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Additional resources for Distributed Operating Systems: Theory and Practice
From these mappings, we defined mappings Fs*t from state functions on S' to state functions on S and F;c from action predicates on A' to functions on A X S. If we examine how all these mappings are actually defined for a real example, we discover that it is the mappings F:t and F;c that are really being defined. This is because we don't know what the actual states are, just the state functions. Let II, ... , In be the state functions of the first system and If, ... , l:n be the state functions of the second system.
The formal definitions are analogous to the ones for Fst and Fs*t, and we won't bother with the details. The mappings F:t and F:c induce a mapping F* from higher-level general predicates to lower-level ones, where, for any higher-level general predicate G, F*(G) is the predicate whose value on (s,a) equals G(Fst(s),Fac(s,a)). l Any general predicate is represented as a function of elementary state and action predicates, so F* can be computed from the F:tUj) and the F:c(A(a)) for the higher-level elementary state functions and action predicates A(a).
We now formally define what it means for the first to implement the second. We call (8, S, A) the lower-level system and its state functions, behaviors, etc. are called lower-level objects; (8', S', A') is said to be the higher-level system and its state functions, etc. are called higher-level objects. 2, where the lower-level system was an assembly language program and the higher-level one was a program written in a higher-level language. We defined a mapping F from lowerlevel behaviors to higher-level behaviors.