2 a Theory of Abstraction Using Abstraction

Korf Korf, 1979] provides a similar computational analysis of the beneets of abstraction for state space search. This analysis is essentially restricted to Abstrips abstractions as it ignores the cost of building abstract proof plans. Like us, Korf assumes no mapping failures, and a constant branching rate. Finally, in Giunchiglia & Walsh, 1989b] we analysed the theoretical and empirical properties of using a propositional abstraction to determine which deenitions to unfold in a proof. However, we did not exploit most of the information in the structure of the abstract proof, only the sequence of deenitions that were unfolded. 8 CONCLUSIONS We have used our theory of abstraction to consider the problem of how an abstract proof can be \mapped back" onto a proof of the original theorem. This analysis has naturally suggested the architecture of a general purpose abstract theorem prover. We have implemented this theorem prover, and begun to investigate its properties. Its behaviour agrees with a simple computational model we have devised for calculating the beneets of abstraction. This provides us with a very complete picture of the theoretical, computational and empirical properties of mapping back. l /l u (l-1)t 1 1 1 0 0 time without abstraction total time constructing proof plan finding abstract proof refining proof plan a t 0 1 Figure 2: The time taken to prove a theorem using one level of abstraction, 1 plotted against the length of the abstract proof, l 1. (l m+1 ? 1)c(b m ; g m ; t m). Note that we assume a constant time to abstract and un-abstract wws. We also ignore mapping failures, abstract proofs which cannot be mapped back. In this sense, our model is optimistic; it computes the best possible savings an abstraction can provide. Thus, we deene (m; n) as follows: We now compare the time to prove the theorem without abstraction, 0 against the time with one level of abstraction, 1 : 0 = c(b 0 ; l 0 ; t 0) as required; 1 = a 0 + c(b 1 ; l 1 ; t 1) + l 1 u 1 + (l 1 ? 1)c(b 0 ; g 0 ; t 0) 1 consists of essentially two competing cost functions: the cost of proving the abstract theorem and the cost of reening the gaps. As we increase the strength of the abstraction, l 1 decreases, the cost of proving …