Logic of temporal attribute implications

Abstract

We study logic for reasoning with if-then formulas describing dependencies between attributes of objects which are observed in consecutive points in time. We introduce semantic entailment of the formulas, show its fixed-point characterization, investigate closure properties of model classes, present an axiomatization and prove its completeness, and investigate alternative axiomatizations and normalized proofs. We investigate decidability and complexity issues of the logic and prove that the entailment problem is NP-hard and belongs to EXPSPACE. We show that by restricting to predictive formulas, the entailment problem is decidable in pseudo-linear time.

Closure structures parameterized by systems of isotone Galois connections

“…Conversely, let c satisfy (31), (34), and (49). Observe that the isotony (32) follows directly by (49) for g = 1.…”

“…(c) The present approach also generalizes an inference system for attribute implications with temporal semantics. Attribute implications annotated by time points have been introduced in [50] and investigated in more detail in [49]. From the point of view of the present formalism, the formulas are expressions of the time units from now).…”

“…Interestingly, in the approach to attribute implications with temporal semantics introduced first in [50] and developed further in [49], we have utilized conceptually similar structures which utilize the notion of being closed under "time shifts." In this paper, we present results showing that most results related to closure structures in [53] and [49] can be handled by a single theory of closure structures defined on complete lattices which are parameterized by systems of isotone Galois connections. In addition, we also show that the approaches in [4,6] result as special cases of the presented formalism.…”

“…In addition, we also show that the approaches in [4,6] result as special cases of the presented formalism. Therefore, the present paper studies closure structures from the perspective of a general class of parameterizations, makes conclusions on a general level, and particular results like those in [4,6,49,53] can be obtained by selecting concrete parameterizations on complete lattices. Our paper is organized as follows.…”

“…In this setting, general fuzzy closure operators with parameterized idempotency conditions appeared. Interestingly, in the approach to attribute implications with temporal semantics introduced first in [50] and developed further in [49],…”