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A finite semantics of simplytyped lambda terms for infinite runs of automata
 Procedings of the 20th international Workshop on Computer Science Logic (CSL ’06), volume 4207 of Lecture Notes in Computer Science
, 2006
"... Vol. 3 (3:1) 2007, pp. 1–23 ..."
The Logical Basis of Evaluation Order and PatternMatching
, 2009
"... for the degree of Doctor of Philosophy. ..."
On Continuous Normalization
 Proc. 11 th CSL 2002 (Edinburgh), volume 2471 of Lecture
"... This work aims at explaining the syntactical properties of continuous normalization, as introduced in proof theory by Mints, and further studied by Ruckert, Buchholz and Schwichtenberg. In an extension of the untyped coinductive calculus by void constructors (socalled repetition rules), a primitiv ..."
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Cited by 6 (2 self)
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This work aims at explaining the syntactical properties of continuous normalization, as introduced in proof theory by Mints, and further studied by Ruckert, Buchholz and Schwichtenberg. In an extension of the untyped coinductive calculus by void constructors (socalled repetition rules), a primitive recursive normalization function is de ned. Compared with other formulations of continuous normalization, this de nition is much simpler and therefore suitable for analysis in a coalgebraic setting. It is shown to be continuous w.r.t. the natural topology on nonwellfounded terms with the identity as modulus of continuity. The number of repetition rules is locally related to the number of reductions necessary to reach the normal form (as represented by the Bohm tree) and the number of applications appearing in this normal form.
A realizability interpretation for classical arithmetic
 In Buss, Hájek, and Pudlák eds., Logic colloquium ’98, AK Peters, 57–90
, 2000
"... Summary. A constructive realizablity interpretation for classical arithmetic is presented, enabling one to extract witnessing terms from proofs of Σ1 sentences. The interpretation is shown to coincide with modified realizability, under a novel translation of classical logic to intuitionistic logic, ..."
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Cited by 5 (4 self)
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Summary. A constructive realizablity interpretation for classical arithmetic is presented, enabling one to extract witnessing terms from proofs of Σ1 sentences. The interpretation is shown to coincide with modified realizability, under a novel translation of classical logic to intuitionistic logic, followed by the FriedmanDragalin translation. On the other hand, a natural set of reductions for classical arithmetic is shown to be compatible with the normalization of the realizing term, implying that certain strategies for eliminating cuts and extracting a witness from the proof of a Σ1 sentence are insensitive to the order in which reductions are applied. 1
Proof Theory of MartinLof Type Theory  An
 Mathematiques et Sciences Humaines, 42 année, n o 165:59 – 99
, 2004
"... We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsisten ..."
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Cited by 4 (2 self)
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We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of MartinLof type theory with Wtype and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of MartinLof type theory with Wtype and a universe closed under the Wtype, and consider the extension of type theory by one Mahlo universe and its prooftheoretic analysis. Finally we repeat the concept of inductiverecursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.
Does Reductive Proof Theory Have A Viable Rationale?
 Erkenntnis
, 2000
"... The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or foundational sch ..."
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The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistencyproof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reducti...
Finite Notations for Infinite Terms
, 1998
"... In [1] Buchholz presented a method to build notation systems for infinite sequentstyle derivations, analogous to wellknown systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive (not " 0 ) recursive function its nth predecessor and e.g. t ..."
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In [1] Buchholz presented a method to build notation systems for infinite sequentstyle derivations, analogous to wellknown systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive (not " 0 ) recursive function its nth predecessor and e.g. the last rule applied. Here we extend the method to the more general setting of infinite (typed) terms, in order to make it applicable in other prooftheoretic contexts as well as in recursion theory. As examples, we use the method to (1) give a new proof of a wellknown tradeoff theorem [6], which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and (2) construct a continuous normalization operator with an explicit modulus of continuity. It is well known that in order to study primitive recursion in higher types it is useful to unfold the primitive recursion operators into infinite terms. A similar phenomenon occurs in proo...
On the computational complexity of cutreduction
, 2007
"... Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theor ..."
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Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way. 1