(Extended Version) Algebraic Characterization of the Class of Languages recognized by Measure Only Quantum Automata

We study a model of one-way quantum automaton where only measurement operations are allowed (MOn-1qfa). We give an algebraic characterization of LMO(Σ), showing that the syntactic monoids of the languages in LMO(Σ) are exactly the J-trivial literally idempotent syntactic monoids, where J is the Green’s relation determined by two-sided ideals. We also prove that LMO(Σ) coincides with the literal variety of literally idempotent piecewise testable languages. This allows us to prove the existence of a polynomial-time algorithm for deciding whether a regular language belongs to LMO(Σ) and to discuss definability issues in terms of the existential first-order logic Σ1[<] and the linear temporal logic without the next operator LTLWN.