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13
InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
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Cited by 821 (23 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
Proof theory of reflection
 Annals of Pure and Applied Logic
, 1994
"... The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. Th ..."
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Cited by 12 (1 self)
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The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. This leads to consistency proofs for the theories KP + Πn–reflection using a small amount of arithmetic (PRA) and the well–foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π1 2 comprehension through transfinite levels of reflection. 1
An Ordinal Analysis of Stability
 ARCHIVE FOR MATHEMATICAL LOGIC
, 2005
"... This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek set theory, KP, plus 1 separation. The strength of the latter theory is encapsulated in the fact that it proves th ..."
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Cited by 6 (1 self)
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This paper is the rst in a series of three which culminates in an ordinal analysis of 1 2 comprehension. On the settheoretic side 1 2 comprehension corresponds to KripkePlatek set theory, KP, plus 1 separation. The strength of the latter theory is encapsulated in the fact that it proves the existence of ordinals such that, for all > , is stable, i.e. L is a 1 elementary substructure of L . The objective of this paper is to give an ordinal analysis of not too complicated stability relations as experience has shown that the understanding of the ordinal analysis of 1 2 comprehension is greatly facilated by explicating certain simpler cases rst. This paper introduces an ordinal representation system based on indescribable cardinals which is then employed for determining an upper bound for the proof{ theoretic strength of the theory KPi + 8 9 is + stable, where KPi is KP augmented by the axiom saying that every set is contained in an admissible set.
Ordinal arithmetic with list structures
 In Logical Foundations of Computer Science
, 1992
"... We provide a set of \natural " requirements for wellorderings of (binary) list structures. We showthat the resultant ordertype is the successor of the rst critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assistedbytheprogrammer givin ..."
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Cited by 5 (0 self)
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We provide a set of \natural " requirements for wellorderings of (binary) list structures. We showthat the resultant ordertype is the successor of the rst critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assistedbytheprogrammer giving a further de nite assertion to be veri ed. This may take the form of a quantity which is asserted todecrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number. In this problem the ordinal might be (n, r)! 2 +(r, s)! + k. A less highbrow form of the same thing would be to give the integer 2 80 (n, r)+2 40 (r, s)+k. Alan M. Turing (1949) 1
The Higher Infinite in Proof Theory
 LOGIC COLLOQUIUM '95. LECTURE NOTES IN LOGIC
, 1995
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Veblen progressions and transfinite iteration of ordinal functions. Submitted
, 2012
"... Abstract In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f ..."
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Cited by 4 (3 self)
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Abstract In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β . These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are leftadditive: given α, β, f α+β = f β f α . Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study cohyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more finegrained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.
Ramified HigherOrder Unification
, 1996
"... While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable uni ..."
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Cited by 1 (0 self)
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While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem even at order 2. However, the decidability result for the pure subsystem indicates that unification terminates more often than general higherorder unification. We present an application to ACA 0 and other expressive subsystems of secondorder Peano arithmetic.
Ramified HigherOrder Unification (Revised Report)
, 1996
"... While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable uni ..."
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While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem even at order 2. However, the decidability result for the pure subsystem indicates that unification terminates more often than general higherorder unification. We present an application to ACA 0 and other expressive subsystems of secondorder Peano arithmetic. This is a revised version of the report 199631, University of Karlruhe; a few bugs are corrected and a "related works" section has been added. 1 Introduction Higherorder logic is one of the most expressive formalisms in which we can express and prove theorems. Bertrand Russell proposed two ways of formalizing it. In ramified type theory [WR27], expressions are stratified in a double hierarchy of types (individuals...
An Ordinal Representation System for ...Comprehension and Related Systems
, 1995
"... The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the prooftheoretic strength of \Pi 1 2 comprehension and related systems. ..."
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The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the prooftheoretic strength of \Pi 1 2 comprehension and related systems.