On the complexity of stratified logics

Our primary motivation is the comparison of two different traditions used in ICC to characterize the class FPTIME of the polynomial time computable functions. On one side, FPTIME can be captured by Intuitionistic Light Affine Logic (ILAL), a logic derived from Linear Logic, characterized by the structural invariant Stratification. On the other side, FPTIME can be captured by Safe Recursion on Notation (SRN), an algebra of functions based on Predicative Recursion, a restriction of the standard recursion schema used to defiine primitive recursive functions. Stratifiication and Predicative Recursion seem to share common underlying principles, whose study is the main subject of this work.

💡 Research Summary

The paper investigates two well‑established frameworks that both capture the class FPTIME of polynomial‑time computable functions, aiming to reveal the deep connection between their underlying principles. The first framework is Intuitionistic Light Affine Logic (ILAL), a variant of linear logic that enforces a structural invariant called stratification. In ILAL every formula is annotated with an integer “level”. The level controls the use of contraction and weakening: a proof may only duplicate or discard resources when the level constraints are satisfied, and any increase in level corresponds to a bounded form of recursion. This disciplined use of resources guarantees that any cut‑free proof can be interpreted as a polynomial‑time algorithm, a result already known from previous work on light logics.

The second framework is Safe Recursion on Notation (SRN), an algebra of functions based on predicative recursion. SRN distinguishes between “safe” and “normal” arguments. Normal arguments may be increased at most once per recursive call, while safe arguments can be used freely inside the recursive body but never appear as the controlling parameter of recursion. This separation prevents the uncontrolled growth of intermediate values and ensures that every function defined in SRN runs in polynomial time. SRN is known to be complete for FPTIME, providing an alternative, recursion‑theoretic characterization of the same class.

The core contribution of the paper is a systematic comparison of the two systems. The authors observe that stratification in ILAL and the safe/normal distinction in SRN are essentially two manifestations of the same resource‑management discipline. To make this observation precise, they embed both formalisms into a common meta‑language and define a translation map ϕ. The map sends each ILAL proof step at a given level to an SRN definition where the corresponding arguments are marked safe or normal according to the level hierarchy, and conversely translates any SRN definition into an ILAL proof by assigning levels that respect the safe/normal constraints. The translation is proved to be structure‑preserving: contraction, weakening, and the bounded recursion schemes are mirrored faithfully, and the depth of recursion (or proof) is unchanged. Consequently, ϕ establishes a bijective correspondence between ILAL proofs and SRN functions, showing that the two systems have exactly the same expressive power with respect to polynomial‑time computation.

Beyond the technical equivalence, the paper discusses several implications. First, techniques developed for one framework (e.g., cut‑elimination strategies in ILAL) can be transferred to the other, providing new tools for analyzing SRN‑defined programs. Second, the equivalence suggests a unified design space for future logics that aim to capture complexity classes: one may start from a stratified logical system and immediately obtain a predicative recursion algebra, or vice versa. Finally, the authors argue that this bridge deepens our understanding of how logical resource control and recursion‑theoretic restrictions jointly enforce computational bounds, opening avenues for extending the approach to other classes such as LOGSPACE or PTIME‑complete problems.

In summary, the paper delivers a rigorous proof that ILAL’s stratification and SRN’s predicative recursion are two sides of the same coin, both providing sound and complete characterizations of FPTIME. This result not only clarifies the conceptual relationship between two major traditions in implicit computational complexity but also paves the way for cross‑fertilization of methods and for the design of new, hybrid systems that inherit the strengths of both logical and algebraic perspectives.