Induction For Formulas Research Articles

Automated induction of heterogeneous proximity measures for supervised spectral embedding.

Spectral embedding methods have played a very important role in dimensionality reduction and feature generation in machine learning. Supervised spectral embedding methods additionally improve the classification of labeled data, using proximity information that considers both features and class labels. However, these calculate the proximity information by treating all intraclass similarities homogeneously for all classes, and similarly for all interclass samples. In this paper, we propose a very novel and generic method which can treat all the intra- and interclass sample similarities heterogeneously by potentially using a different proximity function for each class and each class pair. To handle the complexity of selecting these functions, we employ evolutionary programming as an automated powerful formula induction engine. In addition, for computational efficiency and expressive power, we use a compact matrix tree representation equipped with a broad set of functions that can build most currently used similarity functions as well as new ones. Model selection is data driven, because the entire model is symbolically instantiated using only problem training data, and no user-selected functions or parameters are required. We perform thorough comparative experimentations with multiple classification datasets and many existing state-of-the-art embedding methods, which show that the proposed algorithm is very competitive in terms of classification accuracy and generalization ability.

Investigation on Factors Influencing Workability of Recycled Aggregates Concrete

In order to investigate factors influencing workability of recycled aggregates (RA) concrete, such as water amount, sand ratio and RA replacement percentage, 10 pitch of RA concrete was prepared for measurement of workability and a formula was used for calculating effect of water amount on slump of RA concrete. Experimental results and formula induction indicated that there are logarithmic relationship between flowability of fresh concrete and water amount and ratio of aggregates to cement . The flowability will logarithmically increase with increase of water amount and decrease of aggregates cement ratio. The optimum sand ratio for RA concrete can be determined by measurement of slump. Slump decreases with RA replacement percentage and there is a very good linear relationship between them. When water amount increases, slump increases and when replacement percentage of RA increases, slump decrease. Under different water amount, effect degrees of replacement percentage of RA on slump are very similar and under different replacement percentage of RA, effect degrees of water amount on slump are also very similar.

Systems of explicit mathematics with non-constructive μ-operator. Part II

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET(μ) plus set induction (type induction, formula induction) is proof-theoretically equivalent to Peano arithmetic PA (the second-order system ( Π 0 ∞-CA < ε 0 , the second-order system ( Π 0 ∞-CA) < ε ε 0 ).

Systems of explicit mathematics with non-constructive μ-operator. Part I

Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 (1993) 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: (i) BON(μ) plus set induction is proof-theoretically equivalent to Peano arithmetic PA; (ii) BON(μ) plus formula induction is proof-theoretically equivalent to the system ( Π 0 ∞- CA) <ε0 of second-order arithmetic.

Well-founded induction and the invariance theorem for loops

Well-founded induction is very general. For example the so-called “principle of strong induction” is well-founded induction on the set of natural numbers with their usual d -ordering, while so-called “ transfinite induction” is well-founded induction when the ordering d is total. Proofs using well-founded induction may not proceed as smoothly as one would like when the theorem to be proved is not in the neat form (Vx: x E C: Q.x). We address this inconvenience in what follows by formulating well-founded induction for formulae of the shape (Vx: f.x E C: Q.x) where C is a well-founded set and f is any function (whose range may or may not be a subset of C). We will use the result to make a short formula-driven proof of the fundamental invariance theorem for loops, and indeed this is the immediate motivation of the work. Let (C, <) abbreviate “set C partially ordered by Q “9 and define x<y to be xgy~x#y for all x, y and any partial ordering d . “(C, G ) is well-founded” means that for each subset S of C the following holds:

The relation of A to Prov ˹A˺ in the Lindenbaum sentence algebra

Let U be a consistent axiomatic theory containing Robinson's Q [TMRUT, p. 51]. In order for the results below to be of interest, U must be powerful enough to carry out certain arguments involving versions of the “derivability conditions,” DC(i) to DC(iii) below, of [HBGM, p. 285], [F60, Theorem 4.7], or [L55]. Thus it must contain, at least, mathematical induction for formulas whose prenex normal forms contain at most existential quantifiers. For convenience, U is assumed also to contain symbols for primitive recursive functions and relations, and their defining equations. One of these is used to form the standard provability predicate, Prov ˹A˺, “there exists a number which is the Gödel number of a proof of A.” Upper corners denote numerals for Gödel numbers for the enclosed sentences, and parentheses are often omitted in their presence.This paper contains some results concerning the relation between the sentence A, and the sentence Prov ˹A˺ in the Lindenbaum Sentence Algebra (LSA) for U, the Boolean algebra induced by the pre-order relation A ≤ B ⇔ ⊦A → B. Half of the answer is provided by a theorem of Löb [L55], which states that ⊦Prov ˹A˺ → A ⇔ ⊦A. Hence, in the presence of DC(iii), below, it is never true that Prov ˹A˺ &lt; A in the LSA. However, there is a large and interesting set of sentences, denoted here by Γ, for which A &lt; Prov ⌜A⌝.