Abstract machines for dialogue games

The notion of abstract Boehm tree has arisen as an operationally-oriented distillation of works on game semantics, and has been investigated in two papers. This paper revisits the notion, providing more syntactic support and more examples (like call-by-value evaluation) illustrating the generality of the underlying computing device. Precise correspondences between various formulations of the evaluation mechanism of abstract Boehm trees are established.

💡 Research Summary

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This paper revisits the notion of abstract Böhm trees, a concept that emerged from game‑semantic analyses as an operational distillation of the traditional Böhm‑tree representation of λ‑terms. The authors aim to provide a more syntactically grounded treatment and to demonstrate the breadth of the underlying computational device by covering a range of evaluation strategies, notably call‑by‑value, as well as extensions such as PCF with case statements, control operators à la λµ‑calculus, and Girard’s ludics.

The core technical contribution is the definition of the Geometric Abstract Machine (GAM). A strategy φ represents the term to be evaluated, while a counter‑strategy ψ represents its context (free variables, continuation, etc.). Positions are sequences of moves drawn from an alphabet A together with a distinguished initial move “•”. Even‑length positions (responses) and odd‑length positions (queries) are equipped with pointers that encode the binding relationship between a player’s move and the opponent’s move it refers to. To handle duplication—when a function calls its argument multiple times—the authors introduce multiplexed positions. A multiplexed position carries an extra index j that records the order in which copies of a sub‑position are created; this makes the duplication process explicit rather than implicit.

The GAM’s state is a list Γ of multiplexed positions. The machine alternates between two families of transition rules: (2n) steps that consult the strategy φ and (2n+1) steps that consult the counter‑strategy ψ. At an even step the machine looks up the unique player move that answers the current query, inserts it together with its multiplexing index, and then uses the pointer information to jump back to the earlier opponent position that introduced the queried variable. At an odd step the same process is performed with ψ. Two essential mechanisms are thus interleaved: (i) tree interaction, which relies on the determinacy of φ and ψ to resolve conflicts, and (ii) pointer interaction, which uses the multiplexing indices to decide where the opponent’s move should be placed. This design yields a uniform execution model that simultaneously manages control flow, variable binding, and copying.

Termination of the GAM is characterised by three mutually exclusive cases: (1) the strategy’s root is reached, so no further opponent move exists; (2) a free variable of the counter‑strategy is encountered, leaving no answer; (3) a multiplexed position points to a location that is not in the domain of the current strategy. The authors note that in their earlier work