3 research outputs found
How unprovable is Rabin's decidability theorem?
We study the strength of set-theoretic axioms needed to prove Rabin's theorem
on the decidability of the MSO theory of the infinite binary tree. We first
show that the complementation theorem for tree automata, which forms the
technical core of typical proofs of Rabin's theorem, is equivalent over the
moderately strong second-order arithmetic theory to a
determinacy principle implied by the positional determinacy of all parity games
and implying the determinacy of all Gale-Stewart games given by boolean
combinations of sets. It follows that complementation for
tree automata is provable from - but not -comprehension.
We then use results due to MedSalem-Tanaka, M\"ollerfeld and
Heinatsch-M\"ollerfeld to prove that over -comprehension, the
complementation theorem for tree automata, decidability of the MSO theory of
the infinite binary tree, positional determinacy of parity games and
determinacy of Gale-Stewart games are all
equivalent. Moreover, these statements are equivalent to the
-reflection principle for -comprehension. It follows in
particular that Rabin's decidability theorem is not provable in
-comprehension.Comment: 21 page
Computability and Algorithmic Complexity in Economics
This is an outline of the origins and development of the way computability theory and algorithmic complexity theory were incorporated into economic and finance theories. We try to place, in the context of the development of computable economics, some of the classics of the subject as well as those that have, from time to time, been credited with having contributed to the advancement of the field. Speculative thoughts on where the frontiers of computable economics are, and how to move towards them, conclude the paper. In a precise sense - both historically and analytically - it would not be an exaggeration to claim that both the origins of computable economics and its frontiers are defined by two classics, both by Banach and Mazur: that one page masterpiece by Banach and Mazur ([5]), built on the foundations of Turing’s own classic, and the unpublished Mazur conjecture of 1928, and its unpublished proof by Banach ([38], ch. 6 & [68], ch. 1, #6). For the undisputed original classic of computable economics is RabinÃs effectivization of the Gale-Stewart game ([42];[16]); the frontiers, as I see them, are defined by recursive analysis and constructive mathematics, underpinning computability over the computable and constructive reals and providing computable foundations for the economist’s Marshallian penchant for curve-sketching ([9]; [19]; and, in general, the contents of Theoretical Computer Science, Vol. 219, Issue 1-2). The former work has its roots in the Banach-Mazur game (cf. [38], especially p.30), at least in one reading of it; the latter in ([5]), as well as other, earlier, contributions, not least by Brouwer.