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### Spector bar recursion over finite partial functions [article]

Paulo Oliva, Thomas Powell
2015 arXiv   pre-print
The recursion takes place over finite partial functions u, where the control parameter φ, used in Spector's bar recursion to terminate the computation at sequences s satisfying φ(ŝ)<|s|, now acts as a  ...  We introduce a new, demand-driven variant of Spector's bar recursion in the spirit of the Berardi-Bezem-Coquand functional.  ...  Discrete choice Up until now, we have considered bar recursion over either finite sequences or finite partial functions: in other words over objects of type N → X + 1 with finite support.  ...

### Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis [chapter]

Paulo Oliva
2006 Lecture Notes in Computer Science
In this note bar recursion is presented as a generalisation of a simpler primitive recursive functional needed for the interpretation of a finite (intuitionistic) version of DNS.  ...  Spector's interpretation makes use of a rather abstruse recursion schema, so-called bar recursion, used to interpret the double negation shift DNS.  ...  In the following σ and τ will denote finite partial functions from N to N, i.e. partial functions which are defined on a finite domain.  ...

### Parametrised bar recursion: A unifying framework for realizability interpretations of classical dependent choice [article]

Thomas Powell
2015 arXiv   pre-print
In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with some form of bar recursion, which realizes the negative translation of either countable  ...  From this proof, the soundness of most of the existing bar recursive realizability interpretations of choice, including those based on the Berardi-Bezem-Coquand functional, modified realizability and the  ...  Modified bar recursion (≺ u = ∅) Suppose that we retain the simplification ≺ u = ∅, but now allow ⊳ to range over arbitrary decidable partial orders.  ...

### Page 139 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 63, Issue 2 [page]

1957 American Mathematical Society. Bulletin of the American Mathematical Society
Davis: An intrinsic definition of recursive functional. A definition of recursive functional of arbitrary type is given, in terms of partial recursiveness.  ...  As w takes on integral values, w* ranges over all functionals of finite domain which are of type a.  ...

### Page 760 of Mathematical Reviews Vol. 58, Issue 2 [page]

1979 Mathematical Reviews
Let x, y, z denote elements of A; f, g,h partial functions A”—»A; and ®, ¥, © partial functionals with arguments of type (x, /), where bars signify finite sequences, and values in A. @ is said to be functional  ...  In the first is developed a general notion of recursion over any set A. The second part applies and expands this theory for the case where A is the finite-type structure over the natural numbers.  ...

### Page 11 of Mathematical Reviews Vol. 41, Issue 1 [page]

1971 Mathematical Reviews
Let A,™ be the set of n-ary recursive functions, A,,™ that of n-ary partial recursive functions. f * g denotes the result of substituting g in every argument in place of f.  ...  H. 56 Bases in algebras of partial recursive functions. (Rus- sian) Algebra i Logika 7 (1968), no. 4, 87-105.  ...

### Primitive Recursion and the Chain Antichain Principle

Alexander P. Kreuzer
2012 Notre Dame Journal of Formal Logic
In the course of the proof we refine Howard's ordinal analysis of bar recursion. We also discuss the Erdős-Moser principle, which -taken together with CAC-is equivalent to RT 2 2 .  ...  In detail we show that WKL ω 0 + CAC is Π 0 2 -conservative over PRA and that one can extract primitive recursive realizers for such statements.  ...  In Section 1 we refine Howard's ordinal analysis of bar recursion.  ...

### Solovay's Relative Consistency Proof for FIM and BI [article]

Joan Rand Moschovakis
2021 arXiv   pre-print
Combining this result with Kleene's formalized recursive realizability, he established (in primitive recursive arithmetic PRA) that FIM + MP and BI have the same consistency strength.  ...  In 2002 Robert Solovay proved that a subsystem BI of classical second order arithmetic, with bar induction and arithmetical countable choice, can be negatively interpreted in the neutral subsystem BSK  ...  There are the usual function symbols 0, S, +, ·, plus a finite number of symbols for additional primitive recursive functions and functionals sufficient to formalize the theory of recursive partial functionals  ...

### APPLICATIONS OF INDUCTIVE DEFINITIONS AND CHOICE PRINCIPLES TO PROGRAM SYNTHESIS [chapter]

Ulrich Berger, Monika Seisenberger
2005 From Sets and Types to Topology and Analysis
Such a model can, for example, be constructed from the total elements of the Scott/Ershov hierarchy of partial continuous functionals over the flat domains of partial booleans and partial natural numbers  ...  by the Scott/Ershov model of partial continuous functionals).  ...

### A remark on discovery algorithms for grammars

E. Shamir
1962 Information and Control
If we introduce GSdel numbering, we may alternatively define a discovery algorithm as a partial recursive function f which is defined for all numbers m which are GSdel numbers of a G~ E Ct so that, for  ...  A grammar G is a finite device which classifies every string over a finite set of symbols Z (the alphabet) as accepted or rejected.  ...

### Logical Approaches to Computational Barriers: CiE 2006

A. Beckmann, B. Lowe, D. Normann
2007 Journal of Logic and Computation
575 A Computability Theory of Real Numbers Xizhong Zheng 584 Primitive Recursive Selection Functions over Abstract Algebras Jeffery I.  ...  Peter Schuster, Julia Zappe 481 Inverting Monotone Continuous Functions in Constructive Analysis Helmut Schwichtenberg 490 Partial Recursive Functions in Martin-Löf Type Theory Anton Setzer 505  ...

### Computational foundations of basic recursive function theory

Robert L. Constable, Scott F. Smith
1993 Theoretical Computer Science
When reducing to a class over a bar type, say CT, the reduction function ~ES+T might yield a nonterminating computation, so it is a partial function.  ...  A wide collection of partial functions may be typed with this rule, including all partial recursive functions.  ...

### A computational interpretation of open induction

U. Berger
2004 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.
intuitionistically, open induction and dependent choice are quite different: Unlike dependent choice, open induction is closed under negative-and -translation, and therefore proves the same ¡ £ ¢ ¤ -formulas (over  ...  recursion in finite types.  ...  Note that being not a partial choice function is an open property of partial sequences.  ...

### Continuous Functions on Final Coalgebras

Neil Ghani, Peter Hancock, Dirk Pattinson
2009 Electronical Notes in Theoretical Computer Science
Among the functors we can deal with are those that arise from countable signatures of finite-place untyped operators. These have many applications.  ...  In a previous paper we gave a representation of, and simultaneously a way of programming with, continuous functions on streams, whether discrete-valued functions, or functions between streams.  ...  Yet the principle of Bar induction can clearly be used to model continuous functions on streams, and universal quantification over streams.  ...

### A computational interpretation of Zorn's lemma [article]

Thomas Powell
2020 arXiv   pre-print
This is achieved through G\"odel's functional interpretation, and requires the introduction of a novel form of recursion over chain complete partial orders whose existence in the model of total continuous  ...  We show that a realizer for the functional interpretation of open induction over the lexicographic ordering on sequences follows as a simple application of our main results.  ...  Over the years, they have turned out to form an elegant and robust class of functionals, and in particular are the standard model for bar recursive extensions of the primitive recursive functionals.  ...
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