Noetherianity and Combination Problems

Authors:
Silvio Ghilardi;Enrica Nicolini;Silvio Ranise;Daniele Zucchelli
Affiliations:
Dipartimento di Informatica, Università degli Studi di Milano, Italia;LORIA & INRIA-Lorraine, Nancy, France;LORIA & INRIA-Lorraine, Nancy, France;Dipartimento di Informatica, Università degli Studi di Milano, Italia and LORIA & INRIA-Lorraine, Nancy, France
Venue:
FroCoS '07 Proceedings of the 6th international symposium on Frontiers of Combining Systems
Year:
2007

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Abstract

In abstract algebra, a structure is said to be Noetherian if it does not admit infinite strictly ascending chains of congruences. In this paper, we adapt this notion to first-order logic by defining the class of Noetherian theories. Examples of theories in this class are Linear Arithmetics without ordering and the empty theory containing only a unary function symbol. Interestingly, it is possible to design a non-disjoint combination method for extensions of Noetherian theories. We investigate sufficient conditions for adding a temporal dimension to such theories in such a way that the decidability of the satisfiability problem for the quantifier-free fragment of the resulting temporal logic is guaranteed. This problem is firstly investigated for the case of Linear time Temporal Logic and then generalized to arbitrary modal/temporal logics whose propositional relativized satisfiability problem is decidable.