By H. Andreka, J.D.Monk, I.Nemeti (eds.)

ISBN-10: 0444885439

ISBN-13: 9780444885432

ISBN-10: 9638022574

ISBN-13: 9789638022578

The János Bolyai Mathematical Society held an Algebraic common sense Colloquium among 8-14 August, 1988, in Budapest. An introductory sequence of lectures on cylindric and relation algebras was once given via Roger D. Maddux.

The current quantity isn't constrained to papers provided on the convention. as an alternative, it really is geared toward delivering the reader with a comparatively coherent analyzing on Algebraic common sense (AL), with an emphasis on present study. shall we now not hide the entire of AL, essentially the most very important omission being that the class theoretic models of AL have been handled in simple terms of their connections with Tarskian (or extra conventional) AL. the current quantity was once ready in collaboration with the editors of the complaints of Ames convention on AL (Springer Lecture Notes in machine technology Vol. 425, 1990), and a quantity of Studia Logica dedicated to AL which used to be scheduled to visit press within the fall of 1990. a number of the papers initially submitted to the current quantity look in a single of the latter.

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**Example text**

We call R + := su p R n the transitive closure n> 1 and Д* := I V R+ the reflexive transitive closure of R. R*Q)*R*. Vectors and Points. A relation v with v = vL is called row constant and will always be denoted by lower case letters. In the case of a concrete relation (Boolean matrix) v G BAxB this condition means: Whatever set C and universal relation L G BBxC we choose, an element x G A is either in relation vL to none of the elements 2 G C or to all elements z e C. Relations of this kind may be considered as subsets of the set A, predicates on A, or vectors.

Schmidt 81] and [Schmidt-Strohlein 89]): A (heterogeneous) relation algebra is an algebraic structure (В , V, A, ” , •,T) over a nonempty set В of elements, called (heterogeneous) relations. Every relation R belongs to a subset B r of B, such that the following holds: a) The algebra { B r , V, A, ” ) is a complete atomistic Boolean alge bra with join V, meet A, negation ” , inclusion C, null element O, and universal (or greatest) element L. b) Multiplication (or composition) of relations is a partial associa tive operation and the existence of a product R S implies that Q S is defined for all relations Q G B r .

To avoid the disadvantage of universal quantification we are now looking for the weakest predicate tq that alone already proves a totally correct wrt. the postcondition q and any precon dition p. This leads immediately to the notion of weakest preconditions introduced by Dijkstra. We again distinguish three types of total correct ness. 4. 8 D efinition. Let a be a program with situation transition relation Z, input relation e, and output relation a. Then the vector (predicate on the state set) w p (n , q) := eTJ(Z) A eTCaq is called the weakest precondition (of type 2) of the program a wrt.

### Algebraic Logic by H. Andreka, J.D.Monk, I.Nemeti (eds.)

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