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The Theory of LEGO  A Proof Checker for the Extended Calculus of Constructions
, 1994
"... LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO ..."
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LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO is intended to be used for interactively constructing proofs in mathematical theories presented in these logics. I have developed LEGO over six years, starting from an implementation of the Calculus of Constructions by G erard Huet. LEGO has been used for problems at the limits of our abilities to do formal mathematics. In this thesis I explain some aspects of the metatheory of LEGO's type systems leading to a machinechecked proof that typechecking is decidable for all three type theories supported by LEGO, and to a verified algorithm for deciding their typing judgements, assuming only that they are normalizing. In order to do this, the theory of Pure Type Systems (PTS) is extended and f...
A Computational Architecture for Heterogeneous Reasoning
 Proceedings of the Seventh Conference on Theoretical Aspects of Rationality and Knowledge
, 1998
"... Reasoning, problem solving, indeed the general process of acquiring knowledge, is not an isolated, homogenous affair involving a one agent using a single form of representation, but more typically a complicated, collaborative, heterogeneous activity. This paper describes an effort to expand our unde ..."
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Cited by 5 (0 self)
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Reasoning, problem solving, indeed the general process of acquiring knowledge, is not an isolated, homogenous affair involving a one agent using a single form of representation, but more typically a complicated, collaborative, heterogeneous activity. This paper describes an effort to expand our understanding of such reasoning and to develop tools to enable individuals and groups to use computers more effectively in practical problemsolving tasks. Natural deduction and problem solving A recent article in the New York Times reported the discovery of mass in the neutrino by a team of 120 scientists from 23 research institutions. The discovery involved the design and construction of a massive experiment involving a tank inside a deep zinc mine, filled with 12.5 million gallons of water, and equipped with specially designed light amplifiers covering the inside of the tank. Using this setup as a neutrino detector to compare flavors of neutrinos coming directly from the atmosphere versus those coming through the earth, the scientists were able to determine that some neutrinos changed flavor in passing through the earth. The discovery also had a logical element, with mass being the only plausible explanation for the observations consistent with quantum theory that could not be ruled out in one way or another. As this example illustrates, largescale collaborative projects involve many people reasoning toward the solution of a common problem over an extended period of time. The design and construction of a product, for example, whether a scientific apparatus, a building, or a complex hardware or software system, often involves a multidisciplinary team of clients and engineers working toward a common goal. Such distributed reasoning projects frequently yield less than optimal...
An Interpretation of Kleene's Slash in Type Theory
 Informal Proceedings of the Second Workshop on Logical Frameworks, pages 337342. Esprit Basic Research Action
, 1993
"... Kleene introduced the notion of slash to investigate the disjunction and existence properties under implication for intuitionistic arithmetic. In this paper Kleene's slash is translated to type theory. Besides translations of Kleene's results, the main application of the slash in type theory is that ..."
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Kleene introduced the notion of slash to investigate the disjunction and existence properties under implication for intuitionistic arithmetic. In this paper Kleene's slash is translated to type theory. Besides translations of Kleene's results, the main application of the slash in type theory is that conditions are given for a typable term, containing free variables, to have a normal form beginning with a constructor. 1 Introduction The disjunction and existence properties, that is, ` AB implies ` A or ` B and ` 9xA(x) implies ` A(t) for some term t , respectively, were first proved for intuitionistic arithmetic by Kleene [9] using a modification of recursive realizability. Harrop [8] extended Kleene's result by also considering derivations depending on assumptions. Harrop proved C ` A B implies C ` A or C ` B (ED) C ` 9xA(x) implies C ` A(t) for some term t (EE) where C is a closed formula not containing any strictly positive occurrences of and 9 ; such a formula is called a Har...
DAG PRAWITZ MEANING APPROACHED VIA PROOFS
"... ABSTRACT. According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subar ..."
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ABSTRACT. According to a main idea of Gentzen the meanings of the logical constants are reflected by the introduction rules in his system of natural deduction. This idea is here understood as saying roughly that a closed argument ending with an introduction is valid provided that its immediate subarguments are valid and that other closed arguments are justified to the extent that they can be brought to introduction form. One main part of the paper is devoted to the exact development of this notion. Another main part of the paper is concerned with a modification of this notion as it occurs in Michael Dummett’s book The Logical Basis of Metaphysics. The two notions are compared and there is a discussion of how they fare as a foundation for a theory of meaning. It is noted that Dummett’s notion has a simpler structure, but it is argued that it is less appropriate for the foundation of a theory of meaning, because the possession of a valid argument for a sentence in Dummett’s sense is not enough to be warranted to assert the sentence. 1.