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Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
Functional interpretation and inductive definitions
 Journal of Symbolic Logic
"... Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1. ..."
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Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1.
On the Computational Content of the BolzanoWeierstraß Principle
, 2009
"... We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with nega ..."
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We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with negative translation) as well as the monotone functional interpretation of BW for the product space ∏i∈N[−k i, k i] (with the standard product metric). This results in optimal program and bound extraction theorems for proofs based on fixed instances of BW, i.e. for BW applied to fixed sequences in ∏i∈N[−k i, k i].
The computational content of classical arithmetic ∗
, 2009
"... Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various m ..."
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Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical firstorder arithmetic, and reflects on some of the relationships between them. Variants of the GödelGentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory. 1
PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents
TERM EXTRACTION AND RAMSEY’S THEOREM FOR PAIRS
"... Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This ..."
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Abstract. In this paper we study with prooftheoretic methods the function(al)s provably recursive relative to Ramsey’s theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functionals provably recursive from RCA0 + COH + Π0 1CP are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that WKL0 + Π0 1CP + COH is Π0 2conservative over PRA. Recent work of the first author showed that Π0 1CP + COH is equivalent to a weak variant of the BolzanoWeierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey’s theorem for pairs and two colors (RT2 2) we obtain the upper bounded that the type 2 functionals provable recursive relative to RCA0 + Σ0 2IA+RT2 2 are in T1. This is the fragment of Gödel’s system T containing only type 1 recursion — roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T1bounds from proofs that use RT2 2. Moreover, this yields a new proof of the fact that WKL0 + Σ0 2IA + RT2 2 is Π0 3conservative over RCA0 + Σ0 2IA. The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel’s functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using Π0 1comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard’s ordinal analysis of bar recursion (for RT2 2). 1.
On quantitative versions of theorems due to F. E. Browder and R. Wittmann
, 2010
"... This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the conve ..."
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This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder’s original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann’s proof follows a similar line of reasoning and we adapt our analysis of Browder’s proof to get a quantitative version of Wittmann’s theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder’s theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the socalled demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon’s nonlinear ergodic theorem.