No abstract

Abstract. An Armstrong relation for a set of functional dependencies (FDs) is a relation that satisfies each FD implied by the set but no FD that is not implied by it. The structure and size (number of tuples) of Armstrong relatsons are investigated. Upper and lower bounds on the size of minimal-sized Armstrong relations are derived, and upper and lower bounds on the number of distinct entries that must appear m an Armstrong relation are given. It is shown that the time complexity of finding an Armstrong relation, gwen a set of functional dependencies, is precisely exponential in the number of attributes. Also shown ,s the falsity of a natural conjecture which says that almost all relations obeying a given set of FDs are Armstrong relations for that set of FDs. Finally, Armstrong relations are used to generahze a result, obtained by Demetrovics using quite complicated methods, about the possible sets of keys for a relauon.

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

Abatraet. It is the purpose of this note to show that the question of whether a given propositional formula is intuitionistically valid (in Brouwer's sense, in Kripke's sense, or just provable by Heyting's rules, see Kreisel[7]) is p-space complete (see Stockmeyer [14]). Our result has the following consequences: (a) There is a simple (i.e. polynomial time) translation of intuitionistic propositional logic into classical propositional logic if and only if NP = p-space. (b) The problem of determining if a type of the typed A-calculus is the type of a closed A-term is p-space complete (this will be discussed below). (c) There is a polynomial bounded intuitionistic proof system if and only if NP = p-space (see Cook and Reckhow [2]1.