DPA-Store: DPDA Store Languages

• DPA-Store is the collection of all state-stack pairs produced during a DPDA's accepting computation, encapsulating key structural insights.
• The regularity theorem proves that every DPDA store language is regular, meaning it can be recognized by finite automata like DFAs and NFAs.
• This concept underpins applications such as demonstrating the closure properties of deterministic context-free languages and delineating workspace thresholds in Turing machines.

A DPA-Store, in the context of formal language theory, refers to the store language of a deterministic pushdown automaton (DPDA). The store language comprises all store configurations—combinations of current control state and stack content—that can appear as intermediate stages during some accepting computation of the DPDA. Analysis of store languages illuminates both structural and closure properties in deterministically computable language families and interfaces crucially with regularity, automata classification, and complexity bounds (Ibarra et al., 2017).

1. Formal Model of a DPDA and Store Configurations

A deterministic pushdown automaton is formally represented as a 6-tuple:

$$A = (Q,\, \Sigma,\, \Gamma,\, \delta,\, q_0,\, Z_0,\, F)$$

where:

• $Q$ denotes a finite set of control states.
• $\Sigma$ is the finite input alphabet.
• $\Gamma$ is the stack alphabet.
• $Z_0 \in \Gamma$ is the unique initial stack symbol (bottom-of-stack marker).
• $q_0 \in Q$ is the initial state.
• $F \subseteq Q$ is the set of accepting states.
• $$\delta: Q \times (\Sigma \cup \{\epsilon\}) \times \Gamma \rightarrow Q \times \Gamma^*$$ is the partial transition function.

A configuration is $(q, w, \gamma)$ where $q \in Q$ (current state), $Q$0 (unread input), $Q$1 (stack contents, with the top at the right). The computation step is given by $Q$2 if $Q$3. Acceptance is defined by reachability of $Q$4 for some $Q$5 from $Q$6 (Ibarra et al., 2017).

2. Definition and Structure of Store Language

The store language $Q$7 of a DPDA $Q$8 consists of all state-stack pairs $Q$9 that occur in some accepting computation:

$\Sigma$0

Thus, $\Sigma$1 represents all store configurations $\Sigma$2 attainable “mid-run” during some successful computation, i.e., those from which acceptance is still possible on some suffix of the input (Ibarra et al., 2017).

3. Regularity Theorem for Store Languages

The foundational result, attributed originally to Greibach and systematically presented in standard automata texts (e.g., Hopcroft–Ullman), states that for any (possibly nondeterministic) one-way pushdown automaton, the store language is regular:

• Theorem (Greibach–Hopcroft–Ullman): For any one-way NPDA $\Sigma$3, $\Sigma$4 is regular; hence, for a DPDA, $\Sigma$5.

This regularity implies that no matter the complexity of the original pushdown computation, the abstraction to store languages collapses the possibilities into a set recognizable by a finite automaton (Ibarra et al., 2017).

4. Minimal Automata for Store Languages

Because store languages are regular for DPDAs (and, generally, for NPDAs), the weakest model capable of accepting $\Sigma$6 is a finite automaton. That is, both DFAs and NFAs suffice:

$\Sigma$7

This positions finite automata as sufficient for recognizing the set of all valid DPDA store configurations (Ibarra et al., 2017).

Device Type	Recognizes $\Sigma$8?	Relative Power
DFA or NFA	Yes	Weakest
DPDA	Yes	Stronger
TM (work tape)	Sometimes	Strictly Stronger

The table summarizes the recognition power for $\Sigma$9.

5. Application: Closure of DCFL under Right Quotient

A principal application of store language regularity concerns closure properties. Specifically, deterministic context-free languages (DCFL)—the languages recognized by DPDAs—are closed under right quotient with regular languages:

• Corollary: DCFL is closed under right quotient with regular languages.

The proof:

• Given DPDA $\Gamma$0 and regular language $\Gamma$1 (recognized by DFA $\Gamma$2), construct an auxiliary nondeterministic machine $\Gamma$3 simulating $\Gamma$4 and, at a guessed point, simulating $\Gamma$5 on the remaining input.
• The store language $\Gamma$6 is regular.
• Extract the set $\Gamma$7 of store words where the $\Gamma$8-simulation commences; $\Gamma$9 is regular.
• Construct a DPDA $Z_0 \in \Gamma$0 that simulates $Z_0 \in \Gamma$1 on $Z_0 \in \Gamma$2, and upon finishing, verifies (deterministically) that the stack contents are in $Z_0 \in \Gamma$3.
• $Z_0 \in \Gamma$4 accepts precisely those $Z_0 \in \Gamma$5 for which there exists $Z_0 \in \Gamma$6 with $Z_0 \in \Gamma$7 in $Z_0 \in \Gamma$8, showing $Z_0 \in \Gamma$9.

The same argument applies to models whose store languages are always regular in the nondeterministic case (pushdown, queue, reversal-bounded counter, etc.) (Ibarra et al., 2017).

6. Lower Bounds for Non-Regular Store Languages

While all pushdown stores induce regular store languages, using a Turing work tape instead (either in one- or two-way machines) allows for non-regular store languages, contingent on workspace:

• Two-way Turing machines (2DTM) with $q_0 \in Q$0 workspace only have regular store languages. There exists a 2DTM using $q_0 \in Q$1 workspace whose store language is not regular.
• One-way Turing machines (1DTM) with $q_0 \in Q$2 workspace only have regular store languages. There exists a 1DTM using $q_0 \in Q$3 workspace whose store language is not regular.

This establishes that to obtain a non-regular store language in the Turing machine setting, at least logarithmic (or doubly logarithmic for 2DTM) workspace is necessary. A plausible implication is that the computational power manifested in the store language directly reflects available workspace (Ibarra et al., 2017).

7. Historical Sources and Related Results

The study of store languages originates with Greibach’s 1965 analysis and is elaborated in standard texts such as Hopcroft and Ullman’s “Introduction to Automata Theory.” The results summarized above, along with generalizations to other automata models, are detailed in Ibarra & McQuillan’s work and their 2018 theoretical computer science publication. These results underpin a range of uniform proofs, closure properties, and complexity-theoretic dichotomies in automata theory (Ibarra et al., 2017).