Homing Vector Automata

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 of the deterministic, nondeterministic and blind versions of these real-time 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.

💡 Research Summary

This paper introduces a novel automaton model called the “Homing Vector Automaton” (HVA), which augments a classical finite automaton with a vector as an external storage device. The core operation of an HVA is matrix multiplication: at each step, the current vector is multiplied on the right by a rational-valued matrix determined by the current state, the input symbol, and (in the non-blind version) a check of whether the vector is currently equal to its initial value. The unique acceptance condition requires that the computation ends in an accept state and the vector has returned precisely to its initial value. The authors investigate the computational power of the deterministic (DHVA), nondeterministic (NHVA), blind deterministic (DBHVA), and blind nondeterministic (NBHVA) variants, where “blind” means the vector can only be checked for equality to the initial vector at the very end of the computation.

The analysis reveals a rich hierarchy among these models. A key result (Theorem 1) establishes that the ability to check the vector during computation (non-blindness) grants additional power: there exists a language recognizable by a DHVA(2) that no DBHVA of any dimension can recognize. However, for languages over a unary (single-letter) alphabet, the power of deterministic HVAs is limited to the regular languages (Theorem 2), as the finite state control and periodic behavior of the vector under matrix multiplication ultimately lead to a regular cycle of configurations.

Nondeterminism proves to be a significant source of power. The nondeterministic versions properly contain their deterministic counterparts (Theorem 3). Notably, a blind nondeterministic HVA of dimension 5 (NBHVA(5)) can recognize a reversed version of the NP-complete SUBSETSUM language (Theorem 4). This demonstrates that nondeterministic HVAs can solve problems of substantial computational complexity, leveraging the vector space to encode numbers and nondeterministically select subsets for summation checks.

Throughout the paper, the authors often restrict matrix entries to the set {-1, 0, 1}, showing that even this limited set allows for simulating operations like addition, subtraction, swaps, and resets between vector entries, drawing a connection to real-time multicounter automata. The computational power is shown to be sensitive to both the dimension of the vector (k) and the allowed set of matrix entries.

In addition to the automata theory results, the paper develops a generalized Stern-Brocot encoding method. This technique provides a way to uniquely encode strings over an arbitrary finite alphabet into a single rational number. This method is particularly useful for blind HVAs, as it allows the entire input string to be compressed into a vector entry and verified through a single equality check at the end.

In conclusion, Homing Vector Automata offer a fresh framework for studying computation via linear transformations (matrix multiplications) on a vector state, with a “homing” acceptance condition inspired by notions of state preservation. The model exhibits a non-trivial hierarchy based on determinism, blindness, and dimension, and can recognize languages of high complexity, including NP-complete problems, in its nondeterministic form. The work opens several avenues for future research, including the precise relationship between HVAs and other matrix-based models like quantum automata, and the effect of expanding the allowed set of matrix entries.