Manin, Yu. I., A Course in Mathematical Logic, Springer, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from a much larger set. However, we will exploit the fact that any derived set built starting from N and I can be mapped injectively in N , and that N can be mapped surjectively on any such derived set using standard arithmetic operations based upon primitive recursion only (see, for instance, [Manin:1977]) once we code the domain and codomain of the functions in DN using this maps, we shall obtain exactly the class of recursive functions. Definition 5 The (bijective) encodings # N and # N are defined by # (#n, k#) k and by the standard pairing function (for this latter and its ....

....the domain and codomain of the functions in DN using this maps, we shall obtain exactly the class of recursive functions. Definition 5 The (bijective) encodings # N and # N are defined by # (#n, k#) k and by the standard pairing function (for this latter and its inverse, see [Manin:1977]) Let 0 N the constant zero function, and I the terminal map associated to N . For each derived set X we define the encodings # X N and # # X X : I , then # X 0 and # # X N ; N , then # N # # N . Note that the ....

Yu.I. Manin. A Course in Mathematical Logic. Springer--Verlag, New York, 1977.

....from a much larger set. However, we will exploit the fact that any derived set built starting from N and I can be mapped injectively in N, and that N can be mapped surjectively on any such derived set using standard arithmetic operations based upon primitive recursion only (see, for instance, [Manin:1977]) once we code the domain and codomain of the functions in N using this maps, we shall obtain exactly the class of recursire functions. Definition 5 The (bijective) encodings (y : N N N and (yx: N x N N are defined by (y ( r, k ) 2r k and by the standard pairing function (for this ....

....and codomain of the functions in N using this maps, we shall obtain exactly the class of recursire functions. Definition 5 The (bijective) encodings (y : N N N and (yx: N x N N are defined by (y ( r, k ) 2r k and by the standard pairing function (for this latter and its inverse, see [Manin:1977]) Let 0: I N the constant zero function, and iN: N I the terminal map associated to N. For each derived set X we define the encodings (Yx: X N and (y( N X: if X = I, then (Yx = 0 and (y( IN if X N, then (N (N = o ; if X = Y Z, then (x = o ( y (z) and ....

Yu.I. Manin. A Course in Mathematical Logic. Springer-Verlag, New York, 1977.

....nature of GVT in the GVT manager. Models were developed for each using the Larch Shared Language and Larch C and characteristics verified using the lsl checker, the Larch Prover, and to a lesser extent the lcpp parser. If Yuri Manin s statement a good proof is one that makes us wiser [13] is to be believed, then this specification and verification activity was highly successful. Formal specification and verification were used in an engineering activity in a manner congruent with other engineering disciplines. A pure, complete mathematical model of the Time Warp system was not ....

Y. Manin. A Course in Mathematical Logic. Springer-Verlag, 1977. 13

....# i denotes f Delta Delta Delta f z i times ; g Delta Delta Deltag. 5 Without proof, we state a lemma which is the key to show that sets hypersets are closed under the transitive closure, element removal and unionset operations, and that they fulfill the so called (cf. e.g. [Man77]) separation scheme. Lemma 1. Let G be an uncontractible graph. Whenever defined, G j ) j coincides with G j for a suitable . No two distinct nodes 0 ; 00 can fulfill G j 0 = G j 00 . If 1 ; m ( m 0) are nodes of G, then a hyperset (actually a set when each G ....

Yu. I. Manin. A course in Mathematical Logic. Graduate Texts in Mathematics. Springer-Verlag, 1977.

....generated free group is benign. But we can simplify the problem by working in a two generators group: the Higman Neumann Neumann Embedding Theorem states that we can embed F into a two generators group preserving the recursiveness of presentations (see for the proof the papers by Higman or [8]) Hence, it su#ces to prove that any recursively enumerable normal subgroup N of a free group L = #a, b# is benign in L. 23 We need to precise what we mean by set of words on the generators a and b recursively enumerable . We shall use again Godel s method, assigning to each word w # L ....

Yu. Y. Manin, A Course in Mathematical Logic, Springer-Verlag, New York, Heidelberg, Berlin, 1977.

....the additional assumption that all critical values of the potential U(x) are real. There are several broadly used classes of constructive functions, among them e#ective (algorithmically computable) primitive recursive (defined by a finite number of inductive rules) and elementary functions, see [12]. Our main theorem asserts the strongest form of computability of the upper bound as a function of two natural values n, d. Theorem 1. For any real polynomial U (x) # R[x] of degree n 1 and any di#erential form # = Pdx Qdy of degree d, the number of real ovals # # y 2 U(x) h yielding an ....

Yu. I. Manin, A course in mathematical logic, Graduate Texts in Mathematics, Vol. 53, Springer, New York, 1977. MR 56:15345

.... that maps terms and predicates of the language (the syntactic domain) into objects or sets of objects (or properties) in the domain of interpretation (the semantic domain) whether that domain is taken to be (part of) the real world, a possible world, or intensional objects in an agent s mind [Manin 1977, Shapiro Rapaport 1987] I see two problems with the practice of KRR as I have just outlined it (see also critics of traditional AI, e.g. Harnad 1990] Lakoff 1987] Searle 1980] or recently [Angell 1993] to name just a few) First, the semantic models of KRR systems are purely ....

....of an artificial agent. 11 I do not consider producing or accepting natural language surface forms to constitute interaction; indeed, I would claim the blind agent I described cannot understand the meaning of the color words it might produce or accept. to derive new statements from given ones [Manin 1977]. Any statement s 2 S that can be derived from A using the rules in R is a theorem, and the set of all derivable theorems plus the axioms is a theory. The inference rules R are sound iff given a consistent set A, one cannot derive a contradiction, and the formal system is complete iff all ....

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Manin, Yu. I. (1977), A Course in Mathematical Logic, (Springer Verlag, New York).

....fp i j i 2 Ig a recursive enumerable set of recursive enumerable process expressions is expressible in aprACP r:e: R . Proof: A classic recursion theoretic theorem states that any non empty r.e. set can be obtained as the image of a primitive recursive function. See e.g. Corollary 4. 18 in Manin [7]. It follows that [ Sigma i2I p i ] Sigma i2IN p(n) for certain primitive recursive function p : IN (the closed aprACP r:e: R expressions) Now use the construction from Section 4.2. 2 Proposition 5 The r.e. process expressions denote exactly the r.e. graphs. Proof: It is ....

....process Sigma b2B b0 can be obtained as ae f ( Sigma a2A a0) with f a primitive recursive function (recalling that a primitive recursive function is a special total recursive function, and any total recursive function, seen as a relation, is decidable) This is Corollary 4. 18 in Manin [7]. A similar problem arises for the communication function. There are two ways in which to strengthen the decidability requirement so as to avoid these problems. The renaming operators as well as the communication function should be either effective or coeffective. The requirement of effectivity ....

Yu.I. Manin (1977): A Course in Mathematical Logic, Graduate Texts in Mathematics 53. Springer-Verlag.

....The definition is broad enough: it encompasses the formal proof systems believed to model the formalized versions of proofs of the working mathematician and computer scientist. Formal proofs are usually viewed as chains of deductions according to a small set of formal rules, from a set of axioms ([Man], HC] Formal first order proofs in usual axiom systems such as ZF or Peano arithmetic (cf. Man] are clearly polynomial time verifiable. Our convention, however, is not restricted to such systems. In particular, it does not take sides in the Hilbert Brouwer dispute (formalism vs. ....

....versions of proofs of the working mathematician and computer scientist. Formal proofs are usually viewed as chains of deductions according to a small set of formal rules, from a set of axioms ( Man] HC] Formal first order proofs in usual axiom systems such as ZF or Peano arithmetic (cf. [Man]) are clearly polynomial time verifiable. Our convention, however, is not restricted to such systems. In particular, it does not take sides in the Hilbert Brouwer dispute (formalism vs. intuitionism) proofs in any reasonable deductive system 9 are verifiable in polynomial time. Our ....

Manin, Yu. I.: A Course in Mathematical Logic, Springer Verlag, GTM 53, 1977.

.... leading to x in a suitable logical calculus [8] 4 e.g. it is estimated that a formal demonstration of one of Ramanujan s conjectures assuming set theory and elementary analysis would take about two thousand pages; the length of a deduction from first principles is inconceivable [8] [17]. 3 Further, the sequence of steps that need to be executed to prove a statement x need not be unique 5 . The steps that are required to convince an audience may depend, in addition, on the initial state of the audience. In the initial state (before the execution of a proof) the audience is ....

Y.I.Manin. A Course in Mathematical Logic. Springer-Verlag, 1977.

....the additional assumption that all critical values of the potential U(x) are real. There are several broadly used classes of constructive functions, among them effective (algorithmically computable) primitive recursive (defined by a finite number of inductive rules) and elementary functions, see [11]. Our main theorem asserts the strongest form of computability of the upper bound as a function of two natural values n; d. Theorem 1. For any real polynomial U(x) 2 R[x] of degree n 1 and any differential form = P dx Q dy of degree d the number of real ovals fl ae fy 2 U(x) hg yielding ....

Yu. I. Manin, A course in mathematical logic, Graduate Texts in Mathematics, Vol. 53. Springer, New York, 1977, MR 56 #15345

....contain sums of sets. However, it is clear that any polynomial P built from N, O and I can be mapped injectively to N, and that N itself can be mapped surjectively to any such polynomial P (except O) using standard arithmetic operations (based upon primitive recursion only; see, for instance, [Manin 1977]) such morphisms can be considered computable from any point of view. Once we add these morphisms to the standard class of recursive functions, we do get indeed the maps generated by s and p. This claim will be made formally clear in what follows. Note that the predecessor function allows one to ....

.... Gamma I and the identity 1N : N Gamma N. It is easy to define them in such a way that both they and their inverses are computable (in fact, up to some trivial encodings they are primitive recursive; for an explicit definition of i Theta , the well known pairing function, and its inverse, see [Manin 1977]; the function i can be defined as (2n j 2n 1) By structural induction, we can define for any polynomial P built up from N and I a specified computable domain morphism P Gamma N using i , i Theta , 1N and 0, and a computable codomain morphism N Gamma P using i 1 , i 1 Theta ....

Manin, Yu. I. (1977) A Course in Mathematical Logic. Springer-Verlag, New York.

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Yu. Manin. A Course in Mathematical Logic. Springer Verlag, 1977, pp. xiii+286.

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Yu. Manin. A Course in Mathematical Logic. Springer Verlag, 1977, pp. xiii+286.

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Manin, Yu. I., A Course in Mathematical Logic, Springer, 1977.

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Manin, Yu. I. A Course in Mathematical Logic, New York: SpringerVerlag, 1977.

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Yu. I. Manin, A Course in Mathematical Logic, Berlin: Springer (1976).

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