https://hal.inria.fr/inria-00076024/document

The calculus of constructions

To explain the idea behind the present paper the following fundamental principle is emphasized. Let X = (X 1,…, X n ) be an n-dimensional vector valued random variable, and let µ(x) =µ(x 1…, x n )be its probability measure (defined on euclidean n-space E n ). Suppose that X has the property that µ(x) =µ(gx) for every element g of a group G of order h of transformations of E n into itself. Let f(x) =f(x 1…, x n ) be a µ-integrable complex valued function on E n Then the expected value of f(x) is 

$$Ef\left( X \right) = \smallint f\left( x \right)d\mu \left( x \right) = \smallint \bar f\left( x \right)d\mu \left( x \right)$$

(1.1)

, where 

$$\bar f\left( x \right) = \frac{1}{h}\sum\limits_{g \in G} f \left( {gx} \right)$$

(1.2)

.

/pdf/a-combinatorial-lemma-and-its-application-to-probability-ckojqvfrt2.pdf

A Combinatorial Lemma and its Application to Probability Theory

Fuzzy Relational Systems

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.

/pdf/a-new-recursion-theoretic-characterization-of-the-polytime-3qr62h2f5g.pdf

A new recursion-theoretic characterization of the polytime functions

Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application

/pdf/von-neumann-algebras-and-boolean-valued-analysis-4guhu2agbo.pdf

Von Neumann algebras and Boolean valued analysis

Bounded arithmetic for NC, ALogTIME, L and NL

C*-algebras and Boolean valued analysis

In this article we study relations between various fragments of bounded arithmetic. The fragments of interest bere are the theories S~ and T~ introduced by Buss in [1]. The reader may recall that the principal axioms of S~ resp. of T~ are !t-PIND resp. !t-IND axioms, where the former are instances of polynomially bounded inducton for !t-formulas while the latter are instances of ordinary induction for !t-formulas. Indications of the significance of these theories for complexity theory are two results due to Buss [1] and Krajíèek et al. [7]: (1) In [Ir ít is shown that a function is computable in polynomial time by an oracle machine quering a !r-l-oracle iff it is !t-definable in S~. In particulaT, polynomial-time functions are precisely those !~-definable in S~. (2) In [7] it is shown that T~ = S~+l implies that the polynomial hierarchy collapses to the i + 2 level. These statements show that it is an important problem to establish a relation between the fragments. It is easy to show that S~ ~ T~ ~ S~+l for i ~ 1, In [2] it is proved that S~+l is V!t+l-conservative over T~, and by (2) above (as S~+l is V!t+2-axiomatized) it is not V!t+2-conservative if !r+2 * nr+2. In this paper we are interested in the relation of T~ to S~, whether they are equal or different theories and, assuming that they are different, whether T~ is at least somehow conservative over S~. This is a natural suspicion Dne gets after several unsuccessful attempts to separate these fragments of bounded arithmetic. It is possible that S~ * T~ and stilI that T~ is V!t-conservative over S~. Note that this seems unlikely as it implies, in particular, that all functions !~-definable in S2 are polynomial time computable, a rather powerful statement with strong consequences for complexity theory, cf. [2]. The relation of S~ and T~ was also studied in [6]. It was shown there that the set ofvnt-consequences of T~ is axiomatized over S~ by a single vnt-sentence, namely Con(Gj) consistency of propositional calculus Gj. Thus T~ is vnt-con-

On induction-free provability

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200048337

Gaisi Takeuti

Papers

Von Neumann algebras and Boolean valued analysis

Bounded arithmetic for NC, ALogTIME, L and NL

C*-algebras and Boolean valued analysis

On induction-free provability

A Transfer Principle in Harmonic Analysis