Store Language: Concepts & Applications

Updated 10 July 2026

Store language is a multifaceted concept representing state configurations in automata, mutable heaps in programming, transaction procedures, and tokenized memory in AI.
It transforms reachability and verification issues into language-theoretic problems, leveraging regularity and closure properties for model-checking.
Its diverse implementations—from streaming OLTP systems to semantic memory augmentation—demonstrate practical applications in both theory and industry.

Store language is a polysemous technical term. In automata theory, it denotes the set of store configurations—state plus store contents, but not the input—that can appear in an accepting computation. In programming language semantics, it denotes languages whose semantics includes mutable store, including higher-order references. In transaction processing, it denotes a stored-procedure programming model over streams, windows, and workflows. In recent AI systems, it denotes the use of natural language or semantic tokens as the medium in which experience, catalog structure, or store-specific multimodal knowledge is encoded and retrieved (Ibarra et al., 2017, Sterling et al., 2022, Meehan et al., 2015, Shen et al., 2023).

1. Automata-theoretic store languages

The most precise and historically stable meaning of store language comes from formal language theory. For a machine with finite control and one or more storage devices, a store configuration is the current state together with the contents of the stores, excluding the unread input. The store language $S(M)$ is the set of all such store configurations that occur along some accepting computation (Ibarra et al., 2017). For one-way nondeterministic Turing machines with counters, the paper on finite-visit Turing machines writes configurations as

$(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$

with store configuration

$q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$

and defines $S(M)$ by existential quantification over an accepted input and an accepting computation containing that intermediate store state (Friesen et al., 2 Sep 2025).

A central theorem in this literature is that the store language of every pushdown automaton is regular (Ibarra et al., 2017). The same regularity phenomenon extends well beyond the classical pushdown case. For one-way nondeterministic finite-visit Turing machines, and more generally finite-crossing machines, the store language is regular; for such machines augmented by reversal-bounded counters, the store language is accepted by a machine with only reversal-bounded counters and no worktape (Friesen et al., 2 Sep 2025). The 2020 synthesis on store languages likewise shows that store languages of one-way NPDA, $r$ -flip NPDA, stack automata, reversal-bounded queue automata, and reversal-bounded Turing machines are regular, while store languages of $\NCM$ lie in $\DCM$, and store languages of $\NPCM$ and $r$ -flip $\NPCM$ lie in $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 0 (Ibarra et al., 2020).

This formal notion is exacting because it abstracts away the input while retaining the machine’s operational memory. The result is a language over a configuration alphabet, not over the original input alphabet. That distinction underlies most of the later verification and closure arguments.

2. Reachability, verification, and limits of regularity

The store-language viewpoint is valuable because it converts reachability questions into language-theoretic questions. For a machine $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 1 and a set of configurations $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 2, the 2020 paper defines predecessor and successor sets via the store-transition relation and proves

$(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 3

where $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 4 is the initial store configuration and $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 5 is the regular set of final configurations (Ibarra et al., 2020). This identity ties store languages directly to forward and backward reachability.

From that observation, several model-checking consequences follow. For models whose store languages are regular, predecessor and successor sets of regular configuration sets are regular. For models whose store languages lie in $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 6, predecessor and successor sets of $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 7-definable configuration sets again lie in $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 8. The same paper uses this to decide common-configuration and infinite-common-configuration problems for classes such as $(q, w, x\,\dot{\ }\,y, z_1,\ldots,z_t),$ 9, $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 0-flip $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 1, $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 2, $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 3, and $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 4 (Ibarra et al., 2020).

The 2017 paper derives another consequence: if a deterministic family uses readable store types and the corresponding nondeterministic family has regular store languages, then the deterministic family is closed under right quotient with regular languages (Ibarra et al., 2017). This yields closure under right quotient for deterministic pushdown automata, deterministic $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 5-flip pushdown automata, deterministic stack and checking stack automata, deterministic reversal-bounded queue automata, and deterministic one-way Turing machines with reversal-bounded worktape (Ibarra et al., 2017).

Regularity is not universal. The 2025 paper distinguishes strong finite-crossing from weak finite-crossing. Under the weak notion, store languages can realize arbitrary recursively enumerable complexity; by contrast, under strong finite-visit or finite-crossing restrictions they remain regular (Friesen et al., 2 Sep 2025). The 2017 paper also shows that some deterministic pushdown-plus-counter machines do not accept their own store languages: there exists a $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 6-reversal-bounded $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 7 whose store language cannot be accepted by any $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 8 (Ibarra et al., 2017). These negative results delimit the scope of the regularity phenomenon.

3. Store as mutable state in programming language semantics

In programming language semantics, a store language is a language whose semantics includes a mutable heap. The 2022 denotational-semantics paper defines a store language as a programming language with general or higher-order references, meaning that reference types $q\,x\,\dot{\ }\,y\,C_1^{z_1}\cdots C_t^{z_t},$ 9 may store not only ground data but also values containing references, functions, polymorphic values, and dependent proofs (Sterling et al., 2022). The paper’s setting is an impredicative guarded dependent type theory, iGDTT, extended to iGDTTRef and then to iGDTTRefLRAT.

The technical difficulty is the mutual circularity of worlds, semantic types, and heaps. The paper solves guarded recursive domain equations internally, obtaining the world equation

$S(M)$ 0

so a world is a finite map from locations to delayed semantic types, and semantic types are presheaves over worlds (Sterling et al., 2022). The state effect is interpreted by a monad $S(M)$ 1 indexed by worlds and heaps, and references are interpreted by a connective $S(M)$ 2 selecting locations whose world entry is the delayed type $S(M)$ 3 (Sterling et al., 2022).

This semantics supports polymorphism, recursive types, and proof-relevant logical relations. The paper introduces LRAT, “logical relations as types,” with open and closed modalities and a weak-bisimulation monad, allowing equivalences between imperative abstract data types to be established synthetically (Sterling et al., 2022). In this setting, “store language” no longer means a language of reachable store configurations; it means a typed programming language whose denotation includes higher-order store.

A plausible implication is that the automata-theoretic and semantic traditions address complementary questions. The former classifies which store states can arise operationally; the latter explains how such store-bearing programs can be modeled compositionally, including polymorphism and relational reasoning.

4. Store language as a transactional programming model

In S-Store, “Store language” refers to the programming and transaction model of a streaming OLTP system built as an extension of H-Store (Meehan et al., 2015). The unit of program is the stored procedure, written as a mix of SQL and Java; the unit of execution is a transaction execution, one run of a stored procedure with parameters or an atomic batch from a stream (Meehan et al., 2015).

S-Store extends H-Store by adding streams as time-varying tables, sliding windows as special time-varying tables, PE triggers on stream tables, EE triggers on stream and window tables, a streaming-aware scheduler, and strong versus weak recovery variants (Meehan et al., 2015). A streaming transaction is written formally as

$S(M)$ 4

where $S(M)$ 5 are private stream inputs, $S(M)$ 6 are private window inputs, and $S(M)$ 7 are public table partitions. A workflow is a DAG $S(M)$ 8 of such stored procedures, and correct execution requires both workflow order and stream order: $S(M)$ 9 and

$r$ 0

over rounds $r$ 1 (Meehan et al., 2015).

This programming model unifies streaming and OLTP under ACID transaction executions. Each stored procedure instance is atomic and isolated, while the workflow as a whole is an ordered composition of such transactions rather than a single atomic unit (Meehan et al., 2015). The paper also introduces nested transactions to prevent undesirable interleavings when stronger isolation across several procedures is required.

The term “Store language” in this context is therefore not a formal language of configurations and not natural language. It is the programming abstraction imposed by the store engine: parameterized stored procedures, stream tables, window state, triggers, and recovery semantics.

5. Language as the storage medium in AI systems

A distinct AI usage makes language itself the representational substrate of memory. The egocentric memory-augmentation system of 2023 explicitly proposes “using language as the medium of storage”: first-person RGB frames are encoded into detailed textual descriptions by an egocentric vision-LLM, chunked into overlapping text segments, embedded with text-embedding-ada-002, stored in Chroma, and later retrieved by similarity search and answered over by GPT-4 (Shen et al., 2023). The core mapping is

$r$ 2

followed by chunking into 1024-token windows with 256-token overlap and a minimum chunk size of 350 tokens (Shen et al., 2023). On QA-Ego4D, the best language-encoded system using Ego-LLaVA achieved BLEU $r$ 3, compared with $r$ 4 to $r$ 5 for prior baselines; in a user study, the memory-augmentation system obtained a mean score of $r$ 6 versus $r$ 7 for human participants on delayed episodic-memory questions (Shen et al., 2023).

A related but separate use appears in recommendation. STORE formulates semantic tokenization as a text-to-token task and recommendation as a token-to-token task within a single OPT-base LLM (Liu et al., 2024). Each item is represented by a short code sequence

$r$ 8

obtained by post-pretraining dense “gist-like” tokens and then discretizing them via PCA and K-Means (Liu et al., 2024). In the reported configuration, items use $r$ 9 codes with vocabulary size $\NCM$0 per code position; on MIND, STORE reached Recall@5 $\NCM$1, compared with $\NCM$2 for LC-Rec and $\NCM$3 for TIGER with an OPT backbone (Liu et al., 2024).

Both systems literalize storage as linguistic or tokenized representation. In the memory-augmentation case, the stored object is a chunk of language about lived experience; in the recommendation case, the stored object is a semantic token sequence standing in for an item. This suggests a broader AI interpretation of store language as an internal retrieval-oriented representation layer.

6. Store-centric multimodal and platform language

The phrase also appears in domain-specific multimodal research on physical stores and digital app stores. In food-service and retail stores, Ostrakon-VL defines “store language” as a joint visual-linguistic system covering storefronts, interiors, kitchens, signs, menus, equipment, hygiene cues, business status, and structured output formats such as JSON or multiple choice (Shen et al., 29 Jan 2026). The model is an FSRS-oriented MLLM based on Qwen3-VL-8B, trained with QUAD and evaluated on ShopBench, the first public benchmark for FSRS spanning single images, multi-image sets, and videos (Shen et al., 29 Jan 2026). Ostrakon-VL reports an average ShopBench score of $\NCM$4, exceeding Qwen3-VL-8B at $\NCM$5 and Qwen3-VL-235B-A22B at $\NCM$6; QUAD reduces $\NCM$7 million noisy VQA triplets to $\NCM$8 million high-signal pairs and improves ShopBench performance from $\NCM$9 to $\DCM$0 (Shen et al., 29 Jan 2026).

In the VR app ecosystem, “store language” refers to what app stores require developers to say about privacy, what metadata they display, and what apps actually disclose in privacy policies (Yan et al., 27 Oct 2025). The multi-store study covers $\DCM$1 apps from Oculus, Pico, Viveport, Microsoft, and PlayStation, and finds that there is no standardized, VR-specific, or rigorous language framework for privacy declarations (Yan et al., 27 Oct 2025). One third of apps fail to declare their use of sensitive data, and $\DCM$2 neglect to provide valid privacy policies; the paper also reports that many policies are non-VR-specific and that no store provides dedicated fields such as “Uses eye tracking” or “Uses hand tracking” (Yan et al., 27 Oct 2025).

Here the term is tied neither to formal machine stores nor to heaps. It refers to the visual, textual, and policy vocabularies through which store environments or store platforms are interpreted and regulated. In retail MLLMs the emphasis is semantic grounding and decision support; in VR platforms it is declarative privacy compliance.

Across these literatures, “store language” is not a single theory but a family of representations linking state, admissible updates, and recoverable information. In automata theory, the relevant objects are reachable store configurations. In type theory, they are heaps and references in denotational worlds. In S-Store, they are stored procedures, streams, and windows. In AI systems, they are linguistic traces, semantic identifiers, or domain-specific multimodal vocabularies (Ibarra et al., 2017, Sterling et al., 2022, Meehan et al., 2015, Shen et al., 2023).

A related but distinct line of work studies storage inside language-model activations rather than in external symbolic stores. Under the linear representation hypothesis, a layer computes

$\DCM$3

and linear accessibility requires that a matrix $\DCM$4 decode the features from $\DCM$5. For $\DCM$6-sparse inputs, the paper proves that

$\DCM$7

is required while

$\DCM$8

suffices, showing a quantitative gap between linear accessibility and classical compressed sensing with nonlinear decoding (Garg et al., 11 Feb 2026). This is not a store language in the automata or systems sense, but it sharpens the broader problem of what it means for a model to store many features in a recoverable form.

Several limitations recur. In language-encoded egocentric memory, temporal reasoning and counting remain difficult because the encoder is frame-based and does not directly model temporal relationships (Shen et al., 2023). In FSRS specialization, general benchmark performance drops from $\DCM$9 to $\NPCM$0 average relative to the base model, indicating domain narrowing (Shen et al., 29 Jan 2026). In VR app stores, privacy language is fragmented and often non-specific, with substantial declarative–behavioral mismatch (Yan et al., 27 Oct 2025). In automata theory, relaxing finite-crossing to weak finite-crossing destroys the regularity guarantees and permits non-recursive store languages (Friesen et al., 2 Sep 2025).

A plausible synthesis is that every use of store language asks the same structural question in a different register: how should internal state be encoded so that it can be updated, constrained, queried, verified, or decoded with acceptable computational cost. The answer ranges from regular languages of configurations, to guarded denotational worlds, to ACID stored procedures, to vectorized text memories, to operational vocabularies of shops and platforms.