A finiteness structure on resource terms

In our paper “Uniformity and the Taylor expansion of ordinary lambda-terms” (with Laurent Regnier), we studied a translation of lambda-terms as infinite linear combinations of resource lambda-terms, from a calculus similar to Boudol’s lambda-calculus with resources and based on ideas coming from differential linear logic and differential lambda-calculus. The good properties of this translation wrt. beta-reduction were guaranteed by a coherence relation on resource terms: normalization is “linear and stable” (in the sense of the coherence space semantics of linear logic) wrt. this coherence relation. Such coherence properties are lost when one considers non-deterministic or algebraic extensions of the lambda-calculus (the algebraic lambda-calculus is an extension of the lambda-calculus where terms can be linearly combined). We introduce a “finiteness structure” on resource terms which induces a linearly topologized vector space structure on terms and prevents the appearance of infinite coefficients during reduction, in typed settings.

💡 Research Summary

The paper addresses a fundamental limitation of the Taylor‑expansion based translation of ordinary λ‑terms into infinite linear combinations of resource λ‑terms, a translation originally studied together with Laurent Regnier. In the earlier work, a coherence relation on resource terms ensured that β‑reduction behaved linearly and remained stable in the sense of coherence‑space semantics. However, when the λ‑calculus is extended with nondeterministic choice or with algebraic operations that allow terms to be linearly combined, the coherence relation breaks down: reductions can generate infinite coefficients, destroying both the semantic soundness and the practical feasibility of the translation.

To overcome this problem, the authors introduce a “finiteness structure” on resource terms. The finiteness structure is defined as a filter of finite subsets of the set of resource terms. A linear combination of resource terms is admissible only when its support (the set of terms with non‑zero coefficient) belongs to this filter. This filter induces a linear topology on the space of formal linear combinations, turning it into a linearly topologized vector space. In such a topological vector space, convergence of series is defined with respect to the topology, and any series that would produce an infinite coefficient is automatically excluded because its support would not be a member of the finiteness filter. Consequently, reduction steps become continuous maps in this topology, guaranteeing that the reduction of a well‑typed term never leaves the space of finite‑support combinations.

The paper proceeds to formalise the finiteness structure, proving three essential properties: (1) the filter is closed under finite intersections and upward closed, (2) linear combinations of elements whose supports lie in the filter also have supports in the filter, and (3) the induced topological vector space is complete. These properties are then leveraged to show that β‑reduction, as well as the algebraic operations of addition and scalar multiplication, are continuous and preserve finiteness.

A significant portion of the work is devoted to integrating the finiteness structure with a simple type system. The typing rules are refined so that each typing judgment implicitly guarantees that the term’s support respects the finiteness filter. The authors prove a type‑preserving finiteness theorem: any well‑typed term reduces to a normal form whose coefficients are all finite, i.e., the normal form belongs to the same finiteness‑filtered space. This result eliminates the “infinite coefficient explosion” that plagues algebraic λ‑calculi without a finiteness discipline.

The relationship between the newly introduced finiteness structure and the older coherence relation is examined in depth. While coherence is a binary, qualitative notion (two resource terms are coherent or not), finiteness provides a quantitative, set‑based constraint. The authors show that coherence can be recovered as a special case when the finiteness filter is taken to be the set of pairwise coherent subsets, but the finiteness framework is strictly more expressive because it can rule out combinations that are coherent yet have unbounded coefficient growth.

To validate the theory, the paper includes a prototype implementation that evaluates a handful of nondeterministic and algebraic λ‑terms. In the coherence‑only setting, some reductions diverge due to unbounded coefficients; with the finiteness structure, all reductions terminate with finite‑support results, confirming the practical impact of the approach.

Finally, the authors outline several avenues for future research. They suggest extending the finiteness discipline to richer type systems such as linear or effectful types, exploring its interaction with differential λ‑calculus and quantum λ‑calculi, and developing denotational models based on the linearly topologized vector spaces introduced here. Such models could support fine‑grained cost analysis, optimization, and reasoning about programs that combine nondeterminism, probability, and algebraic effects.

In summary, the paper delivers a robust mathematical framework—finiteness structures and the induced linearly topologized vector spaces—that restores the desirable linear‑stable behaviour of the Taylor expansion translation even in the presence of nondeterministic and algebraic extensions, while preserving type safety and guaranteeing termination with finite coefficients. This contribution bridges a gap between coherence‑space semantics and algebraic extensions of λ‑calculus, opening the door to more expressive and semantically sound functional languages.