A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2020; you can also visit the original URL.
The file type is
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. ...arXiv:1410.6361v3 fatcat:t7l3pg6vg5hstksp6iv6y2rnzi
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. ...doi:10.1007/11780342_44 fatcat:fmwlswznjvg3hdxvgsu2sehuwq
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. ...arXiv:1411.0457v2 fatcat:qheau2jvtjebpofgrk4lojwiom
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. ...
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. ...
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. ...
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. ...doi:10.1215/00294527-1715716 fatcat:zvw37nq6trbjzcalczwkqpoege
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 ...arXiv:2101.05878v1 fatcat:tbn3xgu6rnemhbsxivxtr74jcu
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). ...doi:10.1093/acprof:oso/9780198566519.003.0008 fatcat:n4z453pujrenfgte7abt4tkjhi
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. ...doi:10.1016/s0019-9958(62)90577-6 fatcat:sdin7b3givb3dhqmq57gwkguia
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 ...doi:10.1093/logcom/exm031 fatcat:lv36duerfvd25inr2f2be56qvq
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. ...doi:10.1016/0304-3975(93)90085-8 fatcat:rpt27ublabah5pzlfzuz4eabl4
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. ...doi:10.1109/lics.2004.1319627 dblp:conf/lics/Berger04 fatcat:aihxty33a5evzetc57jlzwbiz4
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. ...doi:10.1016/j.entcs.2009.07.081 fatcat:rj33jvuzknaszcqwb3a3iyjczy
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. ...arXiv:2001.03540v1 fatcat:bq3i543pdfcp3mirjtgm3eq6vu
« Previous Showing results 1 — 15 out of 21,077 results