Abstract
We present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of bounded-error rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unbounded-error model. When focusing only the rational values, we obtain the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.
Highlights
Finite automata are interesting models to study since they express a very natural limitation of finite memory
We present new results on the computational limitations of affine automata (AfAs)
We show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of Affine finite automaton (AfA) in the unbounded-error model
Summary
Finite automata are interesting models to study since they express a very natural limitation of finite memory. We show that using end-marker does not increase the computational power of affine automata with unbounded error or bounded error. We show that the computation of bounded-error rational-valued affine automata is simulated in logarithmic space, and so we answer positively one of the open problems in Dıaz-Caro and Yakaryılmaz (2016). We show that such generalization does not increase the computational power of AfAs with cutpoint language recognition. If we restricted these generalized AfAs to use only rational numbers, we obtain the same result for bounded error language recognition. We show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers or only integers.
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