automata theory designers

Abstract: Counter machines are a particularly simple, Turing-complete model of computation. We study the expressive power of nondeterministic 2-counter machines which are symmetric, in that their set of update transitions is invariant under swapping of counters. Our model is an extension of symmetric 2-VASS, in that we allow modifying both counters within a single transition. As the main result we demonstrate a dramatic impact of symmetry: we show that symmetric 2-counter machines are not Turing-complete, by proving decidability of their reachability problem. We conjecture that the decidability extends to machines with 3 or more counters that are symmetric, meaning the set of update transitions is invariant under any permutation of counters. We leave this problem open for machines with 3 and more counters, while identifying a frontier of the impact of symmetry: we prove undecidability for 3-counter machines invariant under the cyclic permutation group of order 3.

Abstract: We investigate the computational complexity of analyzing the structural and behavioral properties of deterministic k-pebble automata, which represent a natural framework for studying minimal programmable machines. First, we provide an explicit construction of a three-pebble automaton U capable of simulating any deterministic finite automaton (DFA) on a given input word, establishing a link between pebble automata capabilities, Kolmogorov complexity, and automatic complexity. Next, we restrict our architectural framework to two-pebble programmable machines U(2,N) simulating DFAs with at most N states. We analyze the decision problem ANAL(U(2,5)), which asks whether a five-state program can separate two distinct finite words. By establishing a polynomial-time reduction from the quasi-identity checking problem for finite semigroups, we prove that ANAL(U(2,5)) is NP-complete. Finally, we explore the complexity of separating binary strings under the Exponential Time Hypothesis (ETH), showing how computational hardness implies polynomial lower bounds for the size of the smallest DFAs separating binary strings.

Abstract: Graphoid automata evaluate graphs by composing relations between words of states. We examine the equations that make this evaluation independent of the chosen graph expression, and what changes when relations are replaced by matrices over a commutative semiring. We prove that the classical graphoid equations are equivalent to the special commutative Frobenius equations together with the required symmetry laws. The known cospan and hypergraph presentations then describe equality of structural and labelled expressions, with the bone law deleting precisely the isolated internal vertices. We use established partition representations to compare this equality with equality of evaluated matrices. For any nonempty finite state set $Q$ and commutative semiring $K$, we give a uniform criterion for two structural expressions to have the same matrix. The criterion determines which isolated-vertex counts are distinguished and when a vanishing scalar makes matrices from different boundary partitions equal. For the canonical structure, the bone law holds precisely when $|Q|\cdot1_K=1_K$. Preserving all relational compositions is a separate requirement, equivalent to additive idempotency for $|Q|\geq2$. We then examine the characteristic matrices of the known relational structures on $Q$ given by disjoint unions of Abelian groups. Their Frobenius equations hold over every commutative semiring, whereas specialness depends on the order of each group and the bone law depends on the number of groups. These conditions specify which equations remain valid when weights are assigned to a graphoid automaton.