# Reference Data Abstraction and Causal Relation based on Algebraic Expressions

### Susumu Yamasaki, Mariko Sasakura

#### Abstract

This paper is related to algebraic aspects of referential relations in distributed systems, where the sites as states are assumed to contain pages, and each page as reference data involves links to others as well as its own contents. The links among pages are abstracted into causal relations in terms of algebraic expressions. As an algebra for the representation basis of causal relations, more abstract Heyting algebra (a bounded lattice with Heyting implication) is taken rather than the Boolean algebra with classical implication, where the meanings of negatives are different in the two algebras. A standard form may be obtained from any Heyting algebra expression, which may denote causal relations with Heyting negatives. If the evaluation domain is taken from the 3-valued, then the algebraic expressions are abstract enough to represent referential links of pages in a distributed system, where the link may be interpreted as active, inactive and unknown. There is a critical problem to be solved in such a framework as theoretical basis. The model theory is relevant to nonmonotonic function or reasoning in AI, with respect to the mapping associated with the causal relations, such that fixed point theory cannot be always routines. This paper presents a method to inductively construct models of algebraic expressions conditioned in accordance to reference data characters. Then we examine the traverse of states with models of algebraic expressions clustering at states, for metatheory regarding searching the reference data in a distributed system. With abstraction from state transitions, an algebraic structure is refined such that operational aspect of traversing may be well formulated.

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