Sorted multi-adjoint logic programs: termination results and applications C.V. Dam´ asio 1 , J. Medina 2 , and M. Ojeda-Aciego 2 1 Centro Inteligˆ encia Artificial. Universidade Nova de Lisboa. cd@di.fct.unl.pt 2 Dept. Matem´ atica Aplicada. Univ. de M´ alaga. {jmedina,aciego}@ctima.uma.es Abstract. A general framework of logic programming allowing for the combin- ation of several adjoint lattices of truth-values is presented. The main contribu- tion is a new sufficient condition which guarantees termination of all queries for the fixpoint semantics for an interesting class of programs. Several extensions of these conditions are presented and related to some well-known formalisms for probabilistic logic programming. 1 Introduction In the recent years there has been an increasing interest in models of reasoning under “imperfect” information. As a result, a number of approaches have been proposed for the so-called inexact or fuzzy or approximate reasoning, involving either fuzzy or an- notated or similarity-based or probabilistic logic programming. Several proposals have appeared in the literature for dealing with probabilistic information, namely Hybrid Probabilistic Logic Programs [6], Probabilistic Deductive Databases [8], and Probabil- istic Logic Programs with conditional constraints [9]. Residuated and monotonic logic programs [2] and multi-adjoint logic programs [10] were introduced as general frameworks which abstract the particular details of the dif- ferent approaches cited above and focus only on the computational mechanism of infer- ence. This higher level of abstraction makes possible the development of general results about the behaviour of several of the previously cited approaches. The main aim of this paper is to focus on some termination properties of the fixed point semantics of a sorted version of multi-adjoint logic programming. In this sorted approach each sort identifies an underlying lattice of truth-values (weights) which must satisfy the adjoint conditions. Although we restrict to the ground case, we allow infinite programs, and thus there is not loss of generality. The major contribution of this paper is the termination theorems for a general class of sorted multi-adjoint logic programs, complementing results in the literature and en- hancing previous results in [1]. Then, we illustrate the application of the termination theorems to obtain known termination results for some of the previously stated ap- proaches languages. The structure of the paper is as follows. In Section 2, we introduce the preliminary concepts necessary for the definition of the syntax and semantics of sorted multi-adjoint logic programs, presented in Section 3. In Section 4, we state the basic results regarding the termination properties of our semantics, which are applied later in probabilistic settings in Section 5. The paper finishes with some conclusions and pointers to future work. 1