A Process Calculus with Logical Operators

In order to combine operational and logical styles of specifications in one unified framework, the notion of logic labelled transition systems (Logic LTS, for short) has been presented and explored by L"{u}ttgen and Vogler in [TCS 373(1-2):19-40; Inform. & Comput. 208:845-867]. In contrast with usual LTS, two logical constructors $\wedge$ and $\vee$ over Logic LTSs are introduced to describe logical combinations of specifications. Hitherto such framework has been dealt with in considerable depth, however, process algebraic style way has not yet been involved and the axiomatization of constructors over Logic LTSs is absent. This paper tries to develop L"{u}ttgen and Vogler’s work along this direction. We will present a process calculus for Logic LTSs (CLL, for short). The language CLL is explored in detail from two different but equivalent views. Based on behavioral view, the notion of ready simulation is adopted to formalize the refinement relation, and the behavioral theory is developed. Based on proof-theoretic view, a sound and ground-complete axiomatic system for CLL is provided, which captures operators in CLL through (in)equational laws.

💡 Research Summary

The paper introduces a new process calculus, CLL (Calculus for Logic LTS), which integrates operational and logical specification styles within a single formal framework. Building on the notion of Logic LTS (a labelled transition system equipped with an inconsistency predicate F), the authors add logical constructors ∧ (conjunction) and ∨ (disjunction) to the usual process algebraic operators (0, ⊥, prefix α., external choice ⊓, and CSP‑style parallel k_A). The syntax is given by a simple BNF and the operational semantics are defined by a set of SOS rules: 15 transition rules (Ra1–Ra15) and 13 predicate rules (Rp1–Rp13). Negative premises in several rules guarantee τ‑priority, ensuring that the induced transition system is τ‑pure.

The predicate F marks states that are unimplementable; ⊥ is always inconsistent, while 0 is consistent. Conjunction propagates inconsistency both forward (if either operand cannot perform an action) and backward (if all α‑derivatives of a conjunction are inconsistent). Disjunction becomes inconsistent only when both operands are inconsistent. The authors prove that the TSS yields a unique, well‑defined transition model by employing the notion of a supported model and exploiting τ‑purity.

For refinement, the paper adopts ready simulation, a relation that requires matching of τ‑steps, visible actions, and ready sets. They show that ready simulation is a precongruence for all CLL operators, thus supporting compositional reasoning. This means that refining a component preserves the refinement of the whole system.

The second major contribution is an axiomatic system AX CLL. It consists of equational and inequational laws covering all operators, including distributivity of ∧ over ∨, absorption, identity, and interaction between logical and parallel operators. The system is proved sound with respect to the ready‑simulation semantics and ground‑complete: every valid equation between closed terms can be derived from the axioms.

Overall, the paper delivers a rigorous, dual‑view (behavioral and proof‑theoretic) treatment of a process calculus that seamlessly mixes logical constraints with operational behavior, providing a solid foundation for specifying and verifying heterogeneous concurrent systems.