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The modal logic of provability: Cut-elimination
Journal of Philosophical Logic, 1983\textit{D. Leivant} [J. Symb. Logic 46, 531--538 (1981; Zbl 0464.03019)] outlined a cut-elimination procedure for the formulation of the provability logic GL based on the only modal rule \(X,\square X,\square A\to A/\square X\to \square A\). The author and \textit{G. Sambin} [J. Philos.
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Cut-elimination and interpolation for Ω-logic
Archive for Mathematical Logic, 1988Girard's \(\Omega\)-logic is based on the category of well-founded orders, while his \(\beta\)-logic is based on the category of ordinals. Here Girard's (unpublished) results are extended from \(\beta\)-logic to \(\Omega\)-logic.
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A Connection Between Cut Elimination and Normalization
Archive for Mathematical Logic, 2005Summary: Sequent systems for classical and intuitionistic logic and natural deduction systems for these logics are characterized by two important theorems. Sequent systems are characterized by cut-elimination theorems, and natural deduction systems by normalization theorems.
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2010
Our aim is to compare different methods of cut-elimination. For this aim we need logic-free axioms. The original formulation of LK by Gentzen also served the purpose of simulating Hilbert-type calculi and deriving axiom schemata within fixed proof length. Below we show that there exists a polynomial transformation from an LK-proof with arbitrary axioms
Matthias Baaz, Alexander Leitsch
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Our aim is to compare different methods of cut-elimination. For this aim we need logic-free axioms. The original formulation of LK by Gentzen also served the purpose of simulating Hilbert-type calculi and deriving axiom schemata within fixed proof length. Below we show that there exists a polynomial transformation from an LK-proof with arbitrary axioms
Matthias Baaz, Alexander Leitsch
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2010
In Chapter 5 we analyzed methods which eliminate cuts by stepwise reduction of cut-complexity. These methods always identify the uppermost logical operator in the cut-formula and either eliminate it directly (grade reduction) or indirectly (rank reduction).
Matthias Baaz, Alexander Leitsch
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In Chapter 5 we analyzed methods which eliminate cuts by stepwise reduction of cut-complexity. These methods always identify the uppermost logical operator in the cut-formula and either eliminate it directly (grade reduction) or indirectly (rank reduction).
Matthias Baaz, Alexander Leitsch
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2020
In this chapter we first introduce a standard cut-elimination procedure for the first-order logic and the \(\omega \)-logic, by which we see that the tree-height of the resulting cut-free proofs is bounded by a tower of exponential functions. Such a control of the tree-heights in cut-elimination is one of the most important results in ordinal analysis.
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In this chapter we first introduce a standard cut-elimination procedure for the first-order logic and the \(\omega \)-logic, by which we see that the tree-height of the resulting cut-free proofs is bounded by a tower of exponential functions. Such a control of the tree-heights in cut-elimination is one of the most important results in ordinal analysis.
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Cut elimination with applications
2000The “applications of cut elimination” in the title of this chapter may perhaps be described more appropriately as “applications of cutfree systems”, since the applications are obtained by analyzing the structure of cutfree proofs; and in order to prove that the various cutfree systems are adequate for our standard logics all we need to know is that ...
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2015
We present methods for removing top-level cuts from a sequent calculus or Tait-style proof without significantly increasing the space used for storing the proof. For propositional logic, this requires converting a proof from tree-like to dag-like form, but at most doubles the number of lines in the proof.
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We present methods for removing top-level cuts from a sequent calculus or Tait-style proof without significantly increasing the space used for storing the proof. For propositional logic, this requires converting a proof from tree-like to dag-like form, but at most doubles the number of lines in the proof.
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1999
Preface. Introduction. 1. Categories. 2. Functors. 3. Natural Transformations. 4. Adjunctions. 5. Comonads. 6. Cartesian Categories. Conclusion. References. Index.
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Preface. Introduction. 1. Categories. 2. Functors. 3. Natural Transformations. 4. Adjunctions. 5. Comonads. 6. Cartesian Categories. Conclusion. References. Index.
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