V.A. Uspensky, Kolmogorov and mathematical logic, The Journal of Symbolic Logic, vol. 57 (1992), no. 2, pp. 385--412.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of intuitionistic truth explicit, though informal, by introducing what is now known as the Brouwer Heyting Kolmogorov (BHK) semantics ( 48] 49] 52] The BHK semantics is widely recognized as the intended semantics for intuitionistic logic ( 30] 31] 38] 56] 67] 71] 94] 95] 97] [98], 101] 104] Its description uses the unexplained primitive notions of construction and proof (Kolmogorov used the term problem solution for the latter) It stipulates that ffl a proof of AB consists of a proof of A and a proof of B, Received by the editors September 14, 1998; revised August ....

V.A. Uspensky, Kolmogorov and mathematical logic, The Journal of Symbolic Logic, vol. 57 (1992), no. 2, pp. 385--412.

....made Brouwer s definition of intuitionistic truth explicit, though informal, by introducing what is now known as Brouwer Heyting Kolmogorov (BHK) semantics. BHK semantics is widely recognized as the intended semantics for intuitionistic logic ( 18] 19] 20] 24] 37] 47] 50] 72] 73] [74], 75] 76] BHK semantics gives an informal explanation of the truth of intuitionistic connectives. A statement is true if it has a proof, and a proof of a logically compound statement is given in terms of the proofs of its components. The description uses the unexplained primitive notions of ....

....of . This semantics was partially introduced by Heyting [29] clauses for conjunction and disjunction) and by Kolmogorov [34] clauses for implication and negation) The above formulation of BHK semantics appeared in [30] For further comments one may consult [18] 20] 24] 69] 72] 73] [74]. The natural problem of formalizing BHK semantics and establishing the completeness of Int with respect to this semantics remained open until recently despite a long history of studies in this area (see section 3 of this paper) To be sure, there are many models of different natures known for ....

[Article contains additional citation context not shown here]

V.A. Uspensky, "Kolmogorov and mathematical logic", Journal of Symbolic Logic, 57, No.2, 1992.

....logic within the classical mathematics. Technical Report CFIS 98 06, Cornell University y 627 Rhodes Hall, Cornell University, Ithaca NY, 14853 U.S.A. email:artemov hybrid.cornell.edu and Moscow University, Russia. 1 Kleene realizability [22] Medvedev finite problems [33] and its variants ([50], 51] are regarded (cf. 48] 13] 50] 51] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [25] turned out to be ....

....Technical Report CFIS 98 06, Cornell University y 627 Rhodes Hall, Cornell University, Ithaca NY, 14853 U.S.A. email:artemov hybrid.cornell.edu and Moscow University, Russia. 1 Kleene realizability [22] Medvedev finite problems [33] and its variants ( 50] 51] are regarded (cf. 48] 13] [50], 51] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [25] turned out to be based on an inconsistent theory (cf. ....

V.A. Uspensky, "Kolmogorov and mathematical logic", Journal of Symbolic Logic, 57, No.2, 1992.

....From this, we attempt to investigate the information content of a string when we deal with non deterministic algorithms. We rst identify a class of description modes which captures non deterministic methods of computation and on which we can de ne a notion of entropy. The word entropy, as used in [4, 5], means the additively optimal measure of a given class of description modes. Then, we exhibit a proper hierarchy of entropies, based on an incompressibility lemma. We also give a necessary and sucient condition for the entropy of a class of non deterministic mode to be smaller than the usual ....

....information are made up to some additive constant, it is convenient to introduce the partial order ct on functions over natural numbers N de ned by f ct g if there is a constant c such that f(x) g(x) c. Also f= ct g if f ct g and g ct f , and f ct g if f ct g but g6 ct f . According to [4], an additively optimal mode, or in short optimal mode, O for a class C of description modes is a description mode which is in C and such that for every mode R 2 C, KO ct KR . The entropy of the class C is KO . We shall brie y recall standard de nition and result of description modes restricted ....

V.A. Uspensky. Kolmogorov and mathematical logic. Journal of Symbolic Logic, 57(2), June 1992.

....of problems. Introduction In 1932 Kolmogorov ( 16] gave an informal description of the calculus of problems in classical mathematics and conjectured that it coincides with intuitioinistic propositional logic Int. Kleene realizability [15] Medvedev finite problems [23] and its variants ([36], 37] are regarded (cf. 34] 10] 36] 37] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. In 1933 Godel ( 12] defined Int on the basis of the notion of proof in a ....

....Kolmogorov ( 16] gave an informal description of the calculus of problems in classical mathematics and conjectured that it coincides with intuitioinistic propositional logic Int. Kleene realizability [15] Medvedev finite problems [23] and its variants ( 36] 37] are regarded (cf. 34] 10] [36], 37] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. In 1933 Godel ( 12] defined Int on the basis of the notion of proof in a classical mathematical system, where proof ....

V.A. Uspensky, "Kolmogorov and mathematical logic", Sitzungsberichte der Berliner Mathematischen Gessellschaft, 41-74, Berlin, 1993

....sort of interpretation of Int classical BHK semantics. Classical realizabilities: Kleene realizability [19] function realizability [20] modified realizability [24] Medvedev s calculus of finite problems [31] and its variants, give conditions necessary but not sufficient for Int(cf. 12] 45] [46], 47] Each of them realizes some formulas Department of Mathematics, Cornell University, email:artemov math.cornell.edu and Moscow University, Russia. not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [23] turned out to be based on an inconsistent theory ....

V.A. Uspensky, "Kolmogorov and mathematical logic", Journal of Symbolic Logic, 57, No.2, 1992.

....of interest in the (computational part of the) SP programme has been to understand whether or not all forms of computing and formal reasoning may usefully be understood as information compression by pattern matching, unification and search. The concept of an algorithm in AIT (see, for example, [17]) seems to have little in common with ideas about how functions may be executed by compression (as described in Sections 2 and 5, above, and in [26] Because of the contradiction discussed above, AIT cannot, as it stands, accommodate the idea that computing may itself be a process of ....

Uspensky, V. A. (1992). Kolmogorov and mathematical logic. Journal of Symbolic Logic 57(2): 385-412.

....to be canonical. We have reached a con2 tradiction. Borel [1, 2] was the first author who systematically studied random sequences. The complexity theoretic approach was independently initiated by Kolmogorov [22] and Chaitin [9] For more historical facts see Chaitin [17] A Life in Math) Uspensky [31], Li and Vitanyi [23] and Calude [4] 2 Computers and Complexities Denote by N the set of natural numbers; N = N 0 . If S is a finite set, then #S denotes the cardinality of S. We shall use the following functions: i) rem(m, i) the remainder of the integral division of m by i (m, i # N ....

V. A. Uspensky. Kolmogorov and mathematical logic,J. Symbolic Logic 57(1992), 385-412.

No context found.

V.A. Uspensky, Kolmogorov and mathematical logic, The Journal of Symbolic Logic, vol. 57 (1992), no. 2, pp. 385--412.

No context found.

Uspensky, V. A. `Kolmogorov and mathematical logic,' Journal of Symbolic Logic 57(2):

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC