By Zhaohui Luo
This e-book develops a kind concept, stories its houses, and explains its makes use of in machine technological know-how. The publication focuses specifically on how the examine of variety concept might supply a robust and uniform language for programming, software specification and improvement, and logical reasoning. the sort concept constructed right here displays a conceptual contrast among logical propositions and computational facts kinds. ranging from an advent of the elemental recommendations, the writer explains the that means and use of the type-theoretic language with proof-theoretic justifications, and discusses a variety of matters within the examine of variety conception. the sensible use of the language is illustrated by means of constructing an method of specification and knowledge refinement in style conception, which helps modular improvement of specification, courses, and proofs. scholars and researchers in laptop technology and common sense will welcome this intriguing new publication.
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Additional info for Computation and reasoning. A type theory for computer science
Example text
05[×[0, 1]. In Fig. 1a the data consists of 600 instances per class and the MMSE regression solution results indeed in one of the Pˆe = 0 straight lines. This is the large size case; for large n (say, n > 400 instances per class) one obtains solutions with no misclassified instances, practically always. , with practically no misclassified instances) to largely deviated as in Fig. 1b, exhibiting a substantial number of misclassified instances. Finally, in Fig. 1c, the same dataset as in Fig. 05 added to component x2 of class 1 (’crosses’); this small "noise" value was enough to provoke a substantial departure from a fw∗ solution, in spite of the fact that the data is still linearly separable.
Is it possible to conceive data classification problems where MCE and MEE perform better than MMSE? And where MEE outperforms both MCE and MMSE? The answer to these questions is affirmative, as we shall now show with a simple example of a family of data classification problems, where for an infinite subset of the family MEE provides the correct solution, whereas MMSE and MCE do not [150, 219]. 7. Let us consider a family of two-class datasets in bivariate space R2 , target space T = {−1, 1}. 50) 2 2 where u(x; a, b) is the uniform density in [a, b].
2a. 01 1 produces the evolution shown by Figs. 2d. 9997). A small departure from the asymptotic solution means in 1 The value of η influences the convergence rate. 01 value was chosen so that a convenient number of illustrative intermediary PDFs were obtained. 5 0 −4 0 e e −2 0 (c) 2 4 0 −4 −2 0 2 4 (d) Fig. 1; (b), (c) and (d), show the PDF at iterations 28, 30, and 31, respectively. this case Pe = 0. 5! 32). In short, minimization of the error entropy behaves poorly in this case. 1 illustrates the minimization of theoretical error entropy leading to very different results, depending on the initial choice of parameter vectors.