An Algebraic Characterization of Unary Two-Way Transducers

Input- or output-unary sweeping transducers are weaker than their 2-way counterparts

“…Deterministic = [11] Functional Nondeterministic Input-unary Sweeping [6] 2-way Theorem 5.3 Output-unary 1-way [2,7] 2-way Theorem 4.7…”

“…Input-and output-unary Sweeping [7] Table 1. The most head-move restricted variant (among 1-way, sweeping and 2-way) equivalent to a 2-way transducer by cases.…”

“…The relations accepted by 2-way transducers include strictly the rational relations (see Example 2.6). We recall additional operations that we proved sucient to express relations accepted by 2-way transducers when both alphabets Σ and ∆ are unary [7]. The main result of this paper is that when ∆ is unary and Σ arbitrary these operations do not capture the whole family of relations recognized by 2-way transducers but those and only those relations recognized by sweeping transducers.…”

“…In the majority of the results cited above, the authors consider some subfamilies of transducers where, in some sense, the nondeterminism is restricted. In [7], we studied a dierent kind of restriction, considering the case where both input and output alphabets are unary, i.e., Σ and ∆ are both singletons. Restricting the alphabets to a single letter is usual in Automata Theory.…”

“…In [7], we introduced the Hadamard like operations on relations. As a byproduct it was shown that all unary 2-way transducers are equivalent to sweeping transducers: in other words it is no loss of generality to impose that the transducer changes direction on the endmarkers only.…”