Logic.
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Abstract:
Is it simpler to reason about finite structures or about arbitrary structures? Some of the major results of logic in the twentieth century provide a clear and surprising answer to one precise version of this question. Suppose first that we restrict our reasonings to propositions which are expressible in first order logic. We may then understand the question as asking for a comparison between the complexity of 1. determining whether a first order sentence is valid, that is, true under every interpretation whatsoever, and 2. determining whether a first order sentence is valid in the finite, that is, true under every interpretation with a finite universe of discourse. This question can be formulated more concisely and concretely in terms of Val, the set of valid sentences of L; the first order language with identity and a single binary relation symbol E; and Fval, the set of sentences of L which are valid in the finite, namely: is the decision problem for Fval simpler than the decision problem for Val?
Citations
| 68 | Reachability is harder for directed than for undirected finite graphs – Ajtai, Fagin - 1990 |
