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Dialectica Interpretation of WellFounded Induction
 MATHEMATICAL LOGIC QUARTERLY, 12/11/2007
, 2007
"... From a classical proof that the gcd of natural numbers a1 and a2 is a linear combination of the two, we extract by Gödel’s Dialectica interpretation an algorithm computing the coefficients. The proof uses the minimum principle. We show generally how wellfounded recursion can be used to Dialectica i ..."
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From a classical proof that the gcd of natural numbers a1 and a2 is a linear combination of the two, we extract by Gödel’s Dialectica interpretation an algorithm computing the coefficients. The proof uses the minimum principle. We show generally how wellfounded recursion can be used to Dialectica interpret wellfounded induction, which is needed in the proof of the minimum principle. In the special case of the example above it turns out that we obtain a reasonable extracted term, representing an algorithm close to Euclid’s.
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 ..."
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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...
From Higher Order Terms To Circuits
"... INTRODUCTION In his lecture at the congress, the first author gave a survey on some recent results relevant for computability theory in the context of partial continuous functionals (cf. (Scott, 1982; Ershov, 1977; StoltenbergHansen et al., 1994)): ffl An abstract definition of totality due to Be ..."
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INTRODUCTION In his lecture at the congress, the first author gave a survey on some recent results relevant for computability theory in the context of partial continuous functionals (cf. (Scott, 1982; Ershov, 1977; StoltenbergHansen et al., 1994)): ffl An abstract definition of totality due to Berger (cf. (Berger, 1990; Berger, 1993) and (StoltenbergHansen et al., 1994, Ch. 8.3)), and applications concerning density and effective density theorems. ffl Bounded fixed points: one can have the flexibility of fixed point definitions and termination at the same time (cf. (Schwichtenberg and Wainer, 1995)). ffl A notion of strict functionals as a tool to prove termination of higher order rewrite systems (cf. (van de Pol and Schwichtenberg, 1995)). Since this work is published already, we do not give details here but rather concentrate on another "appli
Finite notations for infinite terms Helmut Schwichtenberg
"... 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 proof theory, where one expands induction axioms. For applications it then is often necessary to code these infinit ..."
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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 proof theory, where one expands induction axioms. For applications it then is often necessary to code these infinite objects by natural numbers. A standard method to design such a coding is to proceed as in Kleene's system O of ordinal notations; cf. [6] for a recursion theoretic and [7] for a proof theoretic application of this method. However, working with such codes is not easy. For instance, to prove that the standard operation reducing the cut rank by one can be represented by a primitive recursive operation on codes requires some careful applications of Kleene's recursion theorem for primitive recursive functions.
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"... ffi) and context unwrapping (denoted V E and typed by requiring V to be of type:Bffi and the evaluation context E[] to be of type B with the `hole ' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and F ..."
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ffi) and context unwrapping (denoted V E and typed by requiring V to be of type:Bffi and the evaluation context E[] to be of type B with the `hole ' of type A). Here we essentially give an exposition of Griffin's result, with some simplifications and extensions based on work of Sabry and Felleisen [18]. In particular we stress its connection with questions of termination of different normalization strategies for minimal, intuitionistic and classical logic, or more precisely their fragments in implicational propositional logic. We also give some examples (due to Hirokawa) of derivations in minimal and classical logic which reproduce themselves under certain reasonable conversion rules.
HELMUT SCHWICHTENBERG AND KARL STROETMANN FROM HIGHER ORDER TERMS TO CIRCUITS
"... In his lecture at the congress, the rst author gave a survey on some recent results relevant for computability theory in the context of partial continuous ..."
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In his lecture at the congress, the rst author gave a survey on some recent results relevant for computability theory in the context of partial continuous
Relational Reasoning about Functions and Nondeterminism
, 1998
"... Reproduction of all or part of this workis permitted for educational or research use on condition that this copyright notice isincluded in any copy. See back inner page for a list of recent BRICS Dissertation Series publications. Copies may be obtained by contacting: ..."
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Reproduction of all or part of this workis permitted for educational or research use on condition that this copyright notice isincluded in any copy. See back inner page for a list of recent BRICS Dissertation Series publications. Copies may be obtained by contacting:
Reasoning about terminating . . .
, 1999
"... This thesis addresses two basic problems with the current crop of mechanized proof systems. The first problem is largely technical: the act of soundly introducing a recursive definition is not as simple and direct as it should be. The second problem is largely social: there is very little codeshar ..."
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This thesis addresses two basic problems with the current crop of mechanized proof systems. The first problem is largely technical: the act of soundly introducing a recursive definition is not as simple and direct as it should be. The second problem is largely social: there is very little codesharing between theorem prover implementations; as a result, common facilities are typically built anew in each proof system, and the overall progress of the field is thereby hampered. We use the application domain of functional programming to explore the first problem. We build a patternmatching style recursive function definition facility, based on mechanically proven wellfounded recursion and induction theorems. Reasoning support is embodied by automatically derived induction theorems, which are customised to the recursion structure of definitions. This provides a powerful, guaranteed sound, definitionandreasoning facility for functions that strongly resemble programs in languages such as ML or Haskell. We demonstrate this package (called TFL) on several wellknown challenge problems. In spite of its power, the approach suffers from a low level of automation, because a termination relation must be supplied at function definition time. If humans are to be largely relieved of the task of proving termination, it must be possible for the act of defining a recursive function to be completely separate from the act of finding a termination relation for it and proving the ensuing termination conditions. We show how this separation can be achieved, while still preserving soundness. Building on this, we present a new way to define program schemes and prove highlevel program transformations.