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A Framework for Defining Logics
 JOURNAL OF THE ASSOCIATION FOR COMPUTING MACHINERY
, 1993
"... The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of ariti ..."
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Cited by 716 (39 self)
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The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higherorder judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logicindependent tools such as proof editors and proof checkers can be constructed.
Using Typed Lambda Calculus to Implement Formal Systems on a Machine
 Journal of Automated Reasoning
, 1992
"... this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of ..."
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Cited by 85 (14 self)
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this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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Cited by 59 (2 self)
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
A Trustworthy Proof Checker
 IN ILIANO CERVESATO, EDITOR, WORKSHOP ON THE FOUNDATIONS OF COMPUTER SECURITY
, 2002
"... ProofCarrying Code (PCC) and other applications in computer security require machinecheckable proofs of properties of machinelanguage programs. The main advantage of the PCC approach is that the amount of code that must be explicitly trusted is very small: it consists of the logic in which predic ..."
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Cited by 31 (7 self)
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ProofCarrying Code (PCC) and other applications in computer security require machinecheckable proofs of properties of machinelanguage programs. The main advantage of the PCC approach is that the amount of code that must be explicitly trusted is very small: it consists of the logic in which predicates and proofs are expressed, the safety predicate, and the proof checker. We have built a minimal proof checker, and we explain its design principles, and the representation issues of the logic, safety predicate, and safety proofs. We show that the trusted computing base (TCB) in such a system can indeed be very small. In our current system the TCB is less than 2,700 lines of code (an order of magnitude smaller even than other PCC systems) which adds to our confidence of its correctness.
Permutability of Proofs in Intuitionistic Sequent Calculi
, 1996
"... We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the sa ..."
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Cited by 24 (4 self)
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We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising. Key words: intuitionistic logic, proof theory, natural deduction, sequent calculus. 1 Introduction There is a folklore theorem that two intuitionistic sequent calculus derivations are "really the same" iff they are interpermutable, using permutations as described by Kleene in [13]. Our purpose here is to make precise and prove such a "permutability theorem". Prawitz [18] showed how intuitionistic sequent calculus derivations determine natural deductions, via a mapping ' from LJ to NJ (here we consider only ...
A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Cited by 24 (7 self)
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...
Faster Proof Checking in the Edinburgh Logical Framework
 In 18th International Conference on Automated Deduction
, 2002
"... This paper describes optimizations for checking proofs represented in the Edinburgh Logical Framework (LF). The optimizations allow large proofs to be checked eciently which cannot feasibly be checked using the standard algorithm for LF. The crucial optimization is a form of result caching. To f ..."
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Cited by 17 (3 self)
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This paper describes optimizations for checking proofs represented in the Edinburgh Logical Framework (LF). The optimizations allow large proofs to be checked eciently which cannot feasibly be checked using the standard algorithm for LF. The crucial optimization is a form of result caching. To formalize this optimization, a path calculus for LF is developed and shown equivalent to a standard calculus.
Generating Proofs from a Decision Procedure
 Proceedings of the FLoC Workshop on RunTime Result Verification
, 1999
"... Fully automatic decision procedures are used to improve performance in many different applications of formal verification. In most cases, the decision procedures are treated as trusted components of the verification system. Because the decision procedures may be experimental and highly complex to ..."
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Cited by 17 (0 self)
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Fully automatic decision procedures are used to improve performance in many different applications of formal verification. In most cases, the decision procedures are treated as trusted components of the verification system. Because the decision procedures may be experimental and highly complex tools, it is desirable to have a way of independently confirming their results.
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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Cited by 5 (2 self)
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Adding Metatheoretic Facilities to FirstOrder Theories
, 1995
"... Generic proof systems like Isabelle provide some limited but useful metatheoretic facilities for declared logics; in particular, users can prove simple derived rules and also `solve' formulae that contain metavariables  a technique useful for, e.g., program synthesis. We show how an arbi ..."
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Cited by 5 (1 self)
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Generic proof systems like Isabelle provide some limited but useful metatheoretic facilities for declared logics; in particular, users can prove simple derived rules and also `solve' formulae that contain metavariables  a technique useful for, e.g., program synthesis. We show how an arbitrary first order theory can be conservatively extended to provide similar facilities, without a supporting metatheory, and examine what the limitations of this approach are. Keywords: computer assisted proof, logical frameworks, metatheoretic extensibility, secondorder logic, proof theory. 1 Introduction Using a generic proof development system like Isabelle [12] to prove theorems of a logic is sometimes easier than using a custombuilt prover. This is because the generic system contains a metalogic in which object logics are encoded and this may allow extra methods of proof construction. In this paper we look at two examples of this. First, in the metalogic, users can derive new rules or...