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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 69 (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 ...
The BoyerMoore Theorem Prover and Its Interactive Enhancement
, 1995
"... . The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. F ..."
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. The socalled "BoyerMoore Theorem Prover" (otherwise known as "Nqthm") has been used to perform a variety of verification tasks for two decades. We give an overview of both this system and an interactive enhancement of it, "PcNqthm," from a number of perspectives. First we introduce the logic in which theorems are proved. Then we briefly describe the two mechanized theorem proving systems. Next, we present a simple but illustrative example in some detail in order to give an impression of how these systems may be used successfully. Finally, we give extremely short descriptions of a large number of applications of these systems, in order to give an idea of the breadth of their uses. This paper is intended as an informal introduction to systems that have been described in detail and similarly summarized in many other books and papers; no new results are reported here. Our intention here is merely to present Nqthm to a new audience. This research was supported in part by ONR Contract N...
A Theorem Prover for a Computational Logic
, 1990
"... We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of line ..."
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We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of linear resolution, rewriting, and arithmetic decision procedures. We describe some applications of the prover, including a proof of the correct implementation of a higher level language on a microprocessor defined at the gate level. We also describe the ongoing project of recoding the entire prover as an applicative function within its own logic.
Functional instantiation in first order logic
 GROUP, UNIVERSITY OF GOTEBORG
, 1991
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A mechanical proof of the unsolvability of the halting problem
 Journal of the Association for Computing Machinery
, 1984
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Proof theory of MartinLöf type theory. An overview
 MATHEMATIQUES ET SCIENCES HUMAINES, 42 ANNÉE, N O 165:59 – 99
, 2004
"... We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent ..."
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Cited by 4 (2 self)
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We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of MartinLöf type theory with Wtype and one microscopic universe containing only two finite sets in carried out. Then we look at the analysis MartinLöf theory with Wtype and a universe closed under the Wtype, and consider the extension of type theory by one Mahlo universe and its prooftheoretic analysis. Finally we repeat the concept of inductiverecursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.
NonConstructive Computational Mathematics
 Journal of Automated Reasoning
, 1995
"... We describe a nonconstructive extension to Primitive Recursive Arithmetic, both abstractly, and as implemented on the BoyerMoore prover. Abstractly, this extension is obtained by adding the unbounded ¯ operator applied to primitive recursive functions; doing so, one can define the Ackermann functi ..."
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We describe a nonconstructive extension to Primitive Recursive Arithmetic, both abstractly, and as implemented on the BoyerMoore prover. Abstractly, this extension is obtained by adding the unbounded ¯ operator applied to primitive recursive functions; doing so, one can define the Ackermann function and prove the consistency of Primitive Recursive Arithmetic. The implementation does not mention the ¯ operator explicitly, but has the strength to define the ¯ operator through the builtin functions EVAL$ and V&C$. x1. INTRODUCTION This paper is a mixture of theory and practice. The theory begins with the notions of constructivism and finitism in the philosophy of mathematics. As with all philosophical notions, these cannot appear directly in a mathematical theorem or a computer program, but they have been useful guides over the past hundred years to discovering mathematical results, and more recently, to designing computer implementations. Informally, a constructivist only believes in...
Towards a Mechanically Checked Theory of Computation: A Progress Report
, 1999
"... Formal mathematical logic is ideally suited to describing computational processes. We discuss the use of a mechanized mathematical logic, namely ACL2 (A Computational Logic for Applicative Common Lisp) to model computational problems and to prove theorems about such models. 1 Prelude In 1961, John ..."
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Formal mathematical logic is ideally suited to describing computational processes. We discuss the use of a mechanized mathematical logic, namely ACL2 (A Computational Logic for Applicative Common Lisp) to model computational problems and to prove theorems about such models. 1 Prelude In 1961, John McCarthy first presented the seminal paper "A Basis for a Mathematical Theory of Computation " [23]. In that paper McCarthy defined the class of computable functions on some set of base functions. He discussed other fundamental issues, such as the role of noncomputable functions, quantification, functionals and what we now call abstract data types. He introduced the notion of "recursion induction." He used his formal system to prove many now classic elementary theorems in his emerging theory of computation, including distributivity of Peano multiplication over Peano addition and the associativity of the list concatenation function. He also clearly laid the basis for the formal establishment...
Primitive Recursive Arithmetic and its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections In Honor of Per MartinLöf on the Occasion of His Retirement
"... We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal ..."
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We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal paper on primitive recursive arithmetic (PRA), “The foundations of arithmetic established by means of the recursive mode of thought, without use of apparent variables ranging over infinite domains ” [1923], that the paper was written in 1919 after he had studied Whitehead and Russell’s Principia Mathematica and in reaction to that work. His specific complaint about the foundations of arithmetic (i.e. number theory) in that work was, as implied by his title, the essential role in it of logic and in particular quantification over infinite domains, even for the understanding of the most elementary propositions of arithmetic such as polynomial equations; and he set about to eliminate these infinitary quantifications by means of the “recursive mode of thought. ” On this ground, not only polynomial equations, but all primitive recursive formulas stand on their own feet without logical underpinning. 2. Skolem’s 1923 paper did not include a formal system of arithmetic, but as he noted in his 1946 address, “The development of recursive arithmetic” [1947], formalization of the methods used in that paper results in one of the many equivalent systems we refer to as PRA. Let me stop here and briefly describe one such system. ∗Is paper is loosely based on the Skolem Lecture that I gave at the University of Oslo in June, 2010. The present paper has profited, both with respect to what it now contains and with respect to what it no longer contains, from the discussion following that lecture. 1 We admit the following finitist types1 of objects: