Cyclic proofs, system t, and the power of contraction

Abstract

We study a cyclic proof system C over regular expression types, inspired by linear logic and non-wellfounded proof theory. Proofs in C can be seen as strongly typed goto programs. We show that they denote computable total functions and we analyse the relative strength of C and Gödel’s system T. In the general case, we prove that the two systems capture the same functions on natural numbers. In the affine case, i.e., when contraction is removed, we prove that they capture precisely the primitive recursive functions—providing an alternative and more general proof of a result by Dal Lago, about an affine version of system T. Without contraction, we manage to give a direct and uniform encoding of C into T, by analysing cycles and translating them into explicit recursions. Whether such a direct and uniform translation from C to T can be given in the presence of contraction remains open. We obtain the two upper bounds on the expressivity of C using a different technique: we formalise weak normalisation of a small step reduction semantics in subsystems of second-order arithmetic: ACA 0 and RCA 0 .

Cyclic Implicit Complexity

“…As shown in previous works (see e.g. [Das21] and [KPP21a]), the progressing criterion is sufficient to guarantee that the partial function computed is, in fact, a well-defined total function:…”

“…Regular proofs, and their corresponding cycle normal forms, have been employed to reason about modal µ-calculus and fixed-point logics [NW96,DHL06b], induction and coinduction [BS11], Kleene algebra [DP17,DP18], linear logic [BDS16], arithmetic [Das18], system T [KPP21a, Das21], and continuous cut-elimination [Min78,FS13]. In particular, [KPP21a] and [Das21] investigate the computational aspects of regular proofs. Due to their coinductive nature, non-wellfounded proofs are able to define any number-theoretic (partial) function.…”

“…If we ignore modalities, the rules of B − can be obtained by adapting to binary notation the rules of CT 0 , the type level 0 fragment of CT from [Das21], which are in turn derivable in the type level 0 fragment of C from [KPP21a]. (See Theorem 22 for further discussion on this point.…”

“…As discussed in [Das21], and implicitly stated in [KPP21a], coderivations can be identified with Kleene-Herbrand-Gödel style equational programs, in general computing partial recursive functionals (see, e.g., [Kle71,§63] for further details). We shall specialise this idea to our two-sorted setting.…”

“…We shall adapt to our setting a well-known 'termination criterion' from non-wellfounded proof theory able to identify certain coderivations (even the non-regular ones) that define total functions. First, let us recall some standard proof theoretic concepts about (co)derivations, similar to those occurring in [Das21,KPP21b].…”