Data linkage algebra, data linkage dynamics, and priority rewriting

We introduce an algebra of data linkages. Data linkages are intended for modelling the states of computations in which dynamic data structures are involved. We present a simple model of computation in which states of computations are modelled as data linkages and state changes take place by means of certain actions. We describe the state changes and replies that result from performing those actions by means of a term rewriting system with rule priorities. The model in question is an upgrade of molecular dynamics. The upgrading is mainly concerned with the features to deal with values and the features to reclaim garbage.

💡 Research Summary

The paper introduces a formal algebraic framework for modelling the states of computations that involve dynamic data structures. The core contribution is the Data Linkage Algebra (DLA), an initial algebra with a single sort DL whose elements represent collections of “data linkages” – abstract objects that correspond to molecules composed of atoms, fields, spots, and values. The signature includes constants for spot‑to‑atom links, partial and full field links, value associations, an empty linkage, and two binary operators: ⊕ (union) and ⊕′ (override). The override operator gives priority to the right‑hand operand when two linkages contain conflicting information about the same spot, field, or value, thereby modelling assignment semantics.

A set of equational axioms (commutativity, associativity, idempotence for ⊕; specific overriding rules for ⊕′) is presented, and it is proved that every closed DLA term can be reduced to a “basic term” consisting only of primitive links combined with ⊕. This normal‑form property enables a clear distinction between deterministic linkages (each spot, field, and value appears at most once) and nondeterministic ones.

To capture state changes, the authors employ priority rewrite systems (PRS). A PRS is a term‑rewriting system together with a partial order on its rules; higher‑priority rules block lower‑priority ones. The paper defines a collection of actions (e.g., new, set, get, dispose) together with PRS rules that describe how an action transforms a current data linkage L into a new linkage L′ and produces a reply (boolean or error code). The priority mechanism ensures that, for example, a “set” action overrides any previous value association before the next rule can fire.

The model is an upgrade of the earlier Molecular Dynamics framework. The upgrade adds two major features: (1) value handling, allowing computations over a finite meadow (a finite field with totalized division, where 0⁻¹ = 0), and (2) garbage collection. Four garbage‑collection modes are provided—full, restricted, safe dispose, and unsafe dispose—each expressed as specific override rules that either delete all links of an atom or adjust reference counts. This makes the algebra capable of modelling realistic memory‑management policies.

The paper also integrates Basic Thread Algebra (BTA) with services. Threads are sequential compositions of actions; after each action the execution environment (a service) returns a reply that determines the next step. Services abstract external resources such as allocators or collectors, and the authors show how to combine BTA with DLD so that program‑level behaviours can be expressed and analysed within the same formalism.

Finally, an alternative set‑theoretic semantics is given: data linkages are interpreted as partial functions (spot → atom, atom·field → atom/value), and ⊕′ corresponds to function overriding. This bridges the algebraic presentation with a concrete functional model, confirming consistency and providing a basis for implementation.

Overall, the work delivers a comprehensive algebraic and operational framework for dynamic data structures, unifying value computation, assignment, and garbage collection under a single rewrite‑based semantics. It opens avenues for formal verification of programs that manipulate mutable structures, for reasoning about memory‑management strategies, and for integrating such reasoning with existing process‑algebraic tools.