Gs89] F. Giunchiglia and A. Smaill. Reeection in Constructive and Non- Constructive Automated Reasoning. 4 Final Considerations 3.2 Solving the Predict Problem

What have we achieved? From a representational point of view, we have a simple model of each state and in each s-context we have only those facts which are \relevant" for the state being modeled (with clear computational advantages). We have a clear distinction between the (FOL) model of the states and the metatheoretic principles allowing us to derive the evolution. In PSC we have a complete knowledge of the whole tree of states: this enables us to consider and apply whatever constraint we want about the evolution. For example, supposing we want to consider (metatheoretic) principles enabling us to cut oo \useless" or \impossible" states (e.g. determined via some principles like those in KC87]), we can simply suppose to mark such states (that is s-contexts) as \no good" (e.g. by having a PSC predicate NOGOOD(c)). Such s-contexts are not to be taken into account while generating the evolution tree, and this can be obtained simply putting in the hypotheses of each axiom of PSC, the condition that we are speaking about \good" context. From a formal point of view, inside each context we have all the nice properties FOL has (e.g. clear semantics and proof theory). Solving the predict problem, we avoid contradiction by isolating each state in separated s-contexts: in GW88] this kind of non-monotonicity has been deened non essential. Solving the complete problem, we make \consistency-based" extension to the theory. Key point in our framework is axiom PSC1 which allows (together with the reeection principles) to safely deene the s-contexts corresponding to non monotonic extension of PSC. We also provide an implementationwithin the GETFOL system GT91]. GETFOL is an interactive system based on Natural Deduction Pra65] extending the FOL system developed at Stanford by Richard Weyhrauch Wey80]. GETFOL turns out to be appropriate for our purposes since, among othe features (e.g. the many sorted language), allows the user to deene and handle multiple contexts. 5 Acknowledgements We are indebted to Fausto Giunchiglia for the many invaluable GETFOL&FOL suggestions we had. Talcott and Richard Weyhrauch are also thanked for having helped the second author in various ways. Considering the situation modeled in context C1 in which H1 > H2, and supposing we want to predict the next relation plausibly holding between the two values, we could assume CBB("H1=H2",PRDCT,"C1"), and assert (via PSC1 and R down) H1=H2 in a context C4. A more reened solution, which obeys the limit …