Multi-dimensional mathematical expressions, called tables, have proven to be useful for documenting digital systems. This paper describes 10 classes of tables, giving their syntax and semantics. Several abbreviations that can be useful in tables are introduced. Simple examples are provided. 1 Introduction In earlier papers, [1,2], we have shown how the documentation required for the professional construction and use of computing systems can consist of descriptions of a set of mathematical relations. Those papers discuss the documents very abstractly; the contents of the documents are specified without restricting the notations or formats to be used. This paper complements the earlier papers by defining multidimensional notations (which we call tabular expressions) that have proven useful for describing the specified mathematical functions in practical applications [3, 4, 5, 6, 7, 8]. A companion paper [9], presents an interpretation of logical expressions that is designed for these...

Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to (n+1) exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars, where the no more than n grammars can each make finitely many mistakes. This contrasts sharply with results of Barzdin and Podnieks and, later, Case and Smith, for learnability from both positive and negative data. A subset principle from a 1980 paper of Angluin is extended to the vacillatory and other criteria of this paper. This principle, provides a necessary condition for circumventing overgeneralization in learning from positive data. It is applied to prove another theorem to the eff...

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A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,

A theory of recursive definitions has been mechanized in Isabelle's Zermelo-Fraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.

A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ff-conversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a fi-contraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...

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this paper we discuss the following three questions. 1. Given a real-valued function f 2 L

In [15, 22, 25, 26] Parnas et al. advocate the use of relational model for documenting the intended behaviour of programs. In this method, tabular expressions (or tables) are used to improve readability so that formal documentation can replace conventional documentation. Parnas [23] describes several classes of tables and provides their formal syntax and semantics. In this paper, an alternative, more general and more homogeneous semantics is proposed. The model covers all known types of tables used in Software Engineering. Contents 1 Introduction 2 2 Introductory examples 4 3 Relations 9 3.1 Cartesian Products, Functions, Relations . . . . . . . . . . . . . . . 9 3.2 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Input-Output Relations . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Raw Table Skeleton 14 5 Cell Connection Graph and Medium Table Skeleton 15 6 Raw and Medium Table Elements 19 Supported by NSERC of Canada Grant 7 Well Do...