Operational State Complexity and Decidability of Jumping Finite Automata

Abstract

We consider jumping finite automata and their operational state complexity and decidability status. Roughly speaking, a jumping automaton is a finite automaton with a non-continuous input. This device has nice relations to semilinear sets and thus to Parikh images of regular sets, which will be exhaustively used in our proofs. In particular, we prove upper bounds on the intersection and complementation. The latter result on the complementation upper bound answers an open problem from [G. J. Lavado, G. Pighizzini, S. Seki: Operational State Complexity of Parikh Equivalence, 2014]. Moreover, we correct an erroneous result on the inverse homomorphism closure. Finally, we also consider the decidability status of standard problems as regularity, disjointness, universality, inclusion, etc. for jumping finite automata.

Properties of right one-way jumping finite automata

“…The non-shaded closure properties for REG are folklore. For ROWJ the closure/non-closure results can be found in [3] and that for the language family JFA in [1,6,7,12].…”

“…Roughly speaking, a jumping finite automaton is an ordinary finite automaton, which is allowed to read letters from anywhere in the input string, not necessarily only from the left of the remaining input. In a series of papers [1,6,7,14] different aspects of jumping finite automata were investigated, such as, e.g., inclusion relations, closure and non-closure results, decision problems, computational complexity of jumping finite automata problems, etc. Shortly after the introduction of jumping automata a variant of this machine model was defined, namely (right) one-way jumping finite automata [3].…”

Generalized Circular One-Way Jumping Finite Automata

“…The automata can jump in either direction to delete the input. The model was extensively studied in [16,17,4,8]. Following [14], several jumping transition models have been introduced and studied in the last few years [5,15,10,2,6,9,13,11,7].…”

Generalized Linear One-Way Jumping Finite Automata

“…The author showed that universality, equivalence and inclusion are undecidable for GJFA, in [15]. Some decision problems of the model jumping finite automata (JFA), which is a restricted version of GJFA, had been discussed in [4]. Results related to the complexity of the model JFA were discussed in [7] and [4].…”

“…Some decision problems of the model jumping finite automata (JFA), which is a restricted version of GJFA, had been discussed in [4]. Results related to the complexity of the model JFA were discussed in [7] and [4]. Following [13], several other jumping transition models have been defined and studied in [9,5,2,6,8,12,10].…”