Abstract: The notions of n-ary pseudo weighted finite automata (i.e., pseudo weighted finite auto- mata with n nonempty finite sets of symbols), crisp n-ary pseudo weighted finite automata, and deterministic n-ary pseudo weighted finite automata are introduced. According to the transition function with empty moves or not on each finite set of symbols, the above automata are classified into four forms: n-ary pseudo weighted finite automata with r-type empty moves, n-ary pseudo weighted finite automata with empty moves, crisp n-ary pseudo weighted finite automata with r-type empty moves, and crisp n-ary pseudo weighted finite automata with empty moves. The languages accepted by the above automata and the relationships among them are studied, and the influence of the transition function with empty moves or not on each finite set of symbols is discussed.

Key words: pseudo semiring, n-ary pseudo weighted finite automata, transition function, empty move, relationship