We give an algorithm AL,T which assigns probabilities to logical sentences. For any simple infinite sequence {φs n } of sentences whose truthvalues appear indistinguishable from a biased coin that outputs "true" with probability p, we have limn→∞ AL,T (sn) = p.

We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s -rank.

This paper defines boolean monadic recursive schemes (BMRSs), a restriction on recursive programs, and shows that when interpreted as transductions on strings they describe exactly the subsequential functions. We discuss how this new result furthers the study of the connections between logic, formal languages and functions, and automata.

There are several ways to understand computability over first-order structures. We may admit functions given by arbitrary recursive definitions, or we may restrict ourselves to "iterative," or tail recursive, functions computable by nothing more complicated than while loops. It is well known that in the classical case of recursion theory over the natural numbers, these two notions of computability coincide (though this is not true for all structures). We ask if there are structures over which recursion and tail recursion coincide in terms of computability, but in which a general recursive program may compute a certain function more efficiently than any tail recursion, according to some natural measure of complexity. We give a positive answer to this question, thus answering an open question in Lynch and Blum (Math. Syst. Theory. 12(1), 205-211 1979).

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