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Cutelimination for a logic with definitions and induction
 Theoretical Computer Science
, 1997
"... In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The l ..."
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Cited by 61 (19 self)
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In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The literature contains many approaches to formally adding these reasoning principles with logic specifications. We choose an approach based on the sequent calculus and design an intuitionistic logic F Oλ ∆IN that includes natural number induction and a notion of definition. We have detailed elsewhere that this logic has a number of applications. In this paper we prove the cutelimination theorem for F Oλ ∆IN, adapting a technique due to Tait and MartinLöf. This cutelimination proof is technically interesting and significantly extends previous results of this kind. 1
A Proof Theory for Generic Judgments
, 2003
"... this paper, we do this by adding the #quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type ..."
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Cited by 60 (14 self)
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this paper, we do this by adding the #quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important
A proof theory for generic judgments: An extended abstract
 In LICS 2003
, 2003
"... A powerful and declarative means of specifying computations containing abstractions involves metalevel, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and t ..."
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Cited by 41 (15 self)
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A powerful and declarative means of specifying computations containing abstractions involves metalevel, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, ∇, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with ∇ satisfies cutelimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the πcalculus and the encoding of objectlevel provability.
Encoding Transition Systems in Sequent Calculus
 Theoretical Computer Science
, 1996
"... Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. I ..."
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Cited by 33 (10 self)
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Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. In both cases, it is possible to relate transition paths to proofs in sequent calculus. In neither encoding, however, does it seem possible to capture properties, such as simulation and bisimulation, that need to consider all possible transitions or all possible computation paths. We consider augmenting both intuitionistic and linear logics with a proof theoretical treatment of definitions. In both cases, this addition allows proving various judgments concerning simulation and bisimulation (especially for noetherian transition systems). We also explore the use of infinite proofs to reason about infinite sequences of transitions. Finally, combining definitions and induction into sequent calculus proofs makes it possible to reason more richly about properties of transition systems completely within the formal setting of sequent calculus.
On the specification of sequent systems
 IN LPAR 2005: 12TH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE AND REASONING, NUMBER 3835 IN LNAI
, 2005
"... Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the metatheory of linear logic can be used to draw conclusions about the specified sequent calculus. For e ..."
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Cited by 13 (6 self)
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Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the metatheory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programmingstyle search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semiautomated proof of cutelimination for Girard’s Logic of Unicity (LU).
Encryption as an abstract datatype: An extended abstract
, 2003
"... At the DolevYao level of abstraction, security protocols can be specified using multisets rewriting. Such rewriting can be modeled naturally using proof search in linear logic. The linear logic setting also provides a simple mechanism for generating nonces and session and encryption keys via eigenv ..."
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Cited by 7 (2 self)
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At the DolevYao level of abstraction, security protocols can be specified using multisets rewriting. Such rewriting can be modeled naturally using proof search in linear logic. The linear logic setting also provides a simple mechanism for generating nonces and session and encryption keys via eigenvariables. We illustrate several additional aspects of this direct encoding of protocols into logic. In particular, encrypted data can be seen naturally as an abstract datatype. Entailments between security protocols as linear logic theories can be surprisingly strong. We also illustrate how the wellknown connection in linear logic between bipolar formulas and general formulas can be used to show that the asynchronous model of communication given by multiset rewriting rules can be understood, more naturally as asynchronous process calculus (also represented directly as linear logic formulas). The familiar proof theoretic notion of interpolants can also serve to characterize communication between a role and its environment.
HigherOrder Quantification and Proof Search
 In Proceedings of the AMAST confrerence, LNCS
, 2002
"... Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because firstorder logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these a ..."
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Cited by 7 (4 self)
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Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because firstorder logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these auxiliaries to be present in any equivalence program.
Lógica Linear E a Especificação De Sistemas Computacionais
, 2001
"... In recent years, intuitionistic logic and type systems have been used in numerous computational logical systems as frameworks for the specification of natural deduction proof systems. As we shall illustrate here, linear logic can be similarly used to specify the more general setting of sequent calcu ..."
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Cited by 6 (2 self)
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In recent years, intuitionistic logic and type systems have been used in numerous computational logical systems as frameworks for the specification of natural deduction proof systems. As we shall illustrate here, linear logic can be similarly used to specify the more general setting of sequent calculus proof systems and provides rich forms of analysis and deduction of properties of the specified systems. We shall present several example encodings of sequent calculus proof systems using the Forum presentation of linear logic: linear logic is a resource conscious logic developed by Girard, and Forum is an abstract logic programming language associated to it, due to Miller. We start by proposing an encoding of sequents, rules and systems. Then a correctness result is proved for these encodings and it is observed that metalevel proofs match closely the objectlevel ones. The encoding of an objectlevel proof system as Forum clauses has certain advantages over encoding them as inference figures. For example, Forum specifications do not deal with context explicitly and instead it only focuses on the formulas that are directly involved in the inference rule. The distinction between making the inference rule additive or multiplicative is achieved in inference rule figures by explicitly presenting contexts and either splitting or copying them. The Forum clause representation achieves the same distinction using metalevel additive or multiplicative connectives. Objectlevel quantifiers can be handled directly using the metalevel quantification. Similarly, the structural rules of contraction and weakening can be captured together using the ? modal. Finally, since the encoding of proof systems is natural and direct, we are able to use the rich metatheory of linear logic to help ...
Collection analysis for Horn clause programs
 In Proceedings of PPDP 2006: 8th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming
, 2006
"... dale.miller [at] inria.fr ..."
Encoding Generic Judgments
 In Proceedings of FSTTCS. Number 2556 in LNCS
, 2002
"... The operational semantics of a computation system is often presented as inference rules or, equivalently, as logical theories. Specifications can be made more declarative and highlevel if syntactic details concerning bound variables and substitutions are encoded directly into the logic using te ..."
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Cited by 3 (2 self)
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The operational semantics of a computation system is often presented as inference rules or, equivalently, as logical theories. Specifications can be made more declarative and highlevel if syntactic details concerning bound variables and substitutions are encoded directly into the logic using termlevel abstractions (#abstraction) and prooflevel abstractions (eigenvariables). When one wishes to reason about relations defined using termlevel abstractions, generic judgment are generally required.