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47
Automating the Meta Theory of Deductive Systems
, 2000
"... not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, a ..."
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Cited by 81 (17 self)
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not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, and experimental results related to the areas of programming languages, type theory, and logics. Design: The metalogical framework extends the logical framework LF [HHP93] by a metalogic M + 2. This design is novel and unique since it allows higherorder encodings of deductive systems and induction principles to coexist. On the one hand, higherorder representation techniques lead to concise and direct encodings of programming languages and logic calculi. Inductive de nitions on the other hand allow the formalization of properties about deductive systems, such as the proof that an operational semantics preserves types or the proof that a logic is is a proof calculus whose proof terms are recursive functions that may be consistent.M +
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 61 (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.
Automated Theorem Proving in a Simple MetaLogic for LF
 PROCEEDINGS OF THE 15TH INTERNATIONAL CONFERENCE ON AUTOMATED DEDUCTION (CADE15
, 1998
"... Higherorder representation techniques allow elegant encodings of logics and programming languages in the logical framework LF, but unfortunately they are fundamentally incompatible with induction principles needed to reason about them. In this paper we develop a metalogic M_2 which allows i ..."
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Cited by 35 (16 self)
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Higherorder representation techniques allow elegant encodings of logics and programming languages in the logical framework LF, but unfortunately they are fundamentally incompatible with induction principles needed to reason about them. In this paper we develop a metalogic M_2 which allows inductive reasoning over LF encodings, and describe its implementation in Twelf, a specialpurpose automated theorem prover for properties of logics and programming languages. We have used Twelf to automatically prove a number of nontrivial theorems, including type preservation for MiniML and the deduction theorem for intuitionistic propositional logic.
Least and greatest fixed points in linear logic Extended Version
, 2007
"... david.baelde at enslyon.org dale.miller at inria.fr Abstract. The firstorder theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addi ..."
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Cited by 35 (12 self)
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david.baelde at enslyon.org dale.miller at inria.fr Abstract. The firstorder theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and?), we add least and greatest fixed point operators. The resulting logic, which we call µMALL = , satisfies two fundamental proof theoretic properties. In particular, µMALL = satisfies cutelimination, which implies consistency, and has a complete focused proof system. This second result about focused proofs provides a strong normal form for cutfree proof structures that can be used, for example, to help automate proof search. We then consider applying these two results about µMALL = to derive a focused proof system for an intuitionistic logic extended with induction and coinduction. The traditional approach to encoding intuitionistic logic into linear logic relies heavily on using the exponentials, which unfortunately weaken the focusing discipline. We get a better focused proof system by observing that certain fixed points satisfy the structural rules of weakening and contraction (without using exponentials). The resulting focused proof system for intuitionistic logic is closely related to the one implemented in Bedwyr, a recent model checker based on logic programming. We discuss how our proof theory might be used to build a computational system that can partially automate induction and coinduction. 1
Combining higher order abstract syntax with tactical theorem proving and (co)induction
 In TPHOLs ’02: Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics
, 2002
"... Abstract. Combining Higher Order Abstract Syntax (HOAS) and induction is well known to be problematic. We have implemented a tool called Hybrid, within Isabelle HOL, which does allow object logics to be represented using HOAS, and reasoned about using tactical theorem proving in general and principl ..."
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Cited by 33 (15 self)
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Abstract. Combining Higher Order Abstract Syntax (HOAS) and induction is well known to be problematic. We have implemented a tool called Hybrid, within Isabelle HOL, which does allow object logics to be represented using HOAS, and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. In this paper we describe Hybrid, and illustrate its use with case studies. We also provide some theoretical adequacy results which underpin our practical work. 1 Introduction Many people are concerned with the development of computing systems which can be used to reason about and prove properties of programming languages. However, developing such systems is not easy. Difficulties abound in both practical implementation and underpinning theory. Our paper makes both a theoretical and practical contribution to this research area. More precisely, this paper concerns how to reason about object level logics with syntax involving variable bindingnote that a programming language can be presented as an example of such an object logic. Our contribution is the provision of a mechanized tool, Hybrid, which has been coded within Isabelle HOL, and provides a form of logical framework within which the syntax of an object
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.
Induction and coinduction in sequent calculus
 Postproceedings of TYPES 2003, number 3085 in LNCS
, 2003
"... Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than sett ..."
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Cited by 23 (8 self)
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Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than settheoretic) notion of definition [13, 20, 25, 51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as “closed ” or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and coinductively about properties of computational system making full use of higherorder abstract syntax. Consistency is guaranteed via cutelimination, where we give the first, to our knowledge, cutelimination procedure in the presence of general inductive and coinductive definitions. 1
A Definitional Approach to Primitive Recursion over Higher Order Abstract Syntax
 In Proceedings of the 2003 workshop on Mechanized
, 2003
"... Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K. ..."
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Cited by 22 (5 self)
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Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.
A Proof Search Specification of the πCalculus
 IN 3RD WORKSHOP ON THE FOUNDATIONS OF GLOBAL UBIQUITOUS COMPUTING
, 2004
"... We present a metalogic that contains a new quantifier (for encoding "generic judgment") and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite πcalculus within this metalogic. Since we ..."
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Cited by 21 (11 self)
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We present a metalogic that contains a new quantifier (for encoding "generic judgment") and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite πcalculus within this metalogic. Since we