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Computational Logic by Ulrich Berger and Helmut Schwichtenberg, editors, Springer-Verlag, 1999, 444 pp

Roy Dyckhoff
2001 Journal of functional programming  
Benl and Schwichtenberg discuss the formal correctness of functional programs.  ... 
doi:10.1017/s0956796801214117 fatcat:5tmswn3qqbglfkio4prxcylahm

Normalization [chapter]

Helmut Schwichtenberg
1991 Logic, Algebra, and Computation  
Normalization Helmut Schwichtenberg Mathematisches Institut, Universität München Theresienstraße 39, D-8000 München 2, Germany The aim of this paper is to present a central technique from proof theory,  ...  The simple example treated here is taken from (Schwichtenberg 1982, p. 455 ). The pure types fc are defined inductively by 0 := P (some fixed propositional variable) and k +1 = k -• k.  ... 
doi:10.1007/978-3-642-76799-9_5 fatcat:p2vvxeblszcdjmt6vgcolqmqaa

Mathematische Logik [chapter]

Kurt Schütte, Helmut Schwichtenberg
1990 Ein Jahrhundert Mathematik 1890–1990  
doi:10.1007/978-3-322-80265-1_17 fatcat:kpbnrccu6ja3bk2bpbsrqlzciq

New Developments in Proofs and Computations [chapter]

Helmut Schwichtenberg
2008 New Computational Paradigms  
is in Schwichtenberg (2005) .  ...  This "direct method" has been described in Schwichtenberg (1993) ; in Berger and Schwichtenberg (1995) it has been shown that it gives the same results as the so-called A-translation of Friedman (1978  ... 
doi:10.1007/978-0-387-68546-5_14 fatcat:4z2kntxwqfbyhbiowa2zj5coey

Program Extraction in Constructive Analysis [chapter]

Helmut Schwichtenberg
2009 Logicism, Intuitionism, and Formalism  
We sketch a development of constructive analysis in Bishop's style, with special emphasis on low type-level witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Intermediate Value Theorem and (2) the existence of a continuous inverse to a monotonically increasing continuous function. Using the Minlog proof assistant, the proofs leading to the Intermediate Value
more » ... Theorem are formalized and realizing terms extracted. It turns out that evaluating these terms is a reasonably fast algorithm to compute, say, approximations of √ 2. 2 Real Numbers Reals, Equality of Reals We shall view a real as a Cauchy sequence of rationals with a separately given modulus. Definition. A real number x is a pair ((a n ) n∈N , M ) with a n ∈ Q and M : N → N such that (a n ) n is a Cauchy sequence with modulus M , that is |a n − a m | ≤ 2 −k for n, m ≥ M (k) and M is weakly increasing. M is called a Cauchy modulus of x.
doi:10.1007/978-1-4020-8926-8_13 fatcat:nyhngtujmrdadhifdonrhintzi

A bound for Dickson's lemma [article]

Josef Berger, Helmut Schwichtenberg
2015 arXiv   pre-print
We consider a special case of Dickson's lemma: for any two functions f,g on the natural numbers there are two numbers i<j such that both f and g weakly increase on them, i.e., f_i< f_j and g_i < g_j. By a combinatorial argument (due to the first author) a simple bound for such i,j is constructed. The combinatorics is based on the finite pigeon hole principle and results in a descent lemma. From the descent lemma one can prove Dickson's lemma, then guess what the bound might be, and verify it by
more » ... an appropriate proof. We also extract (via realizability) a bound from (a formalization of) our proof of the descent lemma. Keywords: Dickson's lemma, finite pigeon hole principle, program extraction from proofs, non-computational quantifiers.
arXiv:1503.03325v1 fatcat:vrmlu54jsjfyzkxy57zk74umcy

Program Development by Proof Transformation [chapter]

Ulrich Berger, Helmut Schwichtenberg
1995 Proof and Computation  
doi:10.1007/978-3-642-79361-5_1 fatcat:sueg46547ndrlo3hoywapfytda

Primitive Recursion on the Partial Continuous Functionals [chapter]

Helmut Schwichtenberg
1991 Informatik und Mathematik  
doi:10.1007/978-3-642-76677-0_18 fatcat:ootwm3fl5nfbbgabar3xbdtpju

Minimal Logic for Computable Functions [chapter]

Helmut Schwichtenberg
1993 Logic and Algebra of Specification  
A series presenting the results ofactivities sponsored by the NATO Science Committee, which aims at the dissemination ofadvanced scientific and technological knowiedge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with The electronic index to the NATO ASI Series provides füll bibliographical references (with keywords and/or abstracts) to more than 30000 contributions from international
more » ... published in all sections of the NATO ASI Series. Access to the NATO-PCO DATABASE compiled by the NATO Publication Coordination Office is possible in two ways: -via online FILE 128 (NATO-PCO DATABASE) hosted by ESRIN,
doi:10.1007/978-3-642-58041-3_8 fatcat:y2jscaqpv5dcng7gvzjeeryfzi

Proof Search in Minimal Logic [chapter]

Helmut Schwichtenberg
2004 Lecture Notes in Computer Science  
Case identity, i.e. Q:r = r^C. Then Q:r = r^C =) " QC: Case , i.e. Q: x r = x s^C. We may assume here that the bound variablesx are the same on both sides. Q: xr = x s^C =) " Q8x:r = s^C: Case rigid-rigid, i.e. Q:fr = fs^C. Q:fr = fs^C =) " Q:r =s^C:
doi:10.1007/978-3-540-30210-0_3 fatcat:h2lrsrcemrf4haxumiedhsw7bi

Feasible Computation with Higher Types [chapter]

Helmut Schwichtenberg, Stephen J. Bellantoni
2002 Proof and System-Reliability  
We restrict recursion in finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types ρ → σ and terms λx ρ r as well as ρ σ and λx ρ r. Here we use two sorts of typed variables: complete onesx ρ and incomplete ones x ρ . mod01.tex
doi:10.1007/978-94-010-0413-8_13 fatcat:ogdnkqandvci7ipqpoaslp4pe4

Eine Klassifikation der ɛ0-Rekursiven Funktionen

Helmut Schwichtenberg
1971 Mathematical Logic Quarterly  
in Munster/Westfalen KLEEXE formuliert in [7J das Problem, "whether the ordinal numbers can be used to give a satisfactory classification of the general recursive functions into a hierarchy under some general principle". Der wohl naheliegendste Ansatz zur Konstruktion einer solchen Hierarchie, namlich Beschrankung der Ordnungstypen der fur Rekursionen zugelassenen Wohlordnungen, fuhrt nicht zum Ziel : MYHILL und ROUTLEDGE haben bewiesen, daS jede rekursive Funktion durch elementare Operationen
more » ... nd nur eine Rekursion langs einer elementaren Wohlordnung vom Typ w definiert werden kann ([ll], [16]; s. auch Lru [9]). I n [7] schlagt KLEEKE eine andere Methode vor, rekursive Funktionen mit Ordinalzahlen in Verbindung zu bringen: Man geht aus von einer effektiv erzeugten Funktionenklasse, etwa der Klasse (3 der elementaren Funktionen, konstruiert eine kanonische Aufzahlungsfunktion El (also El $: @), betrachtet die in El elementaren Funktionen, konstruiert fur sie eine kanonische Aufzahlungsfunktion E, (also E , $: @ ( E l ) ) , usw. Diese Konstruktion la& sich transfinit fortsetzen, wenn fur jede Limeszahl eine sie approximierende Fundamentalfolge zur Verfiigung steht. KLEENE verwendet deshalb sein Bezeichnungssystem S, fur konstruktive Ordinalzahlen, und zwar in einer Version, in der nur primitiv rekursive Fundamentalfolgen zugelassen sind ([7], p. 73); eine solche Einschrankung ist notwendig, da man sonst schon auf dem Niveau w alle rekursiven Funktionen erhielte. Aber auch mit dieser Einschrankung kollabiert die Hierarchie: FEFERMAN zeigt in [2], da13 man dann auf dem Niveau 09 alle rekursiven Funktionen erhalt. Wir behandeln hier das Klassifikationsproblem fur einen Teil der rekursiren Funktionen, die "sO-rekursiven" Funktionen; darunter verstehen wir solche Funktionen, die definierbar sind durch elementare Operationen und "elementare %-Rekursionen", A < z0, der Form f ( x , kj) = S(u]-", g" . . . . g r ; x, g) mit einer ,.Standardwohlordnung" < vom Typ A und einem elementaren Funktional F (genauer in 5 1)l). Diese Funktionenklasse fallt mit der von KREISEL in [S] eingefuhrten Klasse der ordinal rekursiven Funktionen zusammen (einfache Folgerung aus § a), enthalt also genau die Funktionen, "deren Rekursivitat in der reinen Zahlentheorie beweisbar ist" ([S], s. auch SHOENFIELD [IS]). Ein erstes KompliziertheitsmaS fiir E,-rekursive Funktionen wird von der Definition nahegelegt (vgl . HEINERMANN [4] ) : 1st f durch elementare Operationen aus g" . . , , g, definiert, und sind g" . . . , gr die "Rekursionszahlen" a" . . . , a, ( < E~) Mit I! bezeichnen wir hier Limeszahlen < c0, mit a, , 9, y , . . . beliebige Ordinalzahlen <F" . Daneben verwenden wir 1 im Rahmen der Cmcmchen Schreibweise 3. t f Q) ftir die Funktion /. Frakturbuchst,aben F, b, 8, . . . stehen fur Variablentupel.
doi:10.1002/malq.19710170113 fatcat:xpldtm6dnfaf7oucfmpkfff64q

An arithmetic for polynomial-time computation

Helmut Schwichtenberg
2006 Theoretical Computer Science  
We define a restriction LHA of Heyting arithmetic HA with the property that all extracted programs are feasible. The restrictions consist in linearity and ramification requirements.
doi:10.1016/j.tcs.2006.03.019 fatcat:waup2znmyfca5j2dypaqbvwdn4

Inverting Monotone Continuous Functions in Constructive Analysis [chapter]

Helmut Schwichtenberg
2006 Lecture Notes in Computer Science  
We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. This term can be applied to concrete continuous functions and arguments, and then normalized to a rational approximation of say a zero of a given function. It turns out that even in the logical term language "normalization by evaluation" is reasonably efficient.
doi:10.1007/11780342_50 fatcat:dck6hofvkbf2xhheluwcvujluu

Strict functionals for termination proofs [chapter]

Jaco van de Pol, Helmut Schwichtenberg
1995 Lecture Notes in Computer Science  
A semantical method to prove termination of higher order rewrite systems (HRS) is presented. Its main tool is the notion of a strict functional, which is a variant of Gandy's notion of a hereditarily monotonic functional 1]. The main advantage of the method is that it makes it possible to transfer ones intuitions about why an HRS should be terminating into a proof: one has to nd a \strict" interpretation of the constants involved in such a way that the left hand side of any rewrite rule gets a
more » ... igger value than the right hand side. The applicability of the method is demonstrated in three examples. An HRS involving map and append. The usual rules for higher order primitive recursion in G odel's T. Derivation terms for natural deduction systems. We prove termination of the rules for {conversion and permutative conversion for logical rules including introduction and elimination rules for the existential quanti er. This has already been proved by Prawitz in 5]; however, our proof seems to be more perspicuous. Technically we build on 7]. There a notion of a strict functional and simultaneously of a strict greater{than relation > str between monotonic functionals is introduced. The main result then is the following. Let M be a term in normal form and 2 FV(M). Then for any strict environment U and all monotonic f and g, one has f > mon g =) M] ] U 7 !f] > str M] ] U 7 !g] . From this van de Pol derives the technique described above for proving termination of higher order term rewrite systems, generalizing a similar approach for rst order rewrite systems (cf. 3, p. 367]). Interesting applications are given in 7]. Here a slight change in the de nition of strictness is exploited (against the original conference paper; cf. 7, Footnote p. 316]). This makes it possible to deal with rewrite rules involving types of level > 2 too, and in particular with proof theoretic applications. In order to do this some theory of strict functionals is developed. We also add product types, which are necessary to treat e.g. the existential quanti er.
doi:10.1007/bfb0014064 fatcat:37aalspibzdd5mdwuttsc7fo2q
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