Peter Schroeder-Heister. Rules of denitional re ection. In M. Vardi, editor, Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science, pages 222-232, Montreal, Canada, June 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is common practice in computer science and mathematics, it is well understood, and it is admissible when reasoning about objects whose types are inductively de ned. But standard induction principles only exist for those types whose constructors satisfy the so called positivity condition [PM93,SH93] This condition requires that the type that is being de ned does not occur in a negative position in any of its constructor types. Types that violate this condition are not inductive and therefore excluded from inductive reasoning. Many concepts however that are prevalent in programming ....

....to formalize concepts from process algebras. They have combined higher order encodings with coinduction principles and implemented their design in Coq [DFH 93] Reasoning by induction is also supported by the calculus of partial inductive de nitions [Hal87] and de nitional re ection [SH93] however, both designs require induction subjects to be closed. 7 Conclusion Higher order encodings supported by logical frameworks such as LF bring many advantages in terms of elegance, eciency, and maintenance, especially when representing complex systems, such as derivation systems, logics, ....

Peter Schroeder-Heister. Rules of denitional re ection. In M. Vardi, editor, Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science, pages 222-232, Montreal, Canada, June 1993.

....as possible worlds (see the completion construction in [9] for N Prolog and [4] for Hypothetical Datalog) 3. By embracing the idea of partiality in inductive de nitions and using the rule of de nitional re ection to incorporate a proof theoretical notion of closure analogous to the completion [22, 14]. None of those approaches are satisfactory for our purposes: most of the predicates we want to negate are open ended; similarly, de nitional re ection is not well behaved, since the version of the latter rule does not t well with the operational semantics required by that very class of ....

P. Schroeder-Heister. Rules of denitional re ection. In M. Vardi, editor, Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science, pages 222-232, Montreal, Canada, June 1993.

....[SP98] It supports higher order encodings, but it does not allow recursion to traverse binders. Consequently, M 2 s expressiveness is limited and those functions that motivate this work are simply not expressible. The calculi of partial inductive de nitions [Hal87] and de nitional re ection [SH93] are purely logical systems. Even when interpreted functionally, their proofs require all arguments to be closed and recursion must not traverse binders. Figure 1 can also be encoded in a logical framework implemented in the meta logic FO IN [MM97] FO IN , however, is a sequent ....

Peter Schroeder-Heister. Rules of denitional re ection. In M. Vardi, editor, Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science, pages 222-232, Montreal, Canada, June 1993.

....only alternative to axioms. Another one is to add non logical deduction rules to predicate logic either taking an introduction and elimination rule for the abstraction symbol in various formulations of set theory [15, 2, 10, 1, 3, 9] or interpreting logic programs or de nitions as deduction rules [11, 16, 17, 13] or in a more general setting [14] Non logical deduction rules and computation rules have some similarities, but we believe that computation rules have some advantages. For instance, non logical deduction rules may blur the notion of cut in natural deduction and extra proof reduction rules have ....

P. Schroeder-Heister. Rules of denitional re ection. Logic in Computer Science, 1993, pp. 222-232.

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