Deterministic Stack Transducers

We introduce and investigate stack transducers, which are one-way stack automata with an output tape. A one-way stack automaton is a classical pushdown automaton with the additional ability to move the stack head inside the stack without altering the contents. For stack transducers, we distinguish between a digging and a non-digging mode. In digging mode, the stack transducer can write on the output tape when its stack head is inside the stack, whereas in non-digging mode, the stack transducer is only allowed to emit symbols when its stack head is at the top of the stack. These stack transducers have a motivation from natural-language interface applications, as they capture long-distance dependencies in syntactic, semantic, and discourse structures. We study the computational capacity for deterministic digging and non-digging stack transducers, as well as for their non-erasing and checking versions. We finally show that even for the strongest variant of stack transducers the stack languages are regular.

HIERARCHIES OF GENERALIZED KOLMOGOROV COMPLEXITIES AND NONENUMERABLE UNIVERSAL MEASURES COMPUTABLE IN THE LIMIT

The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with one-way write-only output tape. This naturally leads to the universal enumerable Solomonoff-Levin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly infinite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briefly mention consequences for universes sampled from such priors.

A Census of Finite Automata

A finite automaton may be thought of as a possible abstraction of a digital computer. Imagine a tape or sequence of letters from some alphabet being fed into a device with a finite number of internal states. When the device is in a particular state and receives an input letter, the system passes to another internal state and prints a letter on an output tape. This operation is continued until the entire input sequence has been processed.Both Harary (6) and Ginsburg (4) have focused attention on the previously unsolved problem of counting the number of equivalence classes of finite automata. In the present paper, this problem is solved completely by proving a variety of theorems about the enumeration of functions.