There have been semantics for logic programs as sets of definite clauses over sequence domains in [2],[6]. The sequence of substitutions caused by resolutions for logic programs can be captured by a fixpoint of a functional [3],[16]. In [15], a functional is regarded as a behaviour of a dataflow network, the semantics over sequence domains induces dataflow computing for logic programs. Also it may provide a transformation of logic programs to functional programs[16]. Motivated by dataflow computing constructions for logic programming, this paper deals with fixpoint semantics over sequence domains for logic programs with equations and negations. A transformation, representing deductions caused by resolutions and narrowings, is associated with a logic program with equations, modified from the operator in [18], so that it may be represented by a continuous functional over a sequence domain, and its least fixpoint is well-defined. An explicit construction of such a continuous functional of sequence variables is necessary for dataflow computing, and we should prove that the functional of sequence variables can exactly represent the transformation concerned with sets. For a general logic program, a functional is constructed over a sequence domain so that it may reflect a consistency-preserving renewal function for the pair of atom sets on the basis of the 3-valued logic approach as in [7], and [11]. It is a problem to construct the domain for the functional representing a generation of atom sets interpreted as true and negation as failure in the generation, for general logic programs. The functional is monotonic over a complete partial order and its least fixpoint is well-defined, although the least fixpoint is not always obtained by the limit of finite computing, because of the functional being not necessarily continuous.

We deal with a fixpoint semantics for normal logic programs by means of an algebraic manipulation of idempotent substitution sets. Because of the negation, the function associated with a given normal logic program, which captures the deductions caused by the program, is in general nonmonotonic, as long as we are concerned with 2-valued logic approach. The demerit of the nonmonotonic function is not to guarantee its fixpoint well, although the fixpoint is regarded as representing the whole behaviour. The stable model as in [6] is fixpoint of nonmonotonic functions, but it is referred to on the assumption of its existences. On the other hand, if we take 3-valued logic approach for normal logic programs as in [5], [9], [11], [14] we have the monotonic function to represent resolutions and negation as failure, and define its fixpoint well, if we permit the fixpoint not to be constructive because of discontinuity. Since the substituitions for variables in the program are essentially significant in the deductions for logic programming, we next focus on the representations by means of substitutions for the deductions, without usual expressions based on atomic formulas. We examine the semantics in terms of abstract interpretations among semantics as surveyed in [9], where an abstraction stands for the capability of representing another semantics. In this paper, in 3-valued logic approach and by means of the substitution manipulation, the semantics is defined to be an abstraction of the semantics in [5], [9]. To construct a semantics based on the idempotent substitution set, the algebraic manipulation of substitutions is significant, whereas the treatment in [10] for the case of definite clause sets is not available because of the restriction of substitutions to some variable domain as most general unifications.