Observation of implicit complexity by non confluence

We propose to consider non confluence with respect to implicit complexity. We come back to some well known classes of first-order functional program, for which we have a characterization of their intentional properties, namely the class of cons-free programs, the class of programs with an interpretation, and the class of programs with a quasi-interpretation together with a termination proof by the product path ordering. They all correspond to PTIME. We prove that adding non confluence to the rules leads to respectively PTIME, NPTIME and PSPACE. Our thesis is that the separation of the classes is actually a witness of the intentional properties of the initial classes of programs.

💡 Research Summary

The paper revisits implicit‑complexity theory by introducing non‑confluence (the possibility that a program may have several distinct reduction paths for the same input) into three well‑studied classes of first‑order functional programs that are known to capture PTIME. The three baseline classes are: (1) cons‑free programs, which forbid the construction of new data structures; (2) programs admitting a (total) interpretation, guaranteeing a monotone decrease of a numeric measure on every recursive call; and (3) programs equipped with a quasi‑interpretation together with a termination proof by the product‑path ordering (PPO). All three have been shown in prior work to be exactly the class of polynomial‑time computable functions.

The authors define, for each class, an “extension with non‑confluence” where the rewrite rules are allowed to be non‑deterministic in the sense that a term may rewrite to two different normal forms. They then prove three separation results:

• Cons‑free + non‑confluence still characterises PTIME. The prohibition on building new cons cells limits the number of distinct reduction branches to a polynomial function of the input size, so even with nondeterministic branching the overall computation remains within polynomial time.

• Interpretation‑admitting + non‑confluence characterises NPTIME. The existence of a decreasing interpretation ensures that every branch makes progress, but the nondeterministic choice among several decreasing branches yields a nondeterministic polynomial‑time computation. Hence the class coincides with the classic nondeterministic polynomial‑time class.

• Quasi‑interpretation + PPO + non‑confluence characterises PSPACE. Quasi‑interpretations bound the size of intermediate values, and PPO guarantees termination, but the presence of many nondeterministic branches can cause the call‑stack depth to grow exponentially while the space used remains polynomially bounded. Consequently, the extended model computes exactly the functions computable in polynomial space.

These results support the authors’ thesis that the intentional restrictions originally imposed on the three PTIME‑capturing classes are the very reasons why they separate from each other when nondeterminism is added. Stronger restrictions (cons‑free) are robust against nondeterminism, medium restrictions (interpretations) become nondeterministic polynomial time, and weaker restrictions (quasi‑interpretations with PPO) blow up to polynomial space.

Beyond the technical theorems, the paper argues that non‑confluence provides a structural way to “lift” implicit‑complexity classes along the classical complexity hierarchy, offering a new perspective compared to traditional relative‑complexity techniques (e.g., bounding recursion depth or memory). The authors also discuss possible extensions: refining the granularity of the hierarchy by limiting the degree of nondeterminism, combining non‑confluence with probabilistic choice, and incorporating these ideas into the design of practical functional languages that expose implicit‑complexity guarantees while allowing controlled nondeterminism.

In summary, the work demonstrates that adding a simple form of nondeterminism to well‑understood implicit‑complexity frameworks yields precisely the expected jumps to NPTIME and PSPACE, thereby confirming that the intentional properties of the original classes are the underlying witnesses of their complexity‑theoretic separation.