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A Class of Reversible Primitive Recursive Functions

Luca Paolini, Mauro Piccolo, Luca Roversi
2016 Electronical Notes in Theoretical Computer Science  
The class RPRF is closed by inversion, can only express bijections on integers -not only natural numbers -, and it is expressive enough to simulate Primitive Recursive Functions, of course, in an effective  ...  We present a class RPRF of reversible functions which holds at bay intensional aspects and emphasizes the extensional side of the reversible computation by following the style of Dedekind-Robinson Primitive  ...  Reversible Primitive Recursive Functions (RPRF). It is the class of functions we propose to fulfill our goals.  ... 
doi:10.1016/j.entcs.2016.03.016 fatcat:h36c4c7nvjh5zjyzgmshllnsf4

Foundations of Online Structure Theory II: The Operator Approach [article]

Rod Downey, Alexander Melnikov, Keng Meng Ng
2021 arXiv   pre-print
but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated  ...  One of the key ideas is that online algorithms can be viewed as a sub-area of computable analysis. Conversely, we also get an enrichment of computable analysis from classical online algorithms.  ...  On the other hand, for a recursion theorist it would be more natural to consider Turing functionals acting on the representation spaces and demand that they are primitive recursive.  ... 
arXiv:2007.07401v4 fatcat:abmc5m2g3zhq5f5gdjk37hvxmi

Introducing Yet Another REversible Language [article]

Claudio Grandi, Dariush Moshiri, Luca Roversi
2019 arXiv   pre-print
Yarel is a core reversible programming language that implements a class of permutations, defined recursively, which are primitive recursive complete.  ...  The current release of Yarel syntax and operational semantics, implemented by compiling Yarel to Java, is 0.1.0, according to Semantic Versioning 2.0.0.  ...  RPP contains total functions and is primitive recursive complete, i.e. every primitive recursive function f can be compiled to an equivalent f • in RPP [17] .  ... 
arXiv:1902.05369v1 fatcat:ib3mvr47tjfrnfkbrb7umnmr54

FOUNDATIONS OF ONLINE STRUCTURE THEORY

NIKOLAY BAZHENOV, ROD DOWNEY, ISKANDER KALIMULLIN, ALEXANDER MELNIKOV
2019 Bulletin of Symbolic Logic  
See also Section 9 for a brief discussion. primitive recursion relative to a class of functions or consider an arbitrary class of functions closed under primitive recursive operators and composition.  ...  A functor is primitive recursive if, essentially, it is given by primitive recursive functionals. Definition 11.2. Let C and D be categories of structures on ω.  ...  In particular, a function f is primitive recursive iff it can be emulated on the universal Turing machine with a primitive recursive time-bound on the steps of approximation.  ... 
doi:10.1017/bsl.2019.20 fatcat:7cs7fag7cjcwdehdo6ji2qd5vu

Certifying algorithms and relevant properties of Reversible Primitive Permutations with Lean [article]

Giacomo Maletto, Luca Roversi
2022 arXiv   pre-print
Reversible Primitive Permutations (RPP) are recursively defined functions designed to model Reversible Computation.  ...  Our reworking of the original proof of that statement is conceptually simpler, fixes some bugs, suggests a new more primitive reversible iteration scheme for RPP, and, in order to keep formalization and  ...  ; -RPP can be naturally extended to become Turing-complete [15] by means of a minimization scheme analogous to the one that extends PRF to the Turing-complete class PR of Partial Recursive functions;  ... 
arXiv:2201.10443v1 fatcat:ktfcbbb45vhrdf4kucht3tf2fy

Page 2003 of Mathematical Reviews Vol. 56, Issue 6 [page]

1978 Mathematical Reviews  
Chapter I and II establish the equivalence of classes of y-recursive functions. Turing com- putable functions and functions computable on register machines.  ...  F.; Weihrauch, K. 15385 A characterization of the classes L, and R, of primitive recursive word functions.  ... 

An algebra and a logic for NC1

Kevin J. Compton, Claude Laflamme
1990 Information and Computation  
In the algebraic characterization a recursion scheme called upward tree recursion is applied to a class of simple functions.  ...  Presented here are an algebra and a logic characterizing the complexity class NC', which consists of functions computed by uniform families of polynomial size, log depth circuits.  ...  (An underlying linear order is not needed here-it can be defined using primitive recursion.) But primitive recursion is just a particular kind of inductive definition.  ... 
doi:10.1016/0890-5401(90)90063-n fatcat:fn746ffmdzco5gaq7p2tlcsek4

Structural Recursion on Ordered Trees and List-Based Complex Objects [chapter]

Edward L. Robertson, Lawrence V. Saxton, Dirk Van Gucht, Stijn Vansummeren
2006 Lecture Notes in Computer Science  
on (nested) sets and bags, coincides with the class of primitive recursive functions on natural numbers [6] .  ...  XQuery allows arbitrary recursive function definitions, resulting in a Turing complete language.  ... 
doi:10.1007/11965893_24 fatcat:fwrviacux5hyjcpebqyx6cyzde

Structural Recursion as a Query Language on Lists and Ordered Trees

Edward L. Robertson, Lawrence V. Saxton, Dirk Van Gucht, Stijn Vansummeren
2008 Theory of Computing Systems  
class of all primitive recursive queries.  ...  In particular, we show that the combination of vertical recursion down a tree combined with horizontal recursion across a list of trees gives rise to a robust class of transformations: it captures the  ...  XQuery allows arbitrary recursive function definitions, resulting in a Turing complete language.  ... 
doi:10.1007/s00224-008-9110-5 fatcat:3vc32pqpf5hkhdpsxj3fopd4jy

The History and Concept of Computability [chapter]

Robert I. Soare
1999 Studies in Logic and the Foundations of Mathematics  
It is presented in a revised and shortened form here with the permission of the Association for Symbolic Logic.  ...  since then, except for the term "computably enumerable," recently introduced, because Turing and Gödel did not explicitly introduce a term for these corresponding sets, but just for the computable functions  ...  He gave a definition of a particular function using double nested recursion and showed that it was not primitive recursive. R.  ... 
doi:10.1016/s0049-237x(99)80017-2 fatcat:j23jinkxfrg57fy55nn3nx3jcy

Why Gödel didn't have church's thesis

Martin Davis
1982 Information and Control  
But Kleene soon showed that the class was very extensive indeed, containing all primitive recursive functions and being closed under minimalization.  ...  his famous example of a function which is not primitive recursive.  ... 
doi:10.1016/s0019-9958(82)91226-8 fatcat:qajogh5tdnhjhgbz2mdvj6k5dy

Linear Recursion [article]

Sandra Alves, Ian Mackie École Polytechnique)
2016 arXiv   pre-print
The extensions are based on unbounded recursion in one case, and bounded recursion with minimisation in the other.  ...  We define two extensions of the typed linear lambda-calculus that yield minimal Turing-complete systems.  ...  As a consequence of Kleene's theorem, we only have to prove that we can encode minimisation of primitive recursive functions in order to show Turing-completeness of L µ , relying on the fact that primitive  ... 
arXiv:1001.3368v4 fatcat:bbzvcea7hvehhjtcfcf7ldoag4

Probabilistic Recursion Theory and Implicit Computational Complexity [chapter]

Ugo Dal Lago, Sara Zuppiroli
2014 Lecture Notes in Computer Science  
and Kleene's partial recursive functions.  ...  We show that probabilistic computable functions, i.e., those functions outputting distributions and computed by probabilistic Turing machines, can be characterized by a natural generalization of Church  ...  function of the machine at hand, call it M , as a (recursive) function on N.  ... 
doi:10.1007/978-3-319-10882-7_7 fatcat:wi2zcjjutjeexfe7cwy3e56ygq

Probabilistic Recursion Theory and Implicit Computational Complexity

Ugo Dal Lago, Sara Zuppiroli, Maurizio Gabbrielli
2014 Scientific Annals of Computer Science  
and Kleene's partial recursive functions.  ...  We show that probabilistic computable functions, i.e., those functions outputting distributions and computed by probabilistic Turing machines, can be characterized by a natural generalization of Church  ...  function of the machine at hand, call it M , as a (recursive) function on N.  ... 
doi:10.7561/sacs.2014.2.177 fatcat:whd4wlyi7jcdpljt6ekodurize

Computability and Recursion

Robert I. Soare
1996 Bulletin of Symbolic Logic  
We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness".  ...  After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of "computability  ...  He gave a definition of a particular function using double nested recursion and showed that it was not primitive recursive. R.  ... 
doi:10.2307/420992 fatcat:yq5vicyjf5aq7ercdrg2jql6zm
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