Regular realizability problems and models of a generalized nondeterminism

Models of a generalized nondeterminism are defined by limitations on nonde- terministic behavior of a computing device. A regular realizability problem is a problem of verifying existence of a special sort word in a regular language. These notions are closely connected. In this paper we consider regular realizability problems for languages consist- ing of all prefixes of an infinite word. These problems are related to the automata on infinite words and to the decidability of monadic second-order theories. The main contribution is a new decidability condition for regular realizability problems and for monadic-second order theories. We also show that decidability of a regular realizability problem is equivalent to decidability of some prefix realizability problem.

💡 Research Summary

The paper investigates regular realizability problems that arise from generalized nondeterminism models (GNA). In a GNA, a multi‑head two‑way automaton is equipped with an auxiliary “guess” memory; the authors focus on the extreme case where there is exactly one possible guess. Consequently, the filter language L consists of all prefixes of a single infinite word W, i.e., L = Pref(W). The regular realizability problem RR(L) then becomes the “prefix realizability problem” Rₚ(W): given a regular language R (presented by a DFA, NFA or regular expression), decide whether R ∩ Pref(W) ≠ ∅. Variants that use factors instead of prefixes and that ask for infinitely many intersections lead to factor realizability (R_f(W)) and Büchi realizability (R∞ₚ(W), R∞_f(W)).

The authors first establish a hierarchy of m‑reductions: R_f(W) ≤_m Rₚ(W) ≤_m R∞ₚ(W) and R∞_f(W) ≤_m R∞ₚ(W). They then connect Büchi realizability to monadic second‑order (MSO) logic: a nondeterministic Büchi automaton A can be effectively translated into an MSO formula φ such that L∞(A) = L∞(φ), and vice‑versa (Büchi’s theorem). Hence, deciding whether a Büchi automaton accepts infinitely many prefixes of W is equivalent to the decidability of the MSO theory MT(ℕ,<,W). Known results (Semenov’s almost‑periodic words, Carton‑Thomas morphic words) therefore yield families of W for which both MSO and prefix realizability are decidable.

The central contribution is a new, simple decidability condition: if every finite word over the alphabet Σ occurs as a factor of the infinite word W, then both the prefix realizability problem Rₚ(W) and the Büchi realizability problem R∞ₚ(W) are decidable. The algorithm simply simulates the input DFA on the computable word W until it reaches either an accepting state or a dead‑lock state. Because W contains a “definitive” factor w_A (a word that forces any automaton to either accept or dead‑lock), the simulation always halts, yielding a correct answer.

To support this condition, the paper introduces the notion of a definitive word for an automaton A: a word w_A such that, starting from any state, reading w_A inevitably leads to an accepting state or a dead‑lock. The authors prove that every finite automaton possesses such a word and that the set D(A) of all definitive words is a non‑empty regular language. They give an explicit construction: for each state q_i, compute a word u_i that reaches an accepting state if possible (or ε otherwise), then concatenate these u_i’s in a specific order to obtain w_n, which is definitive. Consequently, D(A) = ⋃_{q∈Q} L_q Σ* where L_q is the language recognized by A when started from state q with accepting states augmented by dead‑lock states, confirming regularity.

Using definitive words, the decision procedure for Rₚ(W) works as follows: given DFA A, compute D(A); because the global richness condition guarantees that some word from D(A) appears as a factor of W, the simulation of A on W will encounter that factor and thus terminate with a definitive answer. The same reasoning, together with a reduction that swaps accepting and dead‑lock states, yields decidability of R∞ₚ(W) under the same richness condition.

The paper also demonstrates that the converse does not hold: there exists a word W that is prefix‑decidable but not Büchi‑decidable. This construction encodes the halting behavior of all Turing machines into a binary word built from “blocks” b_m = 1 0^m. By carefully interleaving blocks whose ranks correspond to machines that have not halted within a given number of steps, the authors ensure that every finite word appears as a factor (hence prefix‑decidable) while the set of blocks that appear infinitely often encodes the non‑halting problem, making the Büchi realizability problem undecidable.

Finally, the authors summarize the implications: the richness condition unifies several previously known decidable classes (almost‑periodic, morphic) and provides a practical algorithmic framework based on definitive words. They suggest future work on relaxing the condition (e.g., allowing infinite alphabets or partial richness) and on analyzing the computational complexity of the decision procedures.

In summary, the paper establishes a clear and elegant criterion—global factor richness of the infinite word—that guarantees decidability of both regular prefix realizability and Büchi realizability problems, links these problems tightly to MSO theory, introduces definitive words as a constructive tool, and clarifies the precise relationship between different realizability notions.