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Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
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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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Cited by 9 (1 self)
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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
RAMSEY’S THEOREM FOR PAIRS AND PROVABLY RECURSIVE FUNCTIONS
"... Abstract. This paper addresses the strength of Ramsey’s theorem for pairs (RT2 2) over a weak base theory from the perspective of ‘proof mining’. Let RT 2− 2 denote Ramsey’s theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a ..."
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Abstract. This paper addresses the strength of Ramsey’s theorem for pairs (RT2 2) over a weak base theory from the perspective of ‘proof mining’. Let RT 2− 2 denote Ramsey’s theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König’s lemma and a substantial amount of Σ0 1induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of ∀∃theorems. There are two components this work. The first component is a general prooftheoretic result, due to the second author ([13, 14]), that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey’s theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on RT2 2 the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of RT2 2 that are primitive recursive in f. 1.
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
Applied Foundations: Proof Mining in Mathematics
"... A central theme in the foundations of mathematics, dating back to D. Hilbert, can be paraphrased by the following question ‘How is it that abstract methods (‘ideal elements’) can be used to prove ‘real ’ statements e.g. about the natural numbers and is this use necessary in principle?’ Hilbert’s aim ..."
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A central theme in the foundations of mathematics, dating back to D. Hilbert, can be paraphrased by the following question ‘How is it that abstract methods (‘ideal elements’) can be used to prove ‘real ’ statements e.g. about the natural numbers and is this use necessary in principle?’ Hilbert’s aim was to show that the use of such ideal elements can be shown to be consistent by finitistic means (‘Hilbert’s program’). Hilbert’s program turned out to be impossible in the original form by the seminal results of K. Gödel. However, more recent developments show it can be carried out in a partial form in that one can design formal systems A which are sufficient to formalize substantial parts of mathematics and yet can be reduced prooftheoretically to primitive recursive arithmetic PRA, a formal system usually associated with ‘finitism’. These systems