On the Decidability of Formulae Involving Continuous and Closed Functions

The satisfiabil i ty problem for a syllogistic embracing 6, E, Boolean set operations, the Kuratowski topological closure operation ~~, and continuous and closed maps between topological spaces, along w i th the operations of point evaluation, set image, and inverse set image, is solved for formulae that meet a particular syntactic non-circulari ty property. The unquantif ied interpreted language £ to be considered has inf initely many sorts of variables, each corresponding to a different topological space. Three kinds of variables are available, namely, individual, set, and map variables. Ind iv idual variables of a given sort are supposed to range over the universe of that sort, whereas set variables range over the subsets of the appropriate universe. Finally, each map variable ranges over the collection of continuous or closed maps between two appropriate topological spaces. 1 I n t r o d u c t i o n Once a decision a lgor i thm has been found for a formalized mathematical theory T, it is rewarding to discover that the val id i ty problem for some other theory T * , possibly related to a different branch of mathematics, reduces to the val idity problem of T, which one is already able to solve. Instances of this are common in the l i terature. We are therefore motivated in undertaking a quest for new problems whose solution is amenable to 'computable set theory', i.e. to the thick cluster of recently attained methods regarding various portions of classical set theory (see [Cantone, 1988] for an extensive bibliography on this subject). Some indications in this direction have been given in [Cantone et a/., 1987], where the satisfiability problems of two theories dealing wi th monotone functions on tota l ly and part ia l ly ordered sets have been translated into MLS, or "mult ilevel syllogistic" (cf. [Ferro et a/., 1980]), a well-k nown decidable theory. •Work supported by ENI and ENIDATA within the AXL project. * Sub jec t area: B5. Automated deduction. OF FORMULAE INVOLVING CLOSED FUNCTIONS** Eugen io G . O m o d e o ENIDATA-Bologna Viale Aldo Moro, 38 40127 Bologna, I taly Specifically, it was shown there how to reduce to MLS the satisfiability problem for propositional combinations of atoms of type where / stands for a monotone function. In the same paper a decision test was provided for the satisfiability problem of an unqua l i f i ed 'extended Tarsia' theory of reals wi th variables designating continuous real-valued functions, based on Tarski's celebrated result on the real field (see [Tarski, 1951]). Such theory allows real addition and subtraction, mult ipl icat ion and division, and comparison between variables representing real numbers; also, addition and subtraction of (continuous) functions, function evaluation, posit ivity, monotonicity and convexity predicates are allowed. More precisely, a decision test was provided for propositional combinations of atoms of the types