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**11 - 17**of**17**### INTRODUCTION TO THE COMBINATORICS AND COMPLEXITY OF CUT ELIMINATION

"... Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted ..."

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Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted

### Streams and Strings in Formal Proofs

"... Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. ..."

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Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.

### unknown title

"... Proof theory is the branch of mathematical logic that investigates mathematical reasoning and mathematical proofs. This area emanated from Hilbert’s program calling for consistency proofs of formal theories. An interesting aspect of many consistency proofs is that they have a largely constructive ch ..."

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Proof theory is the branch of mathematical logic that investigates mathematical reasoning and mathematical proofs. This area emanated from Hilbert’s program calling for consistency proofs of formal theories. An interesting aspect of many consistency proofs is that they have a largely constructive character, relying on a proof transformation (like cut-elimination) or on a proof translation (like the functional interpretation). It is therefore often possible to regard a consistency proof not only as a general justification of a formal system, but also as a method that can be used to analyze concrete proofs formalized in that system. This viewpoint turned out to be very productive and several interesting mathematical results have been obtained by manually applying these methods to existing proofs. But the manual application of such methods is not the only option: Being of an algorithmic nature they can – in principle – be automated. There are, however, different requirements on a method, depending on wether it shall actually be implemented or merely understood to work in principle. The

### G.Moser R.Zach The Epsilon Calculus and

, 2005

"... Abstract. Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In pa ..."

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Abstract. Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure. Keywords: Hilbert’s ε-calculus, epsilon theorems, Herbrand’s theorem, proof complexity 1.

### Journal of Automated Reasoning manuscript No. (will be inserted by the editor) Classical Logic with Partial Functions

"... the date of receipt and acceptance should be inserted later Abstract We introduce a semantics for classical logic with partial functions, in which illtyped formulas are guaranteed to have no truth value, so that they cannot be used in any form of reasoning. The semantics makes it possible to mix rea ..."

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the date of receipt and acceptance should be inserted later Abstract We introduce a semantics for classical logic with partial functions, in which illtyped formulas are guaranteed to have no truth value, so that they cannot be used in any form of reasoning. The semantics makes it possible to mix reasoning about types and preconditions with reasoning about other properties. This makes it possible to deal with partial functions with preconditions of unlimited complexity. We show that, in spite of its increased complexity, the semantics is still a natural generalization of first-order logic with simple types. If one does not use the increased expressivity, the type system is not stronger than classical logic with simple types. We will define two sequent calculi for our semantics, and prove that they are sound and complete. The first calculus follows the semantics closely, and hence its completeness proof is fairly straightforward. The second calculus is further away from the semantics, but more suitable for practical use because it has better proof theoretic properties. Its completeness can be shown by proving that proofs from the first calculus can be translated.