Results 1  10
of
10
A Feasible Theory for Analysis
, 1994
"... We construct a weak secondorder theory of arithmetic which includes Weak Konig's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with # b 1 graphs) of this theory are the polynomial time computable functions. It is shown that the firstorder strength of this vers ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
We construct a weak secondorder theory of arithmetic which includes Weak Konig's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with # b 1 graphs) of this theory are the polynomial time computable functions. It is shown that the firstorder strength of this version of WKL is exactly that of the scheme of collection for bounded formulae. 1
Theories With SelfApplication and Computational Complexity
 Information and Computation
, 2002
"... Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not ne ..."
Abstract

Cited by 12 (9 self)
 Add to MetaCart
Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not necessarily total. It has turned out that theories with selfapplication provide a natural setting for studying notions of abstract computability, especially from a prooftheoretic perspective.
Choice and uniformity in weak applicative theories
 Logic Colloquium ’01
, 2005
"... Abstract. We are concerned with first order theories of operations, based on combinatory logic and extended with the type W of binary words. The theories include forms of “positive ” and “bounded ” induction on W and naturally characterize primitive recursive and polytime functions (respectively). W ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Abstract. We are concerned with first order theories of operations, based on combinatory logic and extended with the type W of binary words. The theories include forms of “positive ” and “bounded ” induction on W and naturally characterize primitive recursive and polytime functions (respectively). We prove that the recursive content of the theories under investigation (i.e. the associated class of provably total functions on W) is invariant under addition of 1. an axiom of choice for operations and a uniformity principle, restricted to positive conditions; 2. a (form of) selfreferential truth, providing a fixed point theorem for predicates. As to the proof methods, we apply a kind of internal forcing semantics, nonstandard variants of realizability and cutelimination. §1. Introduction. In this paper, we deal with theories of abstract computable operations, underlying the socalled explicit mathematics, introduced by Feferman in the midseventies as a logical frame to formalize Bishop’s style constructive mathematics ([18], [19]). Following a common usage, these theories
Polynomial Time Operations in Explicit Mathematics
 Journal of Symbolic Logic
, 1997
"... In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable fu ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable functions.
A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
On the Proof Theory of Applicative Theories
 PHD THESIS, INSTITUT FÜR INFORMATIK UND ANGEWANDTE MATHEMATIK, UNIVERSITÄT
, 1996
"... ..."
Binary Mod Generated By Their Tally Part
 Archive for Math. Logic
"... We intro du# a class of models of the bou arithmetic theory PV n . These models, which are generated by their tally part, have acu featu : they have endextensions or satisfy B# b n only in case they are closeduose exponentiation. As an application, we show that if I#0 + exp # B#1 then the polyno ..."
Abstract
 Add to MetaCart
We intro du# a class of models of the bou arithmetic theory PV n . These models, which are generated by their tally part, have acu featu : they have endextensions or satisfy B# b n only in case they are closeduose exponentiation. As an application, we show that if I#0 + exp # B#1 then the polynomial hierarchy does not collapse. This paper isc#;A#Ijwith bounded theories ofarithmetic# following Buss (1986). Nonetheless, as opposed to Buss'c#ss';jA setting  where the system of natural numbers forms the standard model  we work with theories that aim todesc#jW e the language {0, 1} # .Henc## we shall use the notation of Ferreira (1990a & 1990b). To help the reader unfamiliar with the notation we briefly desc#y e the (firstorder) stringlanguage that we use. This stringlanguagec#tringl of threec#ree# t symbols #, 0 and 1, two binary func#ry symbols # (for concatenation, usually omitted) and , and a binary relation symbol # (for initialsitia dnes ). The interpretation of these symbols in the standard model 0 1} is c#sI; exc#;# for thefunc##j# symbol : x y is the string xc#Ij#;G;#;I with itself length of y times. Given an element e # 0 1} , we denote by e thec#e## term of the language obtained by # This work wa spa69g supported by project 6E92 of CMAF (Portuga2 1 c#c#WWWWc (via thefunc#KjI symbol #) thec#eqjA ts 0 or 1 ac#AjIAG to the order of the bits in e (for determinateness, we always asso c#so # to the left). We use the following abbreviations: x # # y (s dnes of x with respec# to y) abbreviates #z # y(z#x# y); x # y (the length of x is less than or equal to the length of y) abbreviates 1 x # 1 y; and x # y (x and y have the same length) abbreviates x # y # y # x. The theories studied in this paper are bu...
The Axiom Of Choice And Combinatory Logic
"... We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice. 1. ..."
Abstract
 Add to MetaCart
We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice. 1.
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?
"... Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. ..."
Abstract
 Add to MetaCart
Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.