Homing vector automata

Abstract

We introduce homing vector automata, which are finite automata augmented by a vector that is multiplied at each step by a matrix determined by the current transition, and have to return the vector to its original setting in order to accept the input. The computational power and properties of deterministic, nondeterministic, blind, non-blind, real-time and one-way versions of these machines are examined and compared to various related types of automata. A generalized version of the Stern-Brocot encoding method, suitable for representing strings on arbitrary alphabets, is also developed.

New Results on Vector and Homing Vector Automata

“…A (real-time) k-dimensional deterministic homing vector automaton (DHVA(k)) is defined in [20] as being different from vector automata in two ways: (1) Homing vector automata do not read the right end-marker after reading the input, so there is no chance of postprocessing and, (2) instead of checking the status of the first entry, a homing vector automaton checks whether the complete current vector is identical to the initial vector or not. Formally, the transition function δ is defined as…”

“…In [21] it is shown that the non-context-free language POW r = {a 2 n b n |n ≥ 0} can be recognized by a DBHVA(2) Z 2×2 . It is left open whether there exists a 1NBHVA with integer entries recognizing POW r .…”

“…In many of the models which extend finite automata, a computation is successful if the register is equal to its initial value at the end of the computation. Applying this idea to vector automata, we have introduced the homing vector automaton (HVA) [20], which can check its vector for equivalence to its initial value and accepts a string if the computation ends with the initial vector in an accept state. We have shown that there exist close connections between homing vector automata [21] and extended finite automata [4].…”

“…Applying this idea to vector automata, we have introduced the homing vector automaton (HVA) [20], which can check its vector for equivalence to its initial value and accepts a string if the computation ends with the initial vector in an accept state. We have shown that there exist close connections between homing vector automata [21] and extended finite automata [4]. In fact, homing vector automata can be seen as a special case of rational monoid automata [17].…”

“…We prove that one-way three-dimensional nondeterministic blind homing vector automata (NBHVA) with integer entries can simulate any nondeterministic blind multicounter machine. We answer an open question from [21], by proving that one-way NBHVAs are more powerful than extended finite automata [4] when both are defined on the set of 2 × 2 integer matrices.…”