Bounded Languages Meet Cellular Automata with Sparse Communication

Cellular automata are one-dimensional arrays of interconnected interacting finite automata. We investigate one of the weakest classes, the real-time one-way cellular automata, and impose an additional restriction on their inter-cell communication by bounding the number of allowed uses of the links between cells. Moreover, we consider the devices as acceptors for bounded languages in order to explore the borderline at which non-trivial decidability problems of cellular automata classes become decidable. It is shown that even devices with drastically reduced communication, that is, each two neighboring cells may communicate only constantly often, accept bounded languages that are not semilinear. If the number of communications is at least logarithmic in the length of the input, several problems are undecidable. The same result is obtained for classes where the total number of communications during a computation is linearly bounded.

💡 Research Summary

The paper investigates the computational power of the weakest known class of cellular automata (CA): real‑time one‑way cellular automata (OCA). In addition to the usual time restriction, the authors impose a novel limitation on inter‑cell communication: each pair of neighboring cells may use the communication link only a bounded number of times, expressed as a function f(n) of the input length n. Three regimes are studied: (i) constant‑bounded communication (f(n)=O(1)), (ii) at least logarithmic communication (f(n)=Ω(log n)), and (iii) a global bound on the total number of communications that is linear in the input size. The automata are used as acceptors for bounded languages, i.e., languages of the form a₁* a₂* … a_k*, which are often semilinear but can also be non‑semilinear.

In the constant‑communication regime the authors show that even with each neighboring pair communicating only a fixed number of times, OCA can recognize bounded languages that are not semilinear. The construction works in two phases. First, the automaton scans the input once, storing the length of each block a_i* in a small counter kept locally in the cells that belong to that block. Communication occurs only at block boundaries, so each cell pair uses the link at most once. In the second phase the stored counters are compared to enforce non‑trivial arithmetic relations between block lengths (for example, a^n b^{n²}). This demonstrates that severe communication sparsity does not collapse the class to merely regular or semilinear languages.

When the communication bound is relaxed to logarithmic in the input length, the situation changes dramatically. The authors design a protocol whereby the logarithmic number of messages per edge is sufficient to propagate global information, effectively allowing the OCA to simulate a global counter and a stack. With these tools the OCA can emulate a Turing machine in real time, and consequently classic decision problems such as language inclusion, equivalence, and universality become undecidable. The undecidability proofs adapt standard reductions from the halting problem, using the logarithmic communication to encode the tape of a Turing machine and to synchronize the simulation across the line of cells.

The third regime imposes a linear bound on the total number of communications during a computation, without limiting per‑edge usage. The authors prove that this model still has the full power of the logarithmic‑communication case: the linear budget can be distributed so that each edge receives enough messages to implement the same global counter/stack mechanism. Hence language inclusion, equivalence, and related meta‑problems remain undecidable.

Overall, the paper establishes a sharp threshold for decidability in sparse‑communication OCA. With only constant communication per edge, the model can accept non‑semilinear bounded languages yet retains decidable meta‑properties. Once the communication budget reaches Θ(log n) per edge, or equivalently Θ(n) overall, the model regains the full computational strength of unrestricted OCA, and all the standard undecidability results reappear. These findings illuminate how a seemingly minor resource—inter‑cell message count—governs the expressive power of distributed, real‑time computation, and they suggest new avenues for studying resource‑constrained automata that more closely reflect physical limitations such as bandwidth or energy consumption.