Download Computational Logic and Proof Theory: 5th Kurt Gödel by Leo Bachmair (auth.), Georg Gottlob, Alexander Leitsch, PDF

By Leo Bachmair (auth.), Georg Gottlob, Alexander Leitsch, Daniele Mundici (eds.)

This booklet constitutes the refereed court cases of the fifth Kurt Gödel Colloquium on Computational good judgment and evidence idea, KGC '97, held in Vienna, Austria, in August 1997.
The quantity offers 20 revised complete papers chosen from 38 submitted papers. additionally incorporated are seven invited contributions through top specialists within the quarter. The e-book records interdisciplinary paintings performed within the quarter of desktop technological know-how and mathematical logics via combining study on provability, research of proofs, facts seek, and complexity.

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Extra resources for Computational Logic and Proof Theory: 5th Kurt Gödel Colloquium, KGC '97 Vienna, Austria, August 25–29, 1997 Proceedings

Example text

T h e o r e m 2. The tree isomorphism problem is in alternating logarithmic time. Proof. It will suffice to construct logarithmic depth, Alogtime-uniform, bounded fan-in circuits for the tree isomorphism problem. We present a construction of the circuit; but will omit the straightforward proof that the construction can be made Alogtime uniform. The circuit is essentially a direct implementation of the game described informally above. Fix a tree size n. We may assume that the inputs T1 and T2 both have the same size ITll = IT21 = n, since otherwise they axe not isomorphic.

That is to say~ if trees T1 and T2 are isomorphic under the mapping f , and if $1 is a subtree of T1 and $2 = f(S1) is its isomorphic copy in T2, then $1 and $2 have the same size-signatures in T1 and T2, respectively. By the usual parsing and counting techniques, there is an Alogtime procedure which, given a string representation of a tree T and given a subtree S of T, produces the size signature of S in T. Definition Let T1 ~ T2 be trees. Let S be a subtree of T1. We say that S distinguishes T1 from T2 provided that S is a proper subtree of T and: 23 (1) The logsize of S is strictly less than the logsize o] the parent tree of S, and (2) The number of subtrees of T1 which are similar to S is not equal to the number of subtrees of T2 which are similar to S.

Given a set Z of sentences of Loxo(~,), for every formula q~(x) of Loxo(~): E ~Vxcp(x) iff 1~ ~ Vxcp(x). [] This result corroborates the feeling that 'almost all' requires positive information, otherwise it reduces to 'all'. This observation will become clearer in the context of generic reasoning, to which we now turn. 4. Generic Reasoning We now wish to argue that our logic supports generic reasoning, as in the familiar Tweety example: from the facts that birds 'generally' fly and that Tweety is a 'generic' bird we can conclude that Tweety does fly.

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