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The Taming of the Cut. Classical Refutations with Analytic Cut
 JOURNAL OF LOGIC AND COMPUTATION
, 1994
"... The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of vi ..."
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Cited by 63 (1 self)
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The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of view. Firstly, they cannot represent the use of auxiliary lemmas in proofs. Secondly, they cannot express the bivalence of classical logic. Thirdly, they are extremely inefficient, as is emphasized by the "computational scandal" that such systems cannot polynomially simulate the truthtables. None of these anomalies occurs if the cut rule is allowed. This raises the problem of formulating a proof system which incorporates a cut rule and yet can provide a suitable model of classical analytic deduction. For this purpose we present an alternative refutation system for classical logic, that we call KE. This system, though being "close" to Smullyan's tableau method, is not cutfree but includes a class...
Syntactic Measures of Complexity
, 1999
"... page 14 Declaration  page 15 Notes of copyright and the ownership of intellectual property rights  page 15 The Author  page 16 Acknowledgements  page 16 1  Introduction  page 17 1.1  Background  page 17 1.2  The Style of Approach  page 18 1.3  Motivation  page 19 1.4  Style of ..."
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Cited by 40 (2 self)
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page 14 Declaration  page 15 Notes of copyright and the ownership of intellectual property rights  page 15 The Author  page 16 Acknowledgements  page 16 1  Introduction  page 17 1.1  Background  page 17 1.2  The Style of Approach  page 18 1.3  Motivation  page 19 1.4  Style of Presentation  page 20 1.5  Outline of the Thesis  page 21 2  Models and Modelling  page 23 2.1  Some Types of Models  page 25 2.2  Combinations of Models  page 28 2.3  Parts of the Modelling Apparatus  page 33 2.4  Models in Machine Learning  page 38 2.5  The Philosophical Background to the Rest of this Thesis  page 41 Syntactic Measures of Complexity  page 3  3  Problems and Properties  page 44 3.1  Examples of Common Usage  page 44 3.1.1  A case of nails  page 44 3.1.2  Writing a thesis  page 44 3.1.3  Mathematics  page 44 3.1.4  A gas  page 44 3.1.5  An ant hill  page 45 3.1.6  A car engine  page 45 3.1.7  A cell as part of an organism ...
Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
Wellordering proofs for MartinLöf Type Theory
 Annals of Pure and Applied Logic
, 1998
"... We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , whi ..."
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Cited by 24 (11 self)
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We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is slightly more than the strength of Feferman's theory T 0 , classical set theory KPI and the subsystem of analysis (# 1 2 CA)+(BI). The strength of intensional and extensional version, of the version a la Tarski and a la Russell are shown to be the same. 0 Introduction 0.1 Proof theory and Type Theory Proof theory and type theory have been two answers of mathematical logic to the crisis of the foundations of mathematics at the beginning of the century. Proof theory was originally established by Hilbert in order to prove the consistency of theories by using finitary methods. When Godel showed that Hilbert's program cannot be carried out as originally intended, the focus of proof theory ch...
Basic logic: reflection, symmetry, visibility
 Journal of Symbolic Logic
, 1997
"... Abstract We introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic, quantum and nonmodal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate t ..."
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Abstract We introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic, quantum and nonmodal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterize B positively: reflection, symmetry and visibility. A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection. To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cutelimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube.
Foundations of BQO Theory and Subsystems of SecondOrder Arithmetic
 Pennsylvania State University
, 1993
"... Abstract. We consider the reverse mathematics of wqo and bqo theory. We survey the literature on the subject, which deals mainly with the more advanced results about wqos and bqos, and prove some new results about the elementary properties of these combinatorial structures. We state several open pro ..."
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Abstract. We consider the reverse mathematics of wqo and bqo theory. We survey the literature on the subject, which deals mainly with the more advanced results about wqos and bqos, and prove some new results about the elementary properties of these combinatorial structures. We state several open problems about the axiomatic strength of both elementary and advanced results. A quasiordering (i.e. a reflexive and transitive binary relation) is awqo (well quasiordering) if it contains no infinite descending chains and no infinite sets of pairwise incomparable elements. This concept is very natural, and has been introduced several times, as documented in [19]. The usual working definition of wqo is obtained from the one given above with an application of Ramsey’s theorem: a quasiordering on the setQ is wqo if for every sequence {xn  n ∈ N}of elements ofQ there existm < n such thatxm xn. The notion of bqo (better quasiordering) is a strengthening of wqo which was introduced byNashWilliams in the 1960’s in a sequence of papers culminating in [30] and [31]. This notion has proved to be very useful in showing that specific quasiorderings are indeed wqo. Moreover the property of being bqo is preserved
CutElimination: Experiments with CERES
, 2005
"... Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set of ..."
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Cited by 16 (12 self)
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Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set can then serve as a skeleton of a proof with only atomic cuts. In this paper we present a systematic experiment with the implementation of CERES on a proof of reasonable size and complexity. It turns out that the proof with cuts can be transformed into two mathematically different proofs of the theorem. In particular, the application of positive and negative hyperresolution yield different mathematical arguments. As an unexpected sideeffect the derived clauses of the resolution refutation proved particularly interesting as they can be considered as meaningful universal lemmas. Though the proof under investigation is intuitively simple, the experiment demonstrates that new (and relevant) mathematical information on proofs can be obtained by computational methods. It can be considered as a first step in the development of an experimental culture of computeraided proof analysis in mathematics.
Are Tableaux an Improvement on TruthTables? CutFree proofs and Bivalence
, 1992
"... We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance bet ..."
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Cited by 15 (0 self)
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We show that Smullyan's analytic tableaux cannot psimulate the truthtables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cutfree proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableaulike method without affecting its "analytic" nature. 1 Introduction The truthtable method, introduced by Wittgenstein in his Tractatus LogicoPhilosophicus, provides a decision procedure for propositional logic which is immediately implementable on a machine. However this timehonoured method is usually mentioned only to be immediately dismissed because of its incurable inefficiency. The wellknown tableau method (which is closely related to Gentzen's cutfree sequent calculus) is commonly regarded as a "shortcut" in testing the logical validity of complex propositions...
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 ..."
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Cited by 15 (5 self)
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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