Non-deterministic algebras and algebraization of logics Ana Claudia de Jesus Golzio Marcelo Esteban Coniglio UNICAMP 1 Non-determinism was considered in Computer Science since its beginnings: from non-deterministic Turing machines to models of concurrency, event structures and Petri nets, as well as for variants of process languages and of l-calculus, the use of multifunctions instead of ordinary functions (asigning to each element of the domain a set of possible choices, instead of a single value) has revealed to be a extremely useful conceptual tool. Indeed, there is a need for abstraction when modelling computational procedures, by disregarding irrelevant information. Being so, instead of considering all the dependencies on all the possible parameters, they can be represented by (nondeterministic) choices. In particular, the concept of non-deterministic algebras was introduced in Computer Science in order to deal with nondeterminism. Thus, for instance, non-deterministic algebras were proposed as an - Carvalho, M.; Braida, C.; Salles, J. C.; Coniglio, M. E. terminism in Computer Science from an algebraic perspective can be In the realm of Logic, non-determinism was considered mainly as a tool for obtaining alternative semantics. Non-deterministic matrices constitute a good example of this alternative approach. The non-deterministic matrices (Nmatrices, for short), introdumatrix a valuation assigns to a complex formula can be chosen non-deterministically from a non-empty set options. That is, Nmatrices are based on non-deterministic algebras, in contrast with the usual logical matrices which are based on standard algebras. then such matrices do not provide a good decision procedure for these - ristic Nmatrix and thus obtain metaproperties such as, for example, of truth-funcionality , inherent to the matrix semantics in general and in the "real world", which sometimes may be incomplete, inaccurate use of non-determinism (by means of Nmatrices) in order to weaken the principle of truth-funcionality as a solution to this problem. - - tematic and rigorous study of the algebraic properties of Nmatrices is still missing in the literature. That is, the theory of Nmatrices has not yet been fully developed, from the point of view of its formal properties and expressive power. Principle in which the truth-value of a formula is determined functionally by the truth-value of its immediate sub-formulas. Besides the applications to Computer Science mentioned above, there are few studies on non-deterministic algebras from the perspectiamong others, was not studied with full detail in the non-deterministic context. Thus, in this initial paper we propose the formal study of the theory of Nmatrices from the point of view of universal algebra, with the aim of establishing their potential applications in the realm of algebraic semantics. ces for the more general context of Nmatrices. Thus, many of the known results in the literature on the application of the theory of logic matrices This paper contains the initial notions and results developed in what we call Non-deterministic universal algebra, which is basically a concepts and results in universal algebra in order to adapt them to the non-deterministic context. 2 In this section we present some common results in universal aluniversal algebra. A signature n : n } n n-ary connectives n = 0 constants. The domain | | = n 0}. algebra domain n An + variables formulas schema formulas n = n n ) ) 0 n }. total multifunction M + partial multifunction 1 M 2 M composed multifunction 2 1 M A 2 1 2 1 3 In this section we present the formal notions of non-deterministic algebras (or ND-algebras) and of homomorphisms between ND-algebras, which are fundamental for the development of non-deterministic universal algebra. NDdomain n n 0 . A A A A A 1 n n. = A = {t, t I , I, f, f I }; D = {t, t I , I}; (c) = cA I }). A5 F F F F A F F F F F F F F F F F F F F F F F A F F F F F F A T F F F A T F F F F = A . = A = {t', I', f ' }; D = {t', I'}; (c) = cA the following tables. A A A A A A A = A' = {t', t' I , I', f ', f ' I }; D' = {t', I'}; (c) = cA A A A ' A = A' ' . ' homomorphism h: A n 1 n A 1 n B 1 n )). A B 0 .4 = A = A . . Remember that , if c , we write (c) = cA = {c A }. ' full homomorphism h: A n 1 n B 1 n )) A 1 n A 1 n B 1 n )). n 1 n 4 ' sub- -ND-algebra n 1 n B 1 n A 1 n ). = A = A' ' A' . ' A' . sub-ND-universe n 1 n A 1 n ) = A = A' ' A' ' . A. The sub-universe of A generated by over , A . Proof: n 1 n 1 n A 1 n ) A 1 n ) n E n+1 n A 1 1 n n n n generated - -empty subset of its domain: A. Then SG SG 1 n A 1 n n 1 n A Proof: SG SG SG 1 n A 1 n n 1 n A A. SG SG 1 n A 1 n n SG SG 1 n A 1 n n 1 n SG 1 n A 1 n A 1 n SG 1 n = . SG SG ' n n n Proof: 0 0 n n n+1 n A 1 1 n n h[ A 1 1 n n A 1 1 n n B 1 1 ) n n+1 ' Proof: n n n n ' '' A n 1 n A 1 n ) = A 1 n ' SG SG Proof: n 1 n SG 1 n ) = SG 1 n A 1 n B 1 n B 1 n SG 1 n ). 5 versal algebra. Thus, it will be shown that the category of ND-algebra over a given signature is closed by arbitrary products. 1 = A 1 1 2 = A 2 2 1 A 2 A 1 A 2 A1 A2 11 21 1n 2n ) = A 11 1n ) A 21 2n 1j A 1 2j A 2 0 A1 A2 = A1 A2 . 1 2 : A 1 × A 2 A 1 2 1 A 1 2 A 2 ithcanonical projection 1 × A 2 1 = A 1 1 2 = A 2 2 : A 1 × A 2 A 1 = A 1 1 2 = A 2 2 : A 1 × A 2 A 1 × A 2 . Then A 1 × A 2 1 2 1 A 2 Proof: A 1 × A 2 1 2 ' A A 1 ×A 2 A 1 ×A 2 1 2 B 1 n B 1 n 1 2 B 1 n )} 1 B 1 n 2 B 1 n A B 1 n Ai 1 n )) B 1 n A1 1 1 1 n A2 2 1 2 n )) Ai×Ai 1 n A 1 ×A 2 1 2 1 A 1 ×A 2 2 A 1 ×A 2 h j 1 2 1 = h = h 2 A 1 ×A 2 ) j : A A j j j-th canonical projection A . ) A A A 1 n ) = Ai n 1 n A . j : A A j ) j : A A j A . Then A ) ) 6 M selector A ND-assigment A. )A M interpretation A A A A 0 1 n )A = A 1 n n n A 1 n A . 7 congruence n 1 n 1 n A 1 n A 1 n A 1 n A 1 n } } . Then . n ) 1 n 1 n An A 1 n A 1 n A 1 n A 1 n ND-algebra quotient A/ 1 n A 1 n congruence class The proof is straightforward. 8 ) A A . Proof: n 1 n 1 n A = A j j 1 1 n n A 1 n A 1 n A 1 n ) = A 1 n A A 1 n A 1 n ) = A 1 n A 1 n A 1 n A 1 n ) A )2. The ultraproduct A A . non-deterministic universal algebra. ND- -algebras