S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....together with its Prolog equivalents, is discussed below. To understand the semantics of logic programming, we start with the fragment of the logic for which uniform proofs are complete for logical consequence. Reading proofs from the root upwards, i.e. using the rules as reduction operators [24], uniform proof requires that right rules be applied whenever possible, so that left rules are applied only when the right hand side is atomic. Uniform proofs are said to be simple just in case the implicational left rules are restricted to be essentially unary. For example, in first order ....

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....together with its Prolog equivalents, is discussed below. To understand the semantics of logic programming, we start with the fragment of the logic for which uniform proofs are complete for logical consequence. Reading proofs from the root upwards, i.e. using the rules as reduction operators [24], uniform proof requires that right rules be applied whenever possible, so that left rules are applied only when the right hand side is atomic. Uniform proofs are said to be simple just in case the implicational left rules are restricted to be essentially unary. For example, in first order ....

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....proceeds from a putative conclusion to sucient premisses, regulated by reduction operators, OR , SucientPremiss 1 : SucientPremissm PutativeConclusion OR ; corresponding to (admissible) rules, R. The author believes that this idea of reduction was rst explained in these terms by Kleene [18]. A search is constructed, inductively, by applying instances of reduction operators of this form to putative conclusions of which a proof is desired, thereby yielding a collection of sucient premisses, proofs of which would be sucient to imply the existence of a proof, obtainable by deduction, of ....

Kleene, S.C., \Mathematical Logic", Wiley and Sons, 1968.

.... value of ae ( P k ae k ) Proof: We define t ijk (xxxx) OE k x ijk T ik (xxxx) 30) Because T ik are convex functions and CC is a convex set, the solution x x x x of the problem exists (see, 18] and from the Kuhn Tucker condition it is characterized by the relations (see, e.g. [19]) t ijk ( xxxx) ff ik ; x ijk 0; ff ik ; x ijk = 0; 31) X j x ijk = 1; for all i; k; where ff ik are the Lagrange multipliers. From Definitions (1) 5) to (7) 16) and (30) we have t iik (xxxx) OE k T ik x iik (xxxx) ae k [D(fi j (xxxx) ae k x iik D 0 ....

.... all i; k; T (xxxx) aeD(ae) 15 Proof: We define t ijk (xxxx) mOE k x ijk T k (xxxx) 50) Again, because T k is a convex function and CC is compact, the solution xx x x of the problem exists (see [18] and from the Kuhn Tucker condition it is characterized by the relations (see, e.g. [19]) t ijk (xxxx) ff ik for x ijk such that x ijk 0; ff ik for x ijk such that x ijk = 0: 51) X j x ijk = 1; for all i; k where ff ik are the Lagrange multipliers. From the definitions (1) to (8) 48) and (50) we have t iik (xxxx) mOE k T k x iik = ae k [D(fi i ) fi ....

J. F. Shapiro, Mathematical Programming, Structures and Algorithms, J. Wiley and Sons, 1979.

....consequence of Gamma only if both OE 1 and OE 2 are consequences of Gamma. It should be clear that this amounts to a reading of this instance of the right rule for classical conjunction, R Gamma OE 1 Gamma OE 2 Gamma OE 1 OE 2 as a reduction operator, from conclusion to premisses [16]. 4 ffl OE 1 OE 2 is a consequence of Gamma only if either OE 1 or OE 2 is a consequence of Gamma. It should be clear 4 Note that, for simplicity, we consider here only singleconclusion sequents. 4 that this amounts to a reading of this instance of the right rule for classical ....

S. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....a suitable notion of resolution proof can be defined. A summary of this analysis consists in essentially two, mutually dependent, steps: i) The identification of the class of formulae such that strategy of constructing proofs (considering rules to be reduction operators, in the sense of Kleene [7]) by applying right rules wherever they are applicable is complete, subject to certain exceptions to this strategy, which must be handled, i.e. made goal directed, by the second step. This class of proofs, constructed by the application of the right rules wherever possible, subject to the ....

Kleene, S.C. Mathematical Logic. Wiley and Sons, 1968.

....cannot always push the Omega L above the Omega R rule; but the cases in which this is not possible can be classified. A similar example occurs for the C L rule. Consider the sequent OE OE Omega OE. Clearly, there is a proof in which the Omega R rule precedes the C L rule, i.e. 1 Kleene [14] explains this in the case of classical predicate calculus. 2 OE OE OE OE L L OE OE OE OE Omega R OE; OE OE Omega OE C L. OE OE Omega OE but we cannot apply the rules in the reverse order, as OE is not provable. Thus, as in the previous case, we cannot always push ....

....suitable notion of resolution proof can be defined. A summary of this analysis consists of essentially two, mutually dependent, steps: i) The identification of the class of formulae such that the strategy of constructing proofs (considering rules to be reduction operators, in the sense of Kleene [14]) by applying right rules wherever they are applicable is complete, subject to certain exceptions to this strategy, which must be handled, i.e. made goal directed, by the second step. This class of proofs, constructed by the application of the right rules wherever possible, subject to the ....

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....only to atoms; extensions to the class of negated formulae are possible. Also, for simplicity, we limit implicational definite formulae to exclude G Gammaffi (A 1 : Am ) G Gammaffi V x : A, etc. see [10] Some other extensions are mentioned at the end of this section. 1 Kleene [11] explains this in the case of classical predicate calculus. Definition 2.3 (Definite and Goal Formulae) Let A range over atomic formulae. We define classes of definite formulae and goal formulae as follows: Definite formulae D : A j D D j D Omega D j G Gammaffi A j V x : D j D Goal ....

Kleene, S.C. Mathematical Logic. Wiley and Sons, 1968.

....focus on these interdependencies and the implicit isomorphism among these proof systems, and analyze these issues formally in HOL. We use higher order logic as the metalogic for reasoning about first order proof systems, whereas, usually this kind of reasoning is done in a natural language, e.g. [6, 3, 9]. By using a formal system as the metalogic and formalizing the proofs in a theorem proving environment, we reduce susceptibility to errors. The main theorem we prove here states that whenever a conclusion is provable from hypotheses in Hilbertian axiomatization, it is also provable from the same ....

....is by far not original. For example, Girard et al. 3] note that to a proof of A B in sequent calculus there corresponds a deduction of B under the hypotheses A in natural deduction, and, conversely, a deduction of B from A can be represented in sequent calculus, but not uniquely. Kleene in [6] shows that provability in G4, which is a variant of sequent calculus, implies provability in a Hilbert type formal system H. Also, he proves the deduction theorem for the system H; the latter differs from our proof system HA in taking all logical connectives as basic and, therefore, having three ....

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S. C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....rule Gamma F 1 ; Delta Gamma F 2 ; Delta Gamma F 1 F 2 ; Delta we conclude that the sequent is provable if both p; q p and p; q q are provable. In this way we need to be able to interpret the rules of the sequent calculus as a bottom up method of proof construction (Kleene [16] discusses this in more depth for the classical sequent calculus) In addition, we need to be able to search for goal directed proofs, rather than arbitrary proofs. The best known characterization of this property is the notion of a uniform proof [22] This is defined in terms of proofs in the ....

S. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....scope of this paper, we briefly review the key ideas. In the cut free (linear) sequent calculus, goal directed proof can be achieved via uniform proof [16, 17] The basic idea, introduced in [16] is to use the left and right rules of the sequent calculus as reduction operators (in the sense of [13]) and proceed as follows: if, at any stage, some right rules are applicable, then one of them must be applied; otherwise, i.e. if all goal formulae are atomic, proceed to apply a left rule. In linear logic with multiple conclusions, this basic notion is not quite adequate. A slightly weaker ....

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....( Phi; oe) in which oe is an answer substitution for X and Phi is a proof of the sequent P Goe. Operationally, we must describe how to execute a program when it is supplied with a goal. Recall first that inference rules can be read as reduction operators, from conclusion to premisses. Kleene [22] explains this in the case of the classical predicate calculus. Such operators are the basic units of proof search, or backward chaining, just as inference rules are the basic units of deduction, or forward chaining. A semantics based on goal directed proof search is computationally appealing. ....

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....( Phi; oe) in which oe is an answer substitution for X and Phi is a proof of the sequent P Goe. Operationally, we must describe how execute a program when it is supplied with a goal. Recall first that inference rules can be read as a reduction operators, from conclusion to premisses. Kleene [11] explains this in the case of the classical predicate calculus. Such operators are the basic units of proof search, or backward chaining, just as inference rules are the basic units of deduction, or forward chaining. A semantics based on goal directed proof search is computationally appealing. ....

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

....random variables that sum to 1. Then OE 1 ; OE l Gamma1 have a Dirichlet pdf f if f(OE 1 ; OE l Gamma1 ) Gamma( P l i=1 ff i ) Q l i=1 Gamma(ff i ) l Y i=1 OE ff i Gamma1 i (1) where OE l = 1 Gamma P l Gamma1 i=1 OE i and ff i are positive constants (See, e.g. [De70, Wi62]) We use the following conventions. Suppose f ij g, 1 i k, 1 j n, is a set of positive random variables that sum to 1. Let i Delta ; Deltaj ; I Delta ; DeltaJ ; jji ; ijj ; J ji , and I jj be defined as in the introduction. Consequently, i Delta jji = Deltaj ijj ....

S. Wilks, Mathematical Statistics, 1962, Wiley and Sons, New York.

....into p j only. Assume that the vectors in each of these sequences can be modeled as coming from a multivariate Gaussian distribution and that the vectors are statistically independent. A good clustering solution should have relatively small dispersion within clusters. The within cluster dispersion [7] is defined as W = k X j=1 N j Sigma j where Sigma j is the covariance matrix and N j is the total number of feature vectors in cluster p j . There are several good clustering criteria [6] We prefer to use the determinant of W to measure the goodness of speaker clustering. That is, the ....

Wilks, S., Mathematical Statistics, Wiley and Sons, New York, 1962.

....same as the one given previously. C. Statistical foundation Like the MC estimate, the GC estimate as defined above can be regarded as an estimator of an underlying statistical entity. In the GC case, this entity is a normalized version of the generalized variance of the processes r 1 ; r M [18]. The authors are not aware of any results concerning the performance of the GC estimate in this context (i.e. its bias, consistency, or variance) D. Issues in practical applications Throughout this paper it has been assumed that the GC approach is used to test for the presence of a common ....

S.S. Wilks, Mathematical Statistics, J. Wiley and Sons, 1962.

....a pair (#, #) in which # is an answer substitution for X and # is a proof of the sequent P # G#. Operationally, we must describe how to execute a program when it is supplied with a goal. Recall first that inference rules can be read as reduction operators, from conclusion to premisses. Kleene [21] explains this in the case of the classical predicate calculus. Such operators are the basic units of proof search, or backward chaining, just as inference rules are the basic units of deduction, or forward chaining. A semantics based on goal directed proof search is computationally appealing. ....

S. C. Kleene, Mathematical logic, Wiley and Sons, 1968.

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S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

No context found.

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

No context found.

S.C. Kleene. Mathematical Logic. Wiley and Sons, 1968.

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Wilks, S., Mathematical Statistics, Wiley and Sons, New York, 1962.

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S. Wilks, Mathematical Statistics, Wiley and Sons, New York.

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