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**11 - 15**of**15**### 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

### Reflection Using the Derivability Conditions

"... We extend arithmetic with a new predicate, Pr, giving axioms for Pr based on first-order versions of Lob's derivability conditions. We hoped that the addition of a reflection schema mentioning Pr would then give a non-conservative extension of the original arithmetic theory. The paper investiga ..."

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We extend arithmetic with a new predicate, Pr, giving axioms for Pr based on first-order versions of Lob's derivability conditions. We hoped that the addition of a reflection schema mentioning Pr would then give a non-conservative extension of the original arithmetic theory. The paper investigates this possibility. It is shown that, under special conditions, the extension is indeed non-conservative. However, in general such extensions turn out to be conservative. 1 Introduction In any recursively axiomatized theory of arithmetic, T , one can follow Godel's construction to obtain a `provability predicate', a \Sigma 1 -formula Bew T (x) such that Bew T (pAq) is true if and only if T ` A, where pAq is the Godel number of the formula A. Moreover, if T is sufficiently strong then Bew T satisfies the following predicate (or `uniform') versions of Lob's derivability conditions [7]: (D1) if T ` 8xA then T ` 8xBew T (pAhxiq); (D2) T ` 8x(Bew T (p(A ! B)hxiq) ! (Bew T (pAhxiq) ! Bew T (pBhxiq)...

### A non-well-founded primitive recursive tree provably well-founded

, 2001

"... for co-r.e. sets ..."

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### System Description: CutRes 0.1: Cut Elimination by Resolution

"... . CutRes is a system which takes as input an LK-proof with arbitrary cuts and skolemized end-sequent and gives as output an LK- proof with atomic cuts only. The elimination of cuts is performed in the following way: An unsatisfiable set of clauses C is assigned to a given LK-proof \Pi. Any resol ..."

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. CutRes is a system which takes as input an LK-proof with arbitrary cuts and skolemized end-sequent and gives as output an LK- proof with atomic cuts only. The elimination of cuts is performed in the following way: An unsatisfiable set of clauses C is assigned to a given LK-proof \Pi. Any resolution refutation / of C then serves as a skeleton for an LK-proof \Sigma of the original end-sequent, containing only atomic cuts; \Sigma can be constructed from / and \Pi by projections. Note, that a proof with atomic cuts provides the same information as a cut-free proof. 1 Introduction The use of lemmas in mathematical proofs lies at the very centre of mathematical reasoning; by the use of lemmas it is possible to structure the proof and hence to improve its readability and understandability. Therefore a real mathematical proof (modelled within a sequent calculus) always contains cuts; in the practice of mathematics the elimination of these cuts is, in most cases, not only useless bu...

### Pathways of deduction A. Carbone

"... Cyclic structures underlie formal mathematical reasoning, and replication and folding play a crucial role in the complexity of proofs. These two aspects of the geometry of proofs are discussed. 1 Deductions, foldings and the brain Different models of various regions of the brain have been proposed a ..."

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Cyclic structures underlie formal mathematical reasoning, and replication and folding play a crucial role in the complexity of proofs. These two aspects of the geometry of proofs are discussed. 1 Deductions, foldings and the brain Different models of various regions of the brain have been proposed and they stimulated the discussion on the way our mind works. The essential feature of most of these models is the hierarchical structure which is underlying the organization. What we “see ” is nevertheless not necessarily the basic mechanism. Recent studies in computational complexity and proof theory reveal that hierarchical organizations, even though structurally appealing, are computationally inefficient. In fact, our brain seems to be “fast ” in performing certain tasks (such as perceiving the presence of an animal in the landscape, or intuitively grasping a complicated mathematical idea) and extremely “slow ” in performing others (as the construction of a mathematical