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Volume 28 (1985)

1985 Annals of Pure and Applied Logic  
., Non-closure of the image model and absence of fixed points TERLOUW, J., Reduction of higher type levels by means of an ordinal analysis of finite terms WOOD, C., see SARACINO, D.  ...  ., Fragments of arithmetic SIMPSON, S.G., see FRIEDMAN, H.M. SMITH, R.L., see FRIEDMAN, H.M.  ... 
doi:10.1016/0168-0072(85)90022-3 fatcat:gypuekeywjbijgs27tjo3jyryu

Page 611 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
Mints (RZhMat 1985:12 A25) 87b:03132 Terlouw, Jan (NL-GRON) 87b:03133 Reduction of higher type levels by means of an ordinal analysis of finite terms. Ann. Pure Appl. Logic 28 (1985), no. 1, 73-102.  ...  , say of a functional ®, contains subdefinitions in which auxiliary functionals of higher type levels than ® are introduced by primitive or transfinite recursion, then these ‘detours through higher type  ... 

Formal verification of a partial-order reduction technique for model checking [chapter]

Ching -Tsun Chou, Doron Peled
1996 Lecture Notes in Computer Science  
First, this reduction technique has been implemented in the protocol analysis tool SPIN to significantly speed up the analysis of many practical protocols; hence its correctness has important practical  ...  As a case study, we use the mechanical theorem prover HOL to verify the correctness of a partial-order reduction technique for cutting down the amount of state search performed by model checkers.  ...  Acknowledgements We would like to thank Elsa Gunter for her penetrating remarks (especially regarding the formalization of deterministic transition systems), and Rajeev Alur, Gerard Holzmann, Doug Howe  ... 
doi:10.1007/3-540-61042-1_48 fatcat:mzcv3l3bgbbbvpllnzgynp6zg4

NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM

MARIA HÄMEEN-ANTTILA
2019 Bulletin of Symbolic Logic  
The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays.  ...  Gentzen's method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of  ...  This level of abstraction only involves finite composition of finite sequences in a similar manner to the construction of finite exponentiation of ordinals.  ... 
doi:10.1017/bsl.2018.91 fatcat:aysfqpegergxfirhqrzl2c52a4

Gödel's Functional ("Dialectica") Interpretation [chapter]

Jeremy Avigad, Solomon Feferman
1998 Studies in Logic and the Foundations of Mathematics  
The language of T is said to be of finite type since every type is assigned a finite level by this convention. The pure types (n) are defined by 1. (0) = 0 2. (n + 1) = (n) → 0.  ...  finite type but with values of type 0 only, such that each F in KL is defined by a term in T , while each term t in T is represented by an F in KL either directly (if it is of type 0) or by abstraction  ...  Terms of type Π x∈σ τ denote functions f which take elements a of type σ to elements f (a) of type τ [a/x], and terms of type Σ x∈σ τ denote pairs a, b , where a is an object of type σ and b is an object  ... 
doi:10.1016/s0049-237x(98)80020-7 fatcat:nzhyeybbwjecnhnzg47rea2tnq

Análise de dados ordinais em estudos clínicos e experimentais

Hélio Amante Miot
2020 Jornal Vascular Brasileiro  
Certain phenomena are represented by qualitative data in which each category has a hierarchical relationship to the others (for example, educational level, functional class, phototype, and severity of  ...  This has even more influence on the results if a study's dependent variable is an ordinal variable.  ...  analysis of ordinal data. 4, 13, [15] [16] [17] [18] [19] Table 2 illustrates an analysis of frequency by educational level, using these ordinal methods.  ... 
doi:10.1590/1677-5449.200185 fatcat:bqkiclnhwjgl7k5tt3cdsyvcga

The Axiom of Choice and the Class of Hyperarithmetic Functions

G. Kreisel
1962 Indagationes Mathematicae (Proceedings)  
Research for this paper was partially supported by the Office of Ordnance Research under contract DA-04-200-0RD-997. 21 Series A S[P(n), m] (by the assumed premise of the comprehension axiom) we have (  ...  1.5.1 of [7] , and the special case above is stated in my supplement to [7] .  ...  For, since the range of the quantifiers of level a is well defined only if a is an ordinal, the hierarchies up to co1 use as a means of reduction not only the concept of natural number, but also the (impredicative  ... 
doi:10.1016/s1385-7258(62)50029-2 fatcat:w3w7cbc3afacjk2ikydnf7ompm

The prediction error in CLS and PLS: the importance of feature selection prior to multivariate calibration

Boaz Nadler, Ronald R. Coifman
2005 Journal of Chemometrics  
It is shown, however, that for a finite training set with n samples in p dimensions there are additional error terms that depend on r 2 p 2 =n 2 , where r is the noise level per co-ordinate.  ...  This paper presents an asymptotically exact mathematical analysis of the mean squared error of prediction of CLS and PLS under the linear mixture model commonly assumed in spectroscopy.  ...  The interesting property we find is that, although the 1=n term is typically multiplied by an Oð1Þ coefficient, the 1=n 2 term is multiplied by 2 p 2 , where is the noise per co-ordinate and p is the dimensionality  ... 
doi:10.1002/cem.915 fatcat:lszqt7hvczbvdnnpycz4hv5tfm

Godel's interpretation of intuitionism

W. W. TAIT
2006 Philosophia Mathematica  
He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type.  ...  I received helpful comments on all of these occasions. 1 As for the history of the idea of a hierarchy of types, I know only of these examples: The hierarchy of transfinite types in which the natural numbers  ...  An alternative foundation for T is to introduce the so-called 'minimal term model', in which the objects of type A are just the closed terms of T of type A and the meaning of equations between such terms  ... 
doi:10.1093/philmat/nkj004 fatcat:cpyo4pacyfh2jo5khqebcwvzui

On Relating Theories: Proof-Theoretical Reduction [chapter]

Michael Rathjen, Michael Toppel
2019 Mathesis Universalis, Computability and Proof  
The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition.  ...  Thus we have a beautiful example of Hilbert's program at work, exemplifying the Hilbertian goal of moving from the ideal to the real by eliminating ideal elements.  ...  was supported by a University Research Scholarship of the University of Leeds.  ... 
doi:10.1007/978-3-030-20447-1_16 fatcat:loptt6g2jfbmtdczu2qypbk2vu

Analysis of a guard condition in type theory [chapter]

Roberto M. Amadio, Solange Coupet-Grimal
1998 Lecture Notes in Computer Science  
We show that the proposed type system subsumes those studied by Coquand and Gimenez while still enjoying the basic syntactic properties of subject reduction and strong normalization with respect to a confluent  ...  These recursive definitions are needed to introduce 'infinite' and 'total' objects of coinductive type such as an infinite stream, a digital transducer, or a nonterminating process.  ...  Acknowledgement The first author would like to thank Eduardo Gimenez for providing the simply typed formulation of Iris system and explaining its motivations, and Alexandra Bac for a number of discussions  ... 
doi:10.1007/bfb0053541 fatcat:dlvmnffdtrf4jhqlikgvabl6by

Type-based termination of generic programs

Andreas Abel
2009 Science of Computer Programming  
They are partially ordered; the order is given by • ≤ p and p ≤ p. Kinds κ are the "types" of type constructors.  ...  In F ω , size expressions belong to the type language and have kind ord. κ ::= * types | ord ordinals | pκ 1 → κ 2 co-/contra-/mixed-variant constructor transformers Notation: κ p  ...  This research has been supported by the coordination action TYPES (510996) and thematic network Applied Semantics II (IST-2001-38957) of the European Union and the project Cover of the Swedish Foundation  ... 
doi:10.1016/j.scico.2008.01.004 fatcat:b53ufp6afbhnzjm2d3tqcojzqq

Ordinal Arithmetic: A Case Study for Rippling in a Higher Order Domain [chapter]

Louise A. Dennis, Alan Smaill
2001 Lecture Notes in Computer Science  
Ordinal arithmetic provides a natural set of higher order examples on which transfinite induction may be attempted using rippling.  ...  Ordinal arithmetic provides a natural set of higher order examples on which transfinite induction may be attempted using rippling.  ...  ∀α : ordl. α is an ordinal We introduce here the type of ordinals, ordl; as is well known, this cannot correspond to a set. Definition 5.  ... 
doi:10.1007/3-540-44755-5_14 fatcat:bndrfnhgdngg7liu7kusje6w2u

Integrating Ontology into SBVR

Dane Sorensen, Anthony Pastiak, Amit Mitra, Amar Gupta
2006 Social Science Research Network  
Representation-the connection between expression and a meaning or a portrayal of a meaning by an expression.  ...  Exclusion Difference scaled & Subtypes Patterns of separation in terms of ratios of separation (eg: Physical distance) Ordinal & Subtypes Patterns of separation in terms of quantitative differences  ... 
doi:10.2139/ssrn.906579 fatcat:vwknuijbtvfi7pv5s56sc43r7u

ERA ranking representability: The missing link between ordinal regression and multi-class classification

Willem Waegeman, Bernard De Baets
2011 2011 11th International Conference on Intelligent Systems Design and Applications  
For the three-class case, this results in a new type of cycle transitivity for pairwise AUCs that can be verified by solving an integer quadratic program.  ...  As performance measure, we use the area under the ROC curve (AUC) and its reformulation in terms of graphs.  ...  ACKNOWLEDGMENT W.W. is supported as a postdoc by the Research Foundation of Flanders (FWO-Vlaanderen).  ... 
doi:10.1109/isda.2011.6121820 dblp:conf/isda/WaegemanB11 fatcat:5uyc2kuh5bg6pmq4drprwhv4ke
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