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317
The Foundation of a Generic Theorem Prover
 Journal of Automated Reasoning
, 1989
"... Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabell ..."
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Cited by 471 (49 self)
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Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabelle is now based on higherorder logic  a precise and wellunderstood foundation. Examples illustrate use of this metalogic to formalize logics and proofs. Axioms for firstorder logic are shown sound and complete. Backwards proof is formalized by metareasoning about objectlevel entailment. Higherorder logic has several practical advantages over other metalogics. Many proof techniques are known, such as Huet's higherorder unification procedure. Key words: higherorder logic, higherorder unification, Isabelle, LCF, logical frameworks, metareasoning, natural deduction Contents 1 History and overview 2 2 The metalogic M 4 2.1 Syntax of the metalogic ......................... 4 2.2 ...
A TypeTheoretic Approach to HigherOrder Modules with Sharing
, 1994
"... The design of a module system for constructing and main taining large programs is a difficult task that raises a number of theoretical and practical issues. A fundamental issue is the management of the flow of information between program units at compile time via the notion of an interface. Experie ..."
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Cited by 271 (27 self)
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The design of a module system for constructing and main taining large programs is a difficult task that raises a number of theoretical and practical issues. A fundamental issue is the management of the flow of information between program units at compile time via the notion of an interface. Experience has shown that fully opaque interfaces are awkward to use in practice since too much information is hidden, and that fully transparent interfaces lead to excessive interdependencies, creating problems for maintenance and separate compilation. The "sharing" specifications of Standard ML address this issue by allowing the programmer to specify equational relationships between types in separate modules, but are not expressive enough to allow the programmer com plete control over the propagation of type information be tween modules.
A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. ..."
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Cited by 238 (49 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
A Judgmental Reconstruction of Modal Logic
 Mathematical Structures in Computer Science
, 1999
"... this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deductio ..."
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Cited by 193 (47 self)
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this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as
Eliminating Array Bound Checking Through Dependent Types
 In Proceedings of ACM SIGPLAN Conference on Programming Language Design and Implementation
, 1998
"... We present a typebased approach to eliminating array bound checking and list tag checking by conservatively extending Standard ML with a restricted form of dependent types. This enables the programmer to capture more invariants through types while typechecking remains decidable in theory and can s ..."
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Cited by 189 (24 self)
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We present a typebased approach to eliminating array bound checking and list tag checking by conservatively extending Standard ML with a restricted form of dependent types. This enables the programmer to capture more invariants through types while typechecking remains decidable in theory and can still be performed efficiently in practice. We illustrate our approach through concrete examples and present the result of our preliminary experiments which support support the feasibility and effectiveness of our approach. 1 Introduction The absence of runtime array bound checks is an infamous source of fatal errors for programs in languages such as C. Nonetheless, compilers offer the option to omit array bound checks, since they can turn out to be expensive in practice (Chow 1983; Gupta 1994). In statically typed languages such as ML, one would like to provide strong guarantees about the safety of all operations, so array bound checks cannot be omitted in general. The same is true for Ja...
Using dependent types to express modular structure
 In Thirteenth ACM Symposium on Principles of Programming Languages
, 1986
"... Several related typed languages for modular programming and data abstraction have been proposed recently, including Pebble, SOL, and ML modules. We review and compare the basic typetheoretic ideas behind these languages and evaluate how they ..."
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Cited by 134 (5 self)
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Several related typed languages for modular programming and data abstraction have been proposed recently, including Pebble, SOL, and ML modules. We review and compare the basic typetheoretic ideas behind these languages and evaluate how they
The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Cited by 130 (0 self)
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
Equality In Lazy Computation Systems
, 1998
"... In this paper we introduce a general class of lazy computation systems and define a natural program equivalence for them. We prove that if an extensionality condition holds of each of the operators of a computation system, then the equivalence relation is a congruence, so that the usual kinds of equ ..."
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Cited by 108 (6 self)
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In this paper we introduce a general class of lazy computation systems and define a natural program equivalence for them. We prove that if an extensionality condition holds of each of the operators of a computation system, then the equivalence relation is a congruence, so that the usual kinds of equality reasoning are valid for it. This condition is a simple syntactic one, and is easy to verify for the various lazy computation systems we have considered so far. We also give conditions under which the equivalence coincides with observational congruence. These results have some important consequences for type theories like those of MartinLöf and Nuprl.