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A new recursion-theoretic characterization of the polytime functions

Spephen Bellantoni, Stephen Cook
1992 Computational Complexity  
We give a recursion-theoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds.  ...  In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 |x|·|y| ) of Cobham.  ...  Acknowledgments In an earlier draft of this paper we used the functions D(a; b) = P (b, a) and E(a; b) = P (b, P (b, a) ) as initial functions. We are grateful to Sam Buss for  ... 
doi:10.1007/bf01201998 fatcat:pacbjbz6l5d2lgysrqbbccwp7q

Notations for exponentiation

Arnold Beckmann
2002 Theoretical Computer Science  
We deÿne a coding of natural numbers-which we will call exponential notations-and interpretations of the less-than-relation, the successor, addition and exponentiation function on exponential notations  ...  As a corollary we obtain that feasible arithmetic can prove the consistency of the canonical equational theory for the language containing the successor, addition and exponentiation function.  ...  There are also algebraic characterizations of the polytime functions.  ... 
doi:10.1016/s0304-3975(01)00143-8 fatcat:ym2ou3yuu5fetbq7pxnsicd32e

A Formalization of Polytime Functions [article]

Sylvain Heraud, David Nowak
2011 arXiv   pre-print
We present a deep embedding of Bellantoni and Cook's syntactic characterization of polytime functions.  ...  Based on our core formalization, we have started developing a library of polytime functions that can be reused to build more complex ones.  ...  We thank anonymous reviewers for their helpful suggestions, all of which improved the paper.  ... 
arXiv:1102.5495v2 fatcat:xzztqrsvznfvdhridrtxrb6lvy

A Formalization of Polytime Functions [chapter]

Sylvain Heraud, David Nowak
2011 Lecture Notes in Computer Science  
We present a deep embedding of Bellantoni and Cook's syntactic characterization of polytime functions.  ...  Based on our core formalization, we have started developing a library of polytime functions that can be reused to build more complex ones.  ...  We thank anonymous reviewers for their helpful suggestions, all of which improved the paper.  ... 
doi:10.1007/978-3-642-22863-6_11 fatcat:hm2nnduvbrduxikbrwqbgn5qge

Safe Weak Minimization Revisited

Dieter Spreen
2002 SIAM journal on computing (Print)  
By adding the new operator to those used by Bellantoni-Cook and Leivant to characterize the polynomial-time computable functions one obtains a characterization of the nondeterministic polynomial-time computable  ...  Thus, the generation of the nondeterministic polytime multifunctions from the deterministic polytime functions parallels the generation of the computable functions from the primitive recursive ones.  ...  Acknowledgement The author is grateful to the referees for a careful reading and the improvement suggestions.  ... 
doi:10.1137/s0097539701387854 fatcat:fv57k2ijanbnhpgghz5fmmzmti

On the Role of Canonicity in Knowledge Compilation

Guy Van den Broeck, Adnan Darwiche
2015 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
Among the latter is the recently introduced language of Sentential Decision Diagrams (SDDs): while a polytime Apply function exists for SDDs, it was unknown whether such a function exists for the important  ...  We consider the problem of bottom-up compilation of knowledge bases, which is usually predicated on the existence of a polytime function for combining compilations using Boolean operators (usually called  ...  This work was supported by ONR grant #N00014-12-1-0423, NSF grants #IIS-1118122 and #IIS-0916161, and the Research Foundation-Flanders (FWO-Vlaanderen). GVdB is also at KU Leuven, Belgium.  ... 
doi:10.1609/aaai.v29i1.9423 fatcat:6g3einlc65cztcmhxjmqsdhz3a

Function operators spanning the arithmetical and the polynomial hierarchy

Armin Hemmerling
2010 RAIRO - Theoretical Informatics and Applications  
A modified version of the classical μ-operator as well as the first value operator and the operator of inverting unary functions, applied in combination with the composition of functions and starting from  ...  the primitive recursive functions, generate all arithmetically representable functions.  ...  I want to thank the referees of earlier versions of this paper for their suggestions improving the style of presentation.  ... 
doi:10.1051/ita/2010020 fatcat:fnqzvcdisbbmjema2vgo3bunci

On sharing, memoization, and polynomial time

Martin Avanzini, Ugo Dal Lago
2018 Information and Computation  
By our invariance theorem, the function algebra is sound for polynomial time, and consequently GRSR characterizes the class of polytime computable functions.  ...  We define a simple functional programming language. The domain of the defined functions is a free algebra formed from constructors.  ...  In other words, characterizing complexity classes by purely recursion-theoretical means is non-trivial.  ... 
doi:10.1016/j.ic.2018.05.003 fatcat:jqh5pmhx5neergx7qholahw76e

Page 5455 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
; Miinster A term rewriting characterization of the polytime functions and related complexity classes.  ...  As a corollary a new proof of Bellantoni’s result stating that predicative recursion is closed under predicative parameter recursion is obtained.  ... 

On the complexity of stratified logics [article]

Luca Vercelli
2010 arXiv   pre-print
On the other side, FPTIME can be captured by Safe Recursion on Notation (SRN), an algebra of functions based on Predicative Recursion, a restriction of the standard recursion schema used to defiine primitive  ...  Our primary motivation is the comparison of two different traditions used in ICC to characterize the class FPTIME of the polynomial time computable functions.  ...  ] characterize the class of polynomial time-calculable functions.  ... 
arXiv:1002.3453v1 fatcat:4vbot5gsk5dthaljil7i4ij3gq

Higher-Order Linear Ramified Recurrence [chapter]

Ugo Dal Lago, Simone Martini, Luca Roversi
2004 Lecture Notes in Computer Science  
On the other, HOLRR looks like a usual recursive language.  ...  Its terms are those of a linear (affine) λ-calculus -every variable occurs at most once -extended with a limited recursive scheme on a word algebra.  ...  It is a hybrid system, where the two components -the recursion-theoretic and the type-theoretic ones-are blent in a smooth way.  ... 
doi:10.1007/978-3-540-24849-1_12 fatcat:kcjjomcfpfbvlkzmzp4g46a32u

Page 5475 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
The paper is about a machine-independent characterization of a polytime-space hierarchy {7-%yp}yp<w (g. p = 1) whose functions can be computed by a register machine in polynomial time and space, simultaneously  ...  A remarkable aspect of the work is its implicit proof-theoretical flavor, given by a sort of classification of the clauses defining BC: there are basic instructions that play the role of built-in functions  ... 

Realizability models for BLL-like languages

M. Hofmann, P.J. Scott
2004 Theoretical Computer Science  
We give a realizability model of Girard-Scedrov-Scott's Bounded Linear Logic (BLL). This gives a new proof that all numerical functions representable in that system are polytime.  ...  Our analysis naturally justiÿes the design of the BLL syntax and suggests further extensions.  ...  The main theorem in [5] is that the number-theoretic functions representable in BLL are polytime.  ... 
doi:10.1016/j.tcs.2003.10.019 fatcat:e3dh65w42vf2tilbcvkgdwh3iq

Realizability models for BLL-like languages

M HOFMANN
2004 Theoretical Computer Science  
We give a realizability model of Girard-Scedrov-Scott's Bounded Linear Logic (BLL). This gives a new proof that all numerical functions representable in that system are polytime.  ...  Our analysis naturally justiÿes the design of the BLL syntax and suggests further extensions.  ...  The main theorem in [5] is that the number-theoretic functions representable in BLL are polytime.  ... 
doi:10.1016/s0304-3975(03)00524-3 fatcat:43kveltl75dl3euio63muukreq

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime [article]

Akitoshi Kawamura and Norbert Th. Müller and Carsten Rösnick and Martin Ziegler
2012 arXiv   pre-print
The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics.  ...  It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard.  ...  This gives rise to two causes for noncomputability: a (classical) recursion theoretic and a topological one; cmp. Figure 1b) .  ... 
arXiv:1211.4974v1 fatcat:qdrmh54p3vfydf7627l3pejirm
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