Model-Checking of Linear-Time Properties in Multi-Valued Systems

In this paper, we study model-checking of linear-time properties in multi-valued systems. Safety property, invariant property, liveness property, persistence and dual-persistence properties in multi-valued logic systems are introduced. Some algorithms related to the above multi-valued linear-time properties are discussed. The verification of multi-valued regular safety properties and multi-valued $\omega$-regular properties using lattice-valued automata are thoroughly studied. Since the law of non-contradiction (i.e., $a\wedge \neg a=0$) and the law of excluded-middle (i.e., $a\vee \neg a=1$) do not hold in multi-valued logic, the linear-time properties introduced in this paper have the new forms compared to those in classical logic. Compared to those classical model checking methods, our methods to multi-valued model checking are more directly accordingly. A new form of multi-valued model checking with membership degree is also introduced. In particular, we show that multi-valued model-checking can be reduced to the classical model checking. The related verification algorithms are also presented. Some illustrative examples and case study are also provided.

💡 Research Summary

The paper addresses the problem of model‑checking linear‑time properties in systems whose semantics are expressed in multi‑valued logic rather than classical Boolean logic. Because multi‑valued logics do not satisfy the law of non‑contradiction (a ∧ ¬a = 0) nor the law of excluded middle (a ∨ ¬a = 1), the usual definitions of safety, invariance, liveness, persistence, and dual‑persistence must be reformulated. The authors begin by defining a multi‑valued transition system (MV‑TS) whose states and transitions are labelled with elements of a finite lattice L equipped with a partial order ≤.

Five fundamental linear‑time properties are then introduced in this setting:

• Safety – every execution path avoids a “bad” lattice element below a prescribed threshold;
• Invariant – a property holds at all positions of every path, expressed with lattice conjunctions;
• Liveness – along each infinite path some lattice element above a threshold occurs infinitely often;
• Persistence – after some finite prefix, a desirable lattice value persists forever;
• Dual‑persistence – the complement of persistence, i.e., a “bad” value does not persist.

To verify these properties the paper proposes two automata‑theoretic tools. The first is a Lattice‑Valued Automaton (LVA), a generalisation of Büchi automata where each transition carries a lattice label and acceptance is defined by lattice joins over infinite runs. The second is a Multi‑Valued ω‑Regular Expression (MV‑ω‑regular), which extends classical ω‑regular expressions with lattice operations (∧, ∨, ⊗). Using these devices, the authors show that:

1. Multi‑valued regular safety properties can be reduced to a language‑inclusion test L(MV‑TS) ⊆ L(LVA).
2. Multi‑valued ω‑regular properties can be reduced to an emptiness test of the intersection L(MV‑TS) ∩ L(LVA) = ∅.

Both reductions preserve the lattice structure, allowing the use of standard algorithms for inclusion and emptiness (e.g., SCC decomposition, fix‑point computation).

A key contribution is the reduction of multi‑valued model checking to classical model checking. The authors introduce a two‑step transformation called value decomposition. For each lattice element l ∈ L, a binary transition system S_l is constructed by retaining only those states and transitions whose labels are ≥ l. Classical model‑checking tools (such as SPIN or NuSMV) are then applied to each S_l independently. The results are combined using lattice joins and meets to obtain the overall multi‑valued satisfaction degree. This approach enables practitioners to reuse existing verification infrastructure without implementing new multi‑valued algorithms from scratch.

Furthermore, the paper defines a membership function μ : Σ^ω → L, which maps each infinite word to the degree (a lattice element) to which it satisfies a given property. Unlike Boolean model checking that yields a simple true/false answer, μ provides quantitative information about “how much” a property holds, which is valuable for risk assessment and design optimisation in fuzzy or probabilistic systems.

The theoretical developments are illustrated with an extensive case study of a traffic‑signal controller modelled using a three‑valued logic (normal, warning, error). The authors verify a multi‑valued safety property (no simultaneous green lights) and a persistence property (error states cannot persist indefinitely). By applying value decomposition, three binary models are generated and verified with NuSMV; the lattice‑valued results are then reconstructed, demonstrating that the multi‑valued framework can detect subtle violations that would be invisible in a purely Boolean analysis.

In summary, the paper makes the following original contributions:

• Formal definitions of safety, invariant, liveness, persistence, and dual‑persistence in a lattice‑valued setting.
• Introduction of Lattice‑Valued Automata and MV‑ω‑regular expressions as verification engines.
• Proof that multi‑valued model checking can be systematically reduced to classical model checking via value decomposition.
• A quantitative membership‑degree semantics that enriches the traditional true/false outcome.
• Detailed algorithms, complexity discussion, and a realistic case study that validates the practicality of the approach.

These results broaden the applicability of model checking to fuzzy control systems, probabilistic protocols, and any domain where uncertainty is naturally expressed by more than two truth values.