polyadic and ....

N. Mart-Oliet and J. Meseguer. General Logics and Logical Frameworks. In D. M. Gabbay, editor, What is a Logical System, pages 355-391, Clarendron Press, Oxford (Oxford Science Publications), 1994.

....Church Rosser checking of IDENTITY 3 Checking solution : The specification is Coherent . 3.2. 3 Sequent Calculus for Propositional Logic Let us consider now the description of the representation within rewriting logic of a sequential calculus for classical propositional logic, as rst given in [13] and then in [4] where the basic atoms are denoted by identi ers. We can take advantage of several properties enjoyed by classical logic in order to have a more succinct and elegant sequent calculus. First, using the double negation and De Morgan laws, we can push negation to the atomic level. ....

Narciso Mart-Oliet and Jose Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355-392. Oxford University Press, 1994.

....AL (or abstract AL) Both the core part and the general part have their category theoretic versions. For that of the core part we refer to works of Michael Makkai, cf. e.g. 8] 7] while for the category theoretic version of the general part we refer to works on institutions theory, cf. e.g. [9]. In the present issue we do not discuss the category theoretic approach (or version) and we note that the above references to it are extremely incomplete. Let us return to the non category theoretic approach. The core part consists of the theories of cylindric algebras, polyadic and ....

N. Mart-Oliet and J. Meseguer. General Logics and Logical Frameworks. In D. M. Gabbay, editor, What is a Logical System, pages 355--391, Clarendron Press, Oxford (Oxford Science Publications), 1994.

....formal method tools have already been designed and implemented. We intend to exploit this fact for the temporal logic framework. Moreover, we will investigate whether the mapping from rewrite theories to temporal logic theories is a map of entailment systems or even a map of logics in the sense of [MOM94]. Acknowledgments. I am especially indebted to Jos e Meseguer who discussed with me the ideas presented in this paper. I have benefited very much from conversations with him, Narciso Mart i Oliet, Carolyn Talcott, Tom Maibaum, Jose Fiadeiro, and Peter Olveczky. The efforts of Hans Dieter Ehrich ....

N. Mart'i-Oliet and J. Meseguer. General Logics and Logical Frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....example. Our example is a standard one in metareasoning: the deduction theorem for minimal logic (of implication) Rewriting logic is not the only candidate for a re ective metalogical framework, but we believe it is a good one. Rewriting logic has been demonstrated to be a good logical framework [10, 23, 24, 30, 36, 37] and it is balanced on a point where it is expressive enough to naturally formalize di erent entailment systems, but it is weak enough so that its theories always have initial models. This means 2 that there are sound induction principles for reasoning with respect to these models. To prove ....

....They are user de nable as an algebraic data type satisfying equational axioms, so that rewriting deduction takes place modulo such axioms. Because of this ecumenical neutrality and the simplicity of the rules of the logic, rewriting logic can be e ectively applied as a logical framework. In [10, 23, 24, 30, 36, 37], many examples of logic representations are given, including rst order linear logic, sequent presentations of modal and propositional logics, Horn logic with equality, the lambda calculus, and higher order pure type systems, among others. In all such examples, the representational distance ....

N. Mart-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355-392. Oxford University Press, 1994.

....in which many different logics and formal systems can be naturally represented and interrelated. Doing justice to this logical side is beyond the scope of this paper, but good evidence, including a good number of examples of logic representations can be found in two joint papers with Mart i Oliet [68, 69]. Additional quite impressive evidence is also provided by research based on the ELAN language [46, 103, 49, 14, 18] that stresses the logical framework applications of rewriting logic. There is also a very fruitful relationship between rewriting logic and the theory of reasoning theories ....

Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....can be faithfully represented by means of conservative maps of logics L Gamma F : A number of logics have been proposed in the recent literature as (meta )logical frameworks of this kind. Several of these proposals offer some variant of higher order type theory as the framework logic (see [97] for references) A simpler alternative recently proposed by Mart i Oliet and Meseguer [96, 97] is to use rewriting logic [119] Rewriting logic is a very flexible, yet simple, logic that can be naturally used as a general logical and semantic framework, that is as a metalogic [96] in which: ffl ....

....A number of logics have been proposed in the recent literature as (meta )logical frameworks of this kind. Several of these proposals offer some variant of higher order type theory as the framework logic (see [97] for references) A simpler alternative recently proposed by Mart i Oliet and Meseguer [96, 97] is to use rewriting logic [119] Rewriting logic is a very flexible, yet simple, logic that can be naturally used as a general logical and semantic framework, that is as a metalogic [96] in which: ffl other logics can be defined; ffl languages can be semantically defined and can be prototyped; ....

[Article contains additional citation context not shown here]

Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....served as a basis for a prototype compiler mapping a subset of Maude into parallel machine instructions, and for a sequential Maude implementation currently under way. The metalogical foundations for declarative specification and programming languages have been developed in a series of papers [48, 49, 57, 11, 56] that use the theory of general logics as a metatheory in which the logic of choice for a language can be varied while important modularity and parameterization properties are preserved. The papers [48, 49] also explore the role of rewriting logic and Maude as a logical framework in which many ....

....and programming languages have been developed in a series of papers [48, 49, 57, 11, 56] that use the theory of general logics as a metatheory in which the logic of choice for a language can be varied while important modularity and parameterization properties are preserved. The papers [48, 49] also explore the role of rewriting logic and Maude as a logical framework in which many other logics can be represented, and as a semantic framework in which to specify and prototype many languages and systems, including parallel ones. The papers [57, 11] give foundations and make specific ....

Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....rewriting logic and present an example. Our example is a standard one in metareasoning: the deduction theorem. Rewriting logic is not the only candidate for a re ective metalogical framework, but we believe it is a good one. Rewriting logic has been demonstrated to be a good logical framework [11, 20, 21, 27] and it is balanced on a point where it is strong enough to naturally formalize di erent entailment systems, but it is weak enough so that its theories always have initial models. This means that there are sound induction principles for reasoning with respect to these models. The key then is to re ....

....by terms. They are user de nable as an algebraic data type satisfying equational axioms, so that rewriting deduction takes place modulo such axioms. Because of this ecumenical neutrality and the simplicity of the rules of the logic, rewriting logic has good properties as a logical framework. In [11, 20, 21], many examples of logic representations are given, including rst order linear logic, sequent presentations of modal and propositional logics, Horn logic with equality, among others. In all such examples, the representational distance between the object logic and its representation in rewriting ....

N. Mart-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355-392. Oxford University Press, 1994. 29

.... for such computational media are, e.g. the Encyclospace [16] the description, categorization and computer based interpretation of music in [17] Regarding the formalisms we refer to [18] for the use of Rewriting Logic in describing architectures for virtual distributed environments and to [19] for relations of Rewriting Logic with other formalisms. 3] presents an an introduction to Labelled Deductive systems. Multi agent systems can be formalized at different levels of abstraction: the calculus [11] models distributed systems as a collection of mobile agents, Troll [20] ofi [21] ....

N. Mart ' i-Oliet and J. Meseguer, "General logics and logical frameworks," in What is a logical system?, D. Gabbay, Ed. Oxford University Press, 1994.

....and properties of the formulae expressed by terms. They are user de nable as an algebraic data type satisfying equational axioms, so that rewriting deduction takes place modulo such axioms. Because of this ecumenical neutrality, rewriting logic has good properties as a logical framework. In [17, 18, 19], many examples of logic representations are given, including rst order linear logic, sequent presentations of modal and propositional logics, Horn logic with equality, and so on. In all such examples, the representational distance between the object logic and its representation is practically ....

N. Mart-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355-392. Oxford University Press, 1994.

....by executing the representation in the Maude language [7] This more concrete version is the basis of a proof assistant for OCC, the open calculus of constructions, an equational extension of the calculus of constructions, that is under development. This case study complements earlier work [20, 21, 22], showing that rewriting logic has good properties as a logical framework to represent a wide range of logics, including linear logic, Horn logic with equality, rst order logic, modal logics, sequent based presentations of logics, and so on. In particular, representations for the calculus, and ....

....shows that this alternative approach scales up to more complex type theories (e.g. extensions of UPTS systems) in a more satisfactory way than the purely functional and equational approaches to type checking. 5. 2 UPTS in Rewriting Logic As shown by an extensive collection of examples in [20, 21, 22], rewriting logic can be used as a logical framework that can naturally represent inference systems of di erent kinds in a logically and operationally satisfying way. In the present section we view a type checker as a particular inference system. In contrast to a (higher order) functional ....

N. Mart-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355-392. Oxford University Press, 1994.

....logic, double categories, 2 categories, partial membership equational logic. 1 Introduction Rewriting logic (rl) 20] is an elegant and expressive semantic framework for the specification of languages and systems, and is also a logical framework in which many other logics can be represented [19]. In it term rewriting is fully exploited as a basic computational paradigm: terms are regarded as states of a concurrent system whose state transitions are given by the rewriting rules of the specification. Moreover, a rewriting system specification is viewed as a logical theory such that each ....

N. Mart'i-Oliet and J. Meseguer, General logics and logical frameworks, in: D. Gabbay, Ed., What is a logical system?, 355--392, Oxford Univ. Press (1994).

....can be faithfully represented by means of conservative maps of logics L Gamma F : A number of logics have been proposed in the recent literature as (meta )logical frameworks of this kind. Several of these proposals offer some variant of higher order type theory as the framework logic (see [33] for references) A simpler alternative recently proposed by Mart i Oliet and Meseguer [32, 33] is to use rewriting logic [40] as a very flexible and natural meta logical framework. In rewriting logic axioms are rewrite rules of the form t t 0 that can be applied modulo given structural axioms ....

....A number of logics have been proposed in the recent literature as (meta )logical frameworks of this kind. Several of these proposals offer some variant of higher order type theory as the framework logic (see [33] for references) A simpler alternative recently proposed by Mart i Oliet and Meseguer [32, 33] is to use rewriting logic [40] as a very flexible and natural meta logical framework. In rewriting logic axioms are rewrite rules of the form t t 0 that can be applied modulo given structural axioms specified with the syntax. Such rewrite rules have both a logical and a computational reading. ....

Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

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Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....term in Module can be entered as a module into Maude s rewrite engine and can then be used to rewrite expressions this automatically gives us an implementation of the logic L in rewriting logic. A good number of examples of logic representations can be found in two joint papers with Mart i Oliet [105, 106]. In particular, the map Phi internalizing a representation map Phi from linear logic to rewriting logic was sketched in [105] and has been implemented in Maude by Manuel Clavel in [29] 6.2 Specifying and Building Formal Methods Tools Theorem provers and other formal methods tools have ....

N. Mart'i-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....The generality and expressiveness of rewriting logic as a semantic framework for concurrent computation has also a logical counterpart. Indeed, rewriting logic is also a promising logical framework in which many different logics and formal systems can be naturally represented and interrelated [208,209]. Using a rewriting logic implementation such representations can then be used to generate a wide range of formal tools. 5.1 Representing, Mapping, and Reasoning about Logics The basic idea is that we can represent a logic L with a finitary syntax and inference system within rewriting logic by ....

.... reasoning principles to prove such metalogical properties [10 12] A good number of examples of representations of logics in rewriting logic have been given by different authors, often in the form of executable specifications, including: 1) the logics represented by Mart i Oliet and Meseguer in [208,209], including equational logic, Horn logic with equality, linear logic, logics with quantifiers, and any sequent calculus presentation of a logic for a very general notion of sequent ; 2) the map LinLogic Gamma RWLogic in [208,209] representing propositional linear logic was subsequently ....

[Article contains additional citation context not shown here]

Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. M. Gabbay, editor, What is a Logical System?, volume 4 of Studies in Logic and Computation, pages 355--392. Oxford University Press, 1994.

....rewrites, which are performed in 12 seconds running on a 450MHz Pentium II. 1. 1 Re ective Design of the Tool The design of the present tool exploits and illustrates the logical framework capabilities of rewriting logic by formally specifying the inference system of each tool as a rewrite theory [27, 28]. The fact that these tools reason about equational theories is not a restriction of the general framework capabilities: inference systems for reasoning about speci cations in any other logic could be represented just as easily. However, the fact that rewriting logic contains equational logic as a ....

Narciso Mart-Oliet and Jose Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355-392. Oxford University Press, 1994.

....But where will the metatheory supporting such meta tools come from To make such tools mathematically rigorous, the first thing obviously needed is to have a mathematical metatheory of logics and of translations between logics. We have been investigating the theory of general logics [47, 44, 52, 11, 16] for this purpose. This theory axiomatizes the proof theoretic and model theoretic facets of logics and their translations, includes the theory of institutions as its modeltheoretic component [30] and is related to other similar metatheories (see the survey [52] But meta tools need more than a ....

N. Mart'i-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....But where will the meta theory supporting such meta tools come from To make such tools mathematically rigorous, the first thing obviously needed is to have a mathematical meta theory of logics and of translations between logics. We have been investigating the theory of general logics [39, 36, 44, 9, 14] for this purpose. This theory axiomatizes the proof theoretic and model theoretic facets of logics and their translations, includes the theory of institutions as its modeltheoretic component [25] and is related to other similar meta theories (see the survey [44] But meta tools need more than ....

....are instead first order. Our work should of course be placed within the context of the above related work, and of experiments carried out in different frameworks to prototype formal systems (for more discussion see the survey [44] 2. 1 Rewriting Logic and Reflection We and other researchers [37, 36, 44, 33, 52, 34, 5, 49, 8, 10, 12] have investigated the suitability of rewriting logic [40] as a logical framework and have found it to have very good properties for this purpose. One important practical advantage is that, what might be called the representational distance between a theory T in the original logic and its ....

N. Mart'i-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....served as a basis for a prototype compiler mapping a subset of Maude into parallel machine instructions, and for a sequential Maude implementation currently under way. The metalogical foundations for declarative specification and programming languages have been developed in a series of papers [55, 56, 75, 13, 20] that use the theory of general logics as a metatheory in which the logic of choice for a language can be varied while important modularity and parameterization properties are preserved. The papers [55, 56] also explore the role of rewriting logic and Maude as a logical framework in which many ....

....and programming languages have been developed in a series of papers [55, 56, 75, 13, 20] that use the theory of general logics as a metatheory in which the logic of choice for a language can be varied while important modularity and parameterization properties are preserved. The papers [55, 56] also explore the role of rewriting logic and Maude as a logical framework in which many other logics can be represented, and as a semantic framework in which to specify and prototype many languages and systems, including parallel ones. The papers [75, 13] give foundations and make specific ....

Narciso Mart-Oliet and Jose Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....example. Our example is a standard one in metareasoning: the deduction theorem for minimal logic (of implication) Rewriting logic is not the only candidate for a re ective metalogical framework, but we believe it is a good one. Rewriting logic has been demonstrated to be a good logical framework [11,23,24,30,36,37] and it is balanced on a point where it is expressive enough to naturally formalize di erent entailment systems, but it is weak enough so that its theories always have initial models. This means that there are sound induction principles for reasoning with respect to these models. To prove ....

....They are user de nable as an algebraic data type satisfying equational axioms, so that rewriting deduction takes place modulo such axioms. Because of this ecumenical neutrality and the simplicity of the rules of the logic, rewriting logic can be e ectively applied as a logical framework. In [11,23,24,30,36,37], many examples of logic representations are given, including rst order linear logic, sequent presentations of modal and propositional logics, Horn logic with equality, the lambda calculus, and higher order pure type systems, among others. In all such examples, the representational distance ....

N. Mart-Oliet and J. Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355-392. Oxford University Press, 1994.

.... [11, 13] In particular many different styles of concurrent objects open software components have been formally modelled in rewriting logic [12, 16, 14] It has also been used as a general logical framework within which logics of very different natures have been naturally and concisely represented [8, 9]. More recently it has been used to demonstrate semantic interoperation of different architectural styles [10] Furthermore rewriting logic is a wide spectrum framework that supports semantics preserving transformations from high level specifications to executable prototypes and from these to ....

Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

....in which many different logics, languages, and systems can be naturally represented. This is very useful for executable specification, rapid prototyping, and formal reasoning. A substantial amount of evidence, advocating the suitability of rewriting logic for these purposes already exists [31, 24, 19, 25]. In [26] a very general method for reifying inside rewriting logic representation maps between others logic and rewriting logic is proposed. This reification takes the form in Maude of an equationally defined function between the abstract data type representing the finitary theories of the logic ....

Narciso Mart'i-Oliet and Jos'e Meseguer. General logics and logical frameworks. In D. Gabbay, editor, What is a Logical System?, pages 355--392. Oxford University Press, 1994.

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