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10
Metalogical Frameworks
, 1992
"... In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the me ..."
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Cited by 60 (18 self)
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In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...
A unified approach to Type Theory through a refined λcalculus
, 1994
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. ..."
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Cited by 14 (13 self)
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic.
An implementation of Type:Type
, 2000
"... This paper is a complement to [5]. Here we were using untyped abstraction and could not use a domain model. 8 References ..."
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This paper is a complement to [5]. Here we were using untyped abstraction and could not use a domain model. 8 References
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
Abstract
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
Abstract
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(Show Context)
In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
Abstract
 Add to MetaCart
(Show Context)
In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y
Craig Interpolation, ProofTheoretically via Nested Sequents
, 2013
"... Formal proofs are becoming increasingly important in a number of domains in computer science and mathematics. The topic of the colloquium is structural proof theory, broadly construed. Some examples of relevant topics: Structure Sequential and parallel structure in proofs; sharing and duplication of ..."
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Formal proofs are becoming increasingly important in a number of domains in computer science and mathematics. The topic of the colloquium is structural proof theory, broadly construed. Some examples of relevant topics: Structure Sequential and parallel structure in proofs; sharing and duplication of proofs; permutation of proof steps; canonical forms; focusing; polarities; graphical proof syntax; proof complexity; cuteliminiation strategies; CERes Proof Checking Generating, transmitting, translating, and checking proof objects; universal proof languages; proof certificates; proof compression; cutintroduction; certification of highperformance systems (SMT, resolution, etc.) Proof Search Automated and interactive proof search in constrained logics (linear, temporal, bunched, probabilistic, etc.); mixing deduction and computation; induction and coinduction; cyclic proofs; computational interpretations List of talks (in presentation order)
A TypeFree Formalization of Mathematics Where Proofs Are Objects
, 1996
"... We present a first order untyped axiomatization of mathematics where proofs are objects in the sense of HeytingKolmogorov functional interpretation. The consistency of this theory is open. ..."
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We present a first order untyped axiomatization of mathematics where proofs are objects in the sense of HeytingKolmogorov functional interpretation. The consistency of this theory is open.
COMPUTING SCIENCE NOTES
, 1992
"... A unified approach to Type Theory through a refined Acalculus by ..."