Algebra of Concurrent Games

We introduce parallelism into the basic algebra of games to model concurrent game algebraically. Parallelism is treated as a new kind of game operation. The resulted algebra of concurrent games can be used widely to reason the parallel systems.

💡 Research Summary

The paper proposes an algebraic framework for modeling concurrent games by extending the Basic Algebra of Games (BAG) with a new parallel composition operator, denoted “∥”. BAG, as originally introduced by Goranko and Venema, consists of a set of atomic games, a distinguished idle game ι, and four primitive operations: choice of the first player (∨), choice of the second player (∧), dualization (d), and sequential composition (○). BAG is governed by a list of thirteen axioms (G₁–G₁₃) that capture associativity, commutativity, distributivity, identity elements, and interaction with dualization. Game terms are built inductively from atomic games using these operators, and each term induces outcome relations ρ₁ and ρ₂ on a game board, satisfying monotonicity and consistency. Inclusion (⊆ᵢ) and equality of terms are defined via inclusion of these outcome relations. Crucially, BAG enjoys two elimination theorems: every term can be rewritten into a canonical form (a disjunction of conjunctions of literals composed with ○) and further into a minimal canonical form where no redundant literals or sub‑terms appear. A completeness theorem states that two minimal canonical terms are equivalent iff they are isomorphic (i.e., can be transformed into each other by permuting conjuncts/disjuncts).

The authors’ main contribution is to augment BAG with a parallel operator ∥, yielding the Algebra of Concurrent Games (ACG). They introduce ∥ into the game language and term formation rules, leaving the outcome conditions and inclusion definitions unchanged. Ten new axioms (C_G₁–C_G₁₁) describe the algebraic behavior of ∥: associativity (C_G₁), interaction with sequential composition (C_G₂–C_G₄), distributivity over both ∨ and ∧ (C_G₅–C_G₈), compatibility with dualization (C_G₉), and identity elements ι on both sides (C_G₁₀, C_G₁₁). These axioms ensure that ∥ behaves like a true parallel composition that can be interleaved with the existing operators without breaking the algebraic structure.

Canonical terms are redefined to accommodate ∥: a canonical term now has the shape “⋁ᵢ ⋀ₖ ∥ₗ gᵢₖₗ ○ Gᵢₖₗ”, where each gᵢₖₗ is a literal (atomic game or its dual) and each Gᵢₖₗ is a sub‑term. Minimal canonical terms are defined analogously to BAG, with additional constraints preventing the idle game ι from appearing inside parallel components and forbidding embedding of one conjunct into another within the same ∧‑cluster, as well as preventing one disjunct from embedding into another.

The paper proves two elimination theorems for ACG. The first shows that any ACG term can be rewritten into a canonical term using structural induction, handling the new case of ∥ via the distributive axioms C_G₅–C_G₈. The second shows that any term can be reduced to a minimal canonical term, again by induction and by applying the embedding definition that respects ∥. The proofs are essentially adaptations of the BAG proofs, with the additional parallel case treated uniformly.

For completeness, the authors translate ACG terms into the modal logic used in the original BAG completeness proof. The translation respects the new operator: if two sub‑terms G₁ and G₂ are in a race condition (denoted G₁ % G₂), then the translation of G₁ ∥ G₂ is either the sequential composition of their translations (m(G₁)·m(G₂) or m(G₂)·m(G₁)); otherwise, the translation is the logical disjunction m(G₁) ∨ m(G₂). This mapping preserves validity, allowing the existing BAG completeness argument to carry over. Consequently, the minimal canonical terms of ACG are equivalent exactly when they are isomorphic, establishing a completeness theorem for the concurrent algebra.

In the concluding section, the authors argue that ACG provides a formal, algebraic tool for reasoning about parallel systems within a game‑theoretic framework. While prior work on concurrent games has focused on semantic or algorithmic aspects, ACG supplies a set of equational laws and normal forms that could support automated reasoning, verification, and synthesis of concurrent protocols. The paper, however, does not present concrete case studies or complexity analyses, and the handling of race conditions in the translation is only sketched, leaving room for further refinement. Overall, the work lays a solid theoretical foundation for algebraic treatment of concurrency in games, opening avenues for future research in formal methods and game‑based system design.