Weakly-Unambiguous Parikh Automata

• Weakly-Unambiguous Parikh Automata are defined by a restricted form of nondeterminism, ensuring each accepted word has a uniquely determined Parikh image.
• They integrate Parikh's theorem with reversal-bounded counter models to enable precise analysis of store languages and verification properties.
• Their design facilitates decidable model checking and robust language verification in systems with controlled memory and reversal limitations.

A two-way reversal-bounded counter machine (2W-CM) is a finite automaton equipped with a finite set of non-negative counters, operating on a two-way read-only input tape, in which each counter is constrained to a fixed number of alternations (“reversals”) between increments and decrements along any accepting computation. This computational model has been intensively studied due to its intricate interplay between finite automata, semilinear storage, and unbounded yet controlled memory. Store languages of such machines, namely the set of intermediate store configurations that occur during accepting computations, play a central role in verification, decidability, and algebraic properties of formal models.

1. Formal Definition and Variants

Let $k,r \ge 0$. A two-way nondeterministic $r$-reversal-bounded $k$-counter machine is defined by the tuple

$M = (Q, \Sigma, \delta, q_0, F)$

where:

• $Q$ is a finite set of states,
• $\Sigma$ is the input alphabet,
• $q_0 \in Q$ is the initial state,
• $F \subseteq Q$ is the set of final states,
• $$\delta \subseteq Q \times (\Sigma \cup \{\lhd_{\mathit{left}},\lhd_{\mathit{right}}\}) \times \{0,1\}^k \to \mathcal{P}(Q \times \{-1,0,+1\}^k \times \{L,S,R\})$$ encodes the transitions, with $\lhd_{\mathit{left}}, \lhd_{\mathit{right}}$ as end-markers.

A configuration is a tuple $(q,i,c_1,\ldots,c_k)$ recording the current state, input head position %%%%10%%%%, and counter contents $c_1,\ldots,c_k \in \mathbb{N}_0$. At each step, the transition function specifies—subject to the zero-tests for each counter—how the state, counters, and input head are updated, with the requirement that counters remain non-negative.

A machine is called $r$-reversal-bounded if, on every accepting computation, each counter switches between non-increasing and non-decreasing mode at most $r$ times. The family of all such two-way nondeterministic reversal-bounded counter machines is denoted 2NCM; its deterministic subclass is 2DCM. In some contexts, a bound $t$ is imposed on the number of input head turns (“$t$-turn machines”), leading to families such as $2\mathit{DRCM}(k,r,t)$ for deterministic variants.

A one-way version restricts the input head to move only to the right. Notably, unrestricted two-way counter machines are strictly more potent but vastly less analyzable.

2. Store Configurations and Store Languages

Given a 2W-CM $M$, the store configuration at any point of a computation encodes the current state, input head position, and all counter values, abstracting away from the rest of the input content. Stores are typically represented as strings from an alphabet encoding state, head position, and counters, e.g., $q\,\downarrow_i\,C_1^{c_1} \ldots C_k^{c_k}$.

The store language $S(M)$ is defined as the set of all store configurations that appear at some point during some accepting computation of $M$ on some input. Formally,

$$S(M) = \{ (q, i, c_1,\ldots,c_k) \mid \exists x, \exists \text{ accepting run on } x, \text{ configuration } (q, i, c_1,\ldots,c_k) \text{ appears} \}$$

This concept decouples the observable memory evolution of the machine from specific input insertions, serving as a powerful tool both for complexity analysis and model checking tasks (Friesen et al., 2 Sep 2025).

3. Decidability, Recursion, and Separation by Counter Bounds

A pronounced difference emerges between models with unrestricted counters and those with reversal bounds:

• There exists a deterministic two-way one-counter machine with unrestricted counting whose store language is not recursive. Constructing such a machine can be reduced to simulating a single-tape DTM accepting a non-recursive unary language, encoding acceptance into the counter store [(Friesen et al., 2 Sep 2025), Proposition 7].
• However, when the counter is reversal-bounded, every deterministic two-way one-counter machine has a recursive store language, with decidable membership. For such $M$, given a store $y$, one constructs a reversal-bounded machine that accepts precisely those inputs on whose run $y$ appears and $M$ accepts—thus reducing to the emptiness problem, which is decidable for reversal-bounded one-counter automata (Ibarra–Jiang) [(Friesen et al., 2 Sep 2025), Proposition 10].
• For two or more reversal-bounded counters, the situation reverses: emptiness is undecidable for two-way reversal-bounded two-counter machines, making membership in the corresponding store languages undecidable [(Friesen et al., 2 Sep 2025), Proposition 8].

This delineates a precise boundary in expressive and decision-theoretic power: a single reversal-bounded counter keeps store properties analyzable, while multiple reversal-bounded counters make core verification questions undecidable.

4. Closure Properties and Effective Characterization

When store languages $S(M)$ of 2W-CMs fall within the well-understood class of deterministic one-way reversal-bounded counter machine languages (DCM), strong closure properties are inherited. A key result is:

• If $M$ is a finite-crossing ($t$-turn) two-way $r$-reversal-bounded $k$-counter machine, then $S(M) \in L(\mathrm{DCM})$. This yields closure under union, intersection, complementation, concatenation, Kleene star, homomorphism, inverse homomorphism, and intersection with regular languages (Friesen et al., 2 Sep 2025).

The finite-crossing condition (bounded number of left/right head switches) allows “crossing-sequence” constructions: any finite-crossing 2W-CM can be simulated by a one-way reversal-bounded counter machine on its store evolution, which in turn can be determinized without increasing reversal-bound on counters.

Concretely, the inclusion chain is

$$S(\mathrm{2\text{-}way\ finite\text{-}crossing\ NCM}) \subseteq S(\mathrm{1\text{-}way\ NCM}) \subseteq L(\mathrm{DCM})$$

5. Relations to One-Way Counter Machines and Reductions

The analysis of 2W-CMs leverages reductions to one-way reversal-bounded models:

• Given a finite-crossing two-way reversal-bounded counter machine $M$, there exists a one-way reversal-bounded counter machine $M_1$ and a letter-erasing homomorphism $h$ such that $S(M) = h(L(M_1))$ (Friesen et al., 2 Sep 2025).
• The construction proceeds by guessing a crossing sequence (ordered list of states and counter values at particular head crossings), using one-way counter machines to record these and then mapping back to store configurations.

Through such reductions, deep results about one-way models—e.g., closure under inverse deterministic counter transducers [(Eremondi et al., 2019), Theorem 3.6]—can be leveraged. For instance, the store language of any one-way NCM is a DCM language.

6. Non-Closure and Limits of the Model

Despite strong closure under classical operations, certain “insertion” and inverse image operations elude the 2W-CM class:

Operation	Closure Property	Reference
Inverse infix, suffix,	Not closed for $2\mathit{DRCM}(1,1,1)$	(Eremondi et al., 2019)
outfix, $m$-embedding
Concatenation with regular	Not closed for some $2\mathit{DRCM}(1,1,1)$	(Eremondi et al., 2019)
Inverse deterministic	Closed; remains in some $2\mathit{DRCM}$	(Eremondi et al., 2019)
counter transducer

Non-closure under inverse infix and other related ideals reflects the sensitivity of the class to language insertions, contrasting with regular and context-free languages.

7. Summary and Impact

Two-way reversal-bounded counter machines mark the interface between automata theory and decidable fragments of infinite-state verification:

• The store language framework supplies a uniform vocabulary for comparing memory processes of various automata models and for relating model-based verification to algebraic language theory.
• Decidability and closure properties strongly depend on the number of counters and reversal bounds, successively separating strictly analyzable models (e.g., $k=1$ or one-way) from intractable regimes ($k \ge 2$, two-way).
• Techniques to harness crossing sequences and reductions to one-way models demonstrate transfer of semilinear and effective descriptions, while the limitation results clarify boundaries for the applicability of these automata in language-theoretic and verification settings.

For formal systems with store constraints, two-way reversal-bounded counter machines offer a precisely delimited, robust framework, underlying several advances in verification and automata theory (Friesen et al., 2 Sep 2025, Eremondi et al., 2019).