Regular realizability problems and context-free languages

We investigate regular realizability (RR) problems, which are the problems of verifying whether intersection of a regular language -- the input of the problem -- and fixed language called filter is non-empty. In this paper we focus on the case of context-free filters. Algorithmic complexity of the RR problem is a very coarse measure of context-free languages complexity. This characteristic is compatible with rational dominance. We present examples of P-complete RR problems as well as examples of RR problems in the class NL. Also we discuss RR problems with context-free filters that might have intermediate complexity. Possible candidates are the languages with polynomially bounded rational indices.Comment: conference DCFS 201

On the Decidability of Finding a Positive ILP-Instance in a Regular Set of ILP-Instances

International audienceThe regular intersection emptiness problem for a decision problem P ($$int_{\mathrm {Reg}}$$(P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since $$int_{\mathrm {Reg}}$$(P) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the $$int_{\mathrm {Reg}}$$-problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the $$int_{\mathrm {Reg}}$$-problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of $$int_{\mathrm {Reg}}$$(ILP)