Identity-Verification State Machines

• Identity-verification state machines are rigorous computational models that formalize control flow, safety properties, and correctness guarantees for secure identity procedures.
• They utilize formal methods such as reactive designs, temporal logic, and automated theorem proving to ensure correct-by-construction state transitions in complex systems.
• Adaptive testing strategies and layered semantic models enhance their robustness against adversarial behaviors and nondeterministic challenges, supporting real-world deployments.

Identity-verification state machines formalize the control flow, safety properties, and correctness guarantees of computational systems performing identity verification. They arise across domains such as access control, digital authentication, privacy-preserving cloud services, secure smart contracts, network protocol testing, and distributed identity in the physical world. Their design, analysis, and verification require precise definitions of states and transitions, rigorous semantics, formal verification, and codified invariants to ensure robust and predictable identity procedures.

1. Formal Foundations and Reactive Semantics

The foundational approach to identity-verification state machines leverages denotational semantics based on the Unifying Theories of Programming (UTP) as introduced in (Foster et al., 2018). Here, state-machine behaviors are captured by reactive designs—contract-based specifications supporting compositional reasoning over time-varying input/output and internal states. This approach mechanizes the state-machine semantics for both finite and infinite state systems.

The dynamic semantics are typically expressed via guarded iterations. The refinement relation $S \sqsubseteq P$ ensures that a detailed implementation $P$ upholds the desired safety properties $S$ (e.g., deadlock freedom). Induction theorems for iterative processes guarantee invariants are preserved:

$$\forall i\in A:\quad S \sqsubseteq I \mathbin{\blacktriangleleft} [\,b(i)\,] \mathbin{;} P(i)$$

Here, $b(i)$ are guard conditions and $P(i)$ are process bodies; contract satisfaction requires productive transitions and invariant preservation both as base and inductive steps.

Verification is mechanized in Isabelle/UTP by representing reactive designs as well-formed UTP predicates within the Isabelle/HOL theorem prover. All state-machine operators are lifted to the new types, and symbolic evaluation/rewrite theorems support automated, scalable proof discharges, even for systems with complex transitioning behaviors or infinite state spaces.

2. Synthesis and Specification in Blockchain and Smart Contracts

Finite state machines are central to modeling identity verification as part of formally specified smart contracts, most notably in blockchain environments (Suvorov et al., 2019). State machines are defined as tuples $(S, s_\text{init}, E, Z, \delta, \lambda)$, with $S$ the set of states (e.g., Unregistered, Registered, Submitted, UnderReview, Verified, Rejected), $E$ the set of input events (actions by users or verifiers), $\delta$ the transition function, and $\lambda$ the output (side effect) function.

Correctness is specified via Linear Temporal Logic (LTL) formulas that enforce the legal sequencing and safety invariants, for example:

• $$G(\mathrm{register} \to X \neg F\,\mathrm{register})$$: Registration is a one-time event.
• $G(\mathrm{approve} \to F\,\mathrm{verify})$: No approval occurs before verification.
• $G(\mathrm{Verified} \to X G\,\mathrm{Verified})$: The verified state, once reached, persists.

FSM synthesis tools use these formulae with test traces to generate correct-by-construction transition systems. These FSMs are directly compiled into executable code (e.g., Solidity for Ethereum), where guards and require-statements are enforced, guaranteeing LTL-specified safety and precondition-order.

Auditing and formal verification of these synthesized state machines ensure race conditions, unauthorized state advancements, and replay attacks are prevented; these properties are explicitly guaranteed by the LTL-driven FSM synthesis, providing correctness and robustness absent in ad hoc implementations.

3. State Identification, Testing, and Adaptive Distinguishing

In networked and embedded systems, state identification in the presence of nondeterminism or partiality is essential for protocol conformance and security—core to identity verification in practical deployments (Bos et al., 2019). Labeled Transition Systems (LTS) generalize FSMs by allowing partial, nondeterministic input/output. The adaptive test generation problem is solved using a splitting graph algorithm, where state subsets are recursively partitioned until only mutually compatible (indistinguishable) states remain.

The process constructs a splitting graph $Y = (V, E, W)$, where each node is a subset of states, $E$ encodes partitioning, and $W$ assigns distinguishing witnesses (test sequences). The adaptive distinguishing graph extracted from $Y$ produces test cases that cover all incompatible states. This methodology is robust to adversarial behavior; compatible states are only those for which no adaptive input/output experiment can reveal observable differences.

Empirical evaluation demonstrates that these techniques distinguish more than 99% of all incompatible state pairs in large, industrial-scale models. For identity verification, these adaptive test sequences provide rigorous evidence that each protocol state can be operationally and observably validated, even with nondeterministic outputs or partial transitions.

4. Semantic Models: Thinging Machine and Layered Abstractions

Precise semantic modeling of what constitutes a "state" in identity-verification machines is critical for unambiguous design (Al-Fedaghi, 2020). The "thinging machine" (TM) paradigm defines states not just as simple labels, but decomposes them into five fundamental actions: Create, Process, Release, Transfer, Receive. This richer view supports layered system representations:

• Static model $S$: Full flow of things (data, credentials, decisions) and permissible actions.
• Dynamic model $D$: Slices of $S$ representing distinct system states (before/after credential validation, for example).
• Behavioral model $B$: Time-indexed embedding of these changes, supporting precise reasoning about sequences, timeouts, and event-driven transitions.

In layered FSM notation, transitions are formulated as $\delta: S \times E \to S$, e.g.,

$$\delta(\text{Awaiting Credentials},\,\text{submit}) = \text{Processing Credentials}$$

$$\delta(\text{Processing Credentials},\,\text{verify}) = \text{Verified Identity}$$

$$\delta(\text{Processing Credentials},\,\text{fail}) = \text{Verification Failure}$$

The TM approach ensures every state is understood as a transformation pipeline over data flows, rather than merely a name in a transition graph, providing transparency, error detection, and security assurance—particularly when action-level and message-level semantic errors can propagate into security vulnerabilities.

5. Formalizing Industrial Modeling Standards

Identity-verification applications depend heavily on state-machine models constructed in notations such as UML, where natural-language ambiguity in the specification is a significant source of design errors. Formalizing UML state machine semantics for the purpose of automatic verification is essential for security-critical domains (André et al., 24 Jul 2024).

Formalization strategies include translation into well-defined formalisms (ASMs, Petri nets, timed automata) or direct operational semantics given as inference rules:

$$\frac {\langle \text{State}, \text{Env} \rangle \xrightarrow{e / a} \langle \text{State}', \text{Env}' \rangle, \quad \text{guard}(e, c) = \texttt{true}} {\langle c,\, \text{pending } e \rangle \to \langle c',\, \text{Env}' \rangle}$$

This supports automated checking of run-to-completion properties, noninterference, and deadlock-freedom—even in the presence of concurrency (orthogonal regions), history, and time constraints. For example, in a verification workflow, one may formally prove $$\Box(\texttt{Verifying} \rightarrow \Diamond(\texttt{Verified} \lor \texttt{Failed}))$$, i.e., every verification attempt eventually terminates.

Such rigorously defined semantics, supported by model checkers and certified translations, yield strong assurance that the system cannot enter unintended or insecure states, a critical property for identity-verification machinery deployed in regulated industries.

6. Applied Architectures: Privacy-Preserving and Distributed Models

State-machine formalism extends to pragmatically complex systems, such as privacy-preserving national identity systems and large-scale distributed identity infrastructures.

In privacy-preserving digital identity protocols using fully homomorphic encryption (FHE) (Mohan et al., 15 Aug 2024), the verification procedure itself is modeled as a sequence of state transitions. Users, service providers, third-party servers (TPS), and central servers (CS) each run a portion of the overall state machine, where encrypted user data are processed via arithmetic circuits entirely in the encrypted domain. The TPS executes homomorphic computations (e.g., biometric matching, age comparisons), advancing the state machine toward either "verifiably matched" or "rejected/no match" states, while the CS performs a final "extended decryption," a fixed, query-agnostic operation ensuring confidentiality and minimal trusted computing base.

Distributed digital identity architectures for the physical world (Mayrhofer et al., 13 Aug 2025) introduce a Personal Identity Agent (PIA) acting as a local state machine for each individual. The PIA stores credentials, mediates data flows between sensors and verifiers, and enforces both privacy and protocol correctness. Transactional states span "bootstrapped," "subscribed," "sensor detection," "credentials presented," and "authorization completed," and transitions are governed by multiset rewrite rules, as formalized in the Tamarin modeling tool. Security invariants (e.g., biometric attribute matching) are encoded as lemmas, such as

$$\forall A,i,a,b,ba.\; (CredentialIssued(A,i,a,b,ba) \land SensorDetected(A, b)) \implies (b=ba)$$

These distributed state machines provide privacy and resistance against central points of attack, at the cost of increased communication and protocol complexity.

7. Security Testing, Soundness, and Future Developments

Identity-verification state machines must be robust to adversarial behaviors, system nondeterminism, and emerging computational attacks. Advanced protocols leverage relativistic zero-knowledge proofs (RZKPs), which utilize spatially separated provers and the no-signaling principle for quantum-safe authentication (Ma et al., 18 Jul 2025). Here, the state machine includes phases such as commitment, challenge, and response, across multiple, potentially entangled provers:

• Commitment (e.g., $C \leftarrow H(m \oplus r)$) secures secrets via quantum-resistant hash functions and randomized inputs.
• Challenges and responses are algebraically constructed ($R = f(P, ch)$) and are validated under stringent relativistic constraints (e.g., spatial separation).

For multi-prover protocols, the state machine encodes redundant verification steps to detect collusion or quantum-enabled cheating, and the soundness parameter $\epsilon$ is formally bounded (e.g., $\epsilon \leq 2^{-λ}$) in terms of the number of rounds and instance hardness. Stability and scalability improvements—such as reduced spatial separation and robust synchronization—extend the applicability of such state machines to real-world deployments where quantum capabilities may be present.

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Identity-verification state machines thus represent a confluence of formal semantic modeling, rigorous correctness verification, specification-driven synthesis, adaptive testing, and cryptographic engineering. Through mechanization, formalization, and careful protocol design, these state machines provide the backbone for secure, reliable, and scalable identity management in both digital and physical domains.