Σ1-) formula of L. Thus the notions of Δ0- and Σ1-set may be regarded as generalizations of the notions of recursive and recursively enumerable set, respectively.
We have above noted the important fact that in all arithmetical theories F containing Q, a set is strongly representable in F if and only if it is recursive, and a set is recursively enumerable if and only if it is weakly representable.
A familiar example is the method of diagonalization, as employed in the proof of the undecidability of the classical Halting Problem (from which it follows that the recursive languages are properly included in the recursively enumerable ones).
There are only countably many recursive functions, so there are only countably many recursively enumerable sets of basic sets.
A set is recursively enumerable iff it is the range of a (total or partial) recursive function on the natural numbers (Boolos et al. 2003: §8.3).
Quite independently of the particular formal system chosen, exactly the decidable, or recursive, sets (relations) are strongly representable, and exactly the semi-decidable, or recursively enumerable sets (relations) are weakly representable.
But the process was slow and laborious.Wizard spellingGoogle applies this principle of recursively learning from the data to many of its services, including the humble spell-check, for which it used a pioneering method that produced perhaps the world's best spell-checker in almost every language.
This restricted definition serves as the base-clause for truth-conditional recursion-clauses given at the second stage, at which the truth-values of non-elementary, or molecular, truthbearers are explained recursively in terms of their logical structure and the truth-values of their simpler constituents.
all continuous t-norms are not recursively axiomatizable (see Montagna 2001 for the final result).
These clauses can be applied recursively to define transformations of sets of ordered tuples in \(D\) (the extensions of two-place predicates), sets of sets of objects in \(D\) (the extensions of unary first-order quantifiers), and so on.
Semantics provides a complete account of sentence meaning for the language, [by] recursively specifying the truth conditions of the sentence of the language.
