Two decidability problems for infinite words

In this short note, we give the exact complexity of the infinite Post Correspondence Problem, showing that it is ${\mathrm{\Pi }}_1^0$-complete. Surprisingly, it turns out that the infinite Post Correspondence Problem is not “more complex” than the Post Correspondence Problem, which is known to be ${\mathrm{\Sigma }}_1^0$-complete, but has the exact dual complexity. This gives an answer to a question of Simonnet [15].

We study the determinization of transducers over finite and infinite words. The first part of the paper is devoted to finite words. We recall the characterization of subsequential functions due to Choffrut. We describe here a known algorithm to determinize a transducer.

In the case of infinite words, we consider transducers with all their states final. We give an effective characterization of sequential functions over infinite words. We describe an algorithm to determinize transducers over infinite words. This part contains the main novel results of the paper.

This paper presents a construction that computes the topological closure of a rational relation on infinite words. Besides showing that the closure is rational itself, it also reduces the question of the continuity and the uniform continuity of a function to the question of the functionality of its closure, which is known to be decidable in polynomial time.

We study relations over ω-words using a representation by tree languages. An ω-word over an alphabet with k letters is considered as a path through the k-ary tree, an n-tuple of ω-words as an n-tuple of paths (coded by an appropriate valuation of the k-ary tree using values in {0,1}ⁿ), and a relation over ω-words as a tree language. In the first part of the paper we give a logical characterization of the “Rabin-recognizable relations” (whose associated tree languages are recognized by Rabin tree automata) in terms of “weak chain logic”, a restriction of monadic second-order logic over trees. In the second part of the paper an extended logic is considered, obtained by adjoining the “equal-level predicate” over trees. We describe the class of relations over ω-words which (in the tree language representation) are definable in this logic, and show that the theory of the k-ary tree in this logic is decidable. It covers tree properties which are not expressible in the monadic second-order logic SkS.