Ambiguity and Communication

The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for an input of size n. For all k, r 2 N we construct languages Lr,k which can be recognized by NFA’s with size k poly(r) and ambiguity O(nk), but Lr,k has only NFA’s with exponential size, if ambiguity o(nk) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, 1989, Leung, 1998).

💡 Research Summary

The paper investigates the quantitative notion of ambiguity in nondeterministic finite automata (NFAs). Ambiguity is defined as the maximum number of accepting computation paths an NFA can have on inputs of length n, and it serves as a key measure of the nondeterministic power of the automaton. While earlier work distinguished between linear ambiguity (O(n)) and polynomial ambiguity (O(n^k) for fixed k ≥ 2), it remained open whether a strict hierarchy exists within polynomial ambiguity itself. This question was explicitly raised by Ravikumar and Ibarra (1989) and later by Leung (1998).

To resolve the issue, the authors construct a family of languages L_{r,k} parameterized by two natural numbers r and k. The construction has two crucial properties. First, there exists an NFA of size k·poly(r) that recognizes L_{r,k} with ambiguity bounded by O(n^k). Here poly(r) denotes a polynomial in r, and the exponent k directly controls the degree of polynomial ambiguity. Second, any NFA that recognizes the same language while maintaining ambiguity o(n^k) must have at least 2^{Ω(r)} states, i.e., an exponential blow‑up in size. Consequently, reducing ambiguity even slightly below the O(n^k) threshold forces the automaton to become exponentially larger.

The technical core of the proof is a novel reduction from communication‑complexity lower bounds to NFA size lower bounds. The authors begin with a basic “distance‑verification” problem and amplify it using a direct‑product construction that creates k independent verification blocks within each input. Each block corresponds to a sub‑instance of a well‑studied set‑disjointness problem, whose randomized communication complexity is known to be linear in the input size. By embedding these sub‑instances into the NFA’s transition structure, they show that any NFA with ambiguity o(n^k) would have to simulate a protocol that communicates less than the known lower bound, which is impossible. Therefore, the NFA must contain exponentially many distinct states to encode the necessary information, establishing the size lower bound.

The main results can be summarized as follows: (1) For every fixed k and sufficiently large r, L_{r,k} is recognized by an NFA of size k·poly(r) with ambiguity O(n^k). (2) Any NFA recognizing L_{r,k} with ambiguity o(n^k) requires at least 2^{Ω(r)} states. (3) Hence, for each integer k there exists a language that separates the ambiguity class O(n^k) from O(n^{k‑1}), yielding a strict, infinite hierarchy of polynomial ambiguity levels. This hierarchy resolves the long‑standing open problem concerning the existence of infinitely many distinct polynomial‑ambiguity classes.

The implications are multifold. From a theoretical standpoint, the work clarifies the trade‑off between nondeterministic branching (ambiguity) and state complexity, providing a precise quantitative relationship. Practically, it suggests that designers of NFAs (e.g., in pattern matching, lexical analysis, or hardware verification) can deliberately allow a controlled amount of ambiguity to keep the automaton compact, but must accept a polynomial blow‑up in the number of accepting paths. Moreover, the communication‑complexity technique introduced here is likely adaptable to other nondeterministic models such as pushdown automata, tree automata, or even quantum finite automata, opening avenues for future research. By finally settling the question posed three decades ago, the paper marks a significant milestone in automata theory and complexity theory.