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129
Lectures on proof theory
 in Proc. Summer School in Logic, Leeds 67
, 1968
"... This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of purely ni ..."
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This is a survey of some of the principal developments in proof theory from its inception in the 1920s, at the hands of David Hilbert, up to the 1960s. Hilbert's aim was to use this as a tool in his nitary consistency program to eliminate the \actual in nite " in mathematics from proofs of purely nitary statements. One of the main approaches that turned out to be the most useful in pursuit of this program was that due to Gerhard Gentzen, in the 1930s, via his calculi of \sequents" and his CutElimination Theorem for them. Following that we trace how and why prima facie in nitary concepts, such as ordinals, and in nitary methods, such as the use of in nitely long proofs, gradually came to dominate prooftheoretical developments. In this rst lecture I will give anoverview of the developments in proof theory since Hilbert's initiative in establishing the subject in the 1920s. For this purpose I am following the rst part of a series of expository lectures that I gave for the Logic Colloquium `94 held in ClermontFerrand 2123 July 1994, but haven't published. The theme of my lectures there was that although Hilbert established his theory of proofs as a part of his foundational program and, for philosophical reasons whichwe shall get into, aimed to have it developed in a completely nitistic way, the actual work in proof theory This is the rst of three lectures that I delivered at the conference, Proof Theory: History
Natural Deduction for NonClassical Logics
, 1996
"... We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke m ..."
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We present a framework for machine implementation of families of nonclassical logics with Kripkestyle semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.
Algorithms for ordinal arithmetic
 In 19th International Conference on Automated Deduction (CADE
, 2003
"... Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is A ..."
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Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1
Admissibility of Structural Rules for ContractionFree Systems of Intuitionistic Logic
 Journal of Symbolic Logic
, 2000
"... We give a direct proof of admissibility of cut and contraction for the contractionfree sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multisuccedent calculus; this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs, i.e. ..."
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We give a direct proof of admissibility of cut and contraction for the contractionfree sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multisuccedent calculus; this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs, i.e. those which use induction on sequent weight or appeal to admissibility of rules in other calculi.
Cutelimination and proofsearch for biintuitionistic logic using nested sequents
, 2008
"... We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant cal ..."
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We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proofsearch strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for biintuitionistic logic. As far as we know, our new calculus is the first sequent calculus for biintuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cutelimination proof, and which can be used naturally for backwards proofsearch.
Choice and uniformity in weak applicative theories
 Logic Colloquium ’01
, 2005
"... Abstract. We are concerned with first order theories of operations, based on combinatory logic and extended with the type W of binary words. The theories include forms of “positive ” and “bounded ” induction on W and naturally characterize primitive recursive and polytime functions (respectively). W ..."
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Abstract. We are concerned with first order theories of operations, based on combinatory logic and extended with the type W of binary words. The theories include forms of “positive ” and “bounded ” induction on W and naturally characterize primitive recursive and polytime functions (respectively). We prove that the recursive content of the theories under investigation (i.e. the associated class of provably total functions on W) is invariant under addition of 1. an axiom of choice for operations and a uniformity principle, restricted to positive conditions; 2. a (form of) selfreferential truth, providing a fixed point theorem for predicates. As to the proof methods, we apply a kind of internal forcing semantics, nonstandard variants of realizability and cutelimination. §1. Introduction. In this paper, we deal with theories of abstract computable operations, underlying the socalled explicit mathematics, introduced by Feferman in the midseventies as a logical frame to formalize Bishop’s style constructive mathematics ([18], [19]). Following a common usage, these theories
Validated ProofProducing Decision Procedures
, 2004
"... A widely used technique to integrate decision procedures (DPs) with other systems is to have the DPs emit proofs of the formulas they report valid. One problem that arises is debugging the proofproducing code; it is very easy in standard programming languages to write code which produces an incorre ..."
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A widely used technique to integrate decision procedures (DPs) with other systems is to have the DPs emit proofs of the formulas they report valid. One problem that arises is debugging the proofproducing code; it is very easy in standard programming languages to write code which produces an incorrect proof. This paper demonstrates how proofproducing DPs may be implemented in a programming language, called RogueSigmaPi (RSP), whose type system ensures that proofs are manipulated correctly. RSP combines the Rogue rewriting language and the Edinburgh Logical Framework (LF). Typecorrect RSP programs are partially correct: essentially, any putative LF proof object produced by a typecorrect RSP program is guaranteed to type check in LF. The paper describes a simple proofproducing combination of propositional satisfiability checking and congruence closure implemented in RSP.
Seeking Explanations: Abduction in Logic, Philosophy of Science and Artifical Intelligence
, 1997
"... 175 Bibliography 177 viii Acknowledgements It is a privilege to have five professors on my reading committee representing the different areas of my Ph.D. program in Philosophy and Symbolic Systems: Computer Science, Linguistics, Logic, Philosophy, and Psychology. Tom Wasow was the first person to p ..."
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175 Bibliography 177 viii Acknowledgements It is a privilege to have five professors on my reading committee representing the different areas of my Ph.D. program in Philosophy and Symbolic Systems: Computer Science, Linguistics, Logic, Philosophy, and Psychology. Tom Wasow was the first person to point me in the direction of abduction. He gave me useful comments on earlier versions of this dissertation, always insisting that it be readable to non experts. It was also a pleasure to work with him this last year coordinating the undergraduate Symbolic Systems program. Dagfinn Føllesdal encouraged me to continue exploring connections between abduction and philosophy of science. Yoav Shoham and Pat Suppes gave me very good advice about future expansions of this work. Jim Greeno chaired my defense and also gave me helpful suggestions. For their help with my dissertation, and for the classes in which they taught me, I am very grateful. To my advisor, Johan van Benthem, I offer my deepest gr...
Ordinal arithmetic in ACL2
 In ACL2 Workshop 2003
, 2003
"... Abstract. Ordinals form the basis for termination proofs in ACL2. Currently, ACL2 uses a rather inefficient representation for the ordinals up to ɛ0 and provides limited support for reasoning about them. We present algorithms for ordinal arithmetic on an exponentially more compact representation tha ..."
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Abstract. Ordinals form the basis for termination proofs in ACL2. Currently, ACL2 uses a rather inefficient representation for the ordinals up to ɛ0 and provides limited support for reasoning about them. We present algorithms for ordinal arithmetic on an exponentially more compact representation than the one used by ACL2. The algorithms have been implemented and numerous properties of the arithmetic operators have been mechanically verified, thereby greatly extending ACL2’s ability to reason about the ordinals. We describe how to use the libraries containing these results, which are currently distributed with ACL2 version 2.7. 1