In this paper, the Lagrangian formulation of variable-rate vector quantization is extended to quantization with simultaneous constraints on entropy and codebook size, including variable- and fixed-rate quantization as special cases. The formulation leads to a Lloyd quantizer design algorithm and generalizations of Gersho's approximations characterizing optimal performance for asymptotically large rate. A variation of Gersho's approach is shown to yield rigorous results partially characterizing the asymptotically optimal performance.

Results are developed for optimal quantization with combined entropy and log codebook size constraints, including high rate characterizations of distortion, entropy, codebook size, and quantizer point density functions.

Probabilistic Turing machines are Turing machines with the ability to flip coins in order to make random decisions. We allow probabilistic Turing machines small but nonzero error probability in computing number-theoretic functions. An example is given of a function computable more quickly by probabilistic Turing machines than by deterministic Turing machines. It is shown how probabilistic linear-bounded automata can simulate nondeterministic linear-bounded automata.

Abstract Subcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets with respect to the complete sets of other reducibilities. It is shown that q-cylindrification is an order-preserving map from the r.e. T-degrees to the r.e. S-degrees. Consequently, T-complete sets are precisely the r.e. sets whose q-cylindrifications are S-complete.