Truth Predicate of Inductive Definitions and Logical Complexity of Infinite-Descent Proofs

Formal reasoning about inductively defined relations and structures is widely recognized not only for its mathematical interest but also for its importance in computer science, and has applications in verifying properties of programs and algorithms. Recently, several proof systems of inductively defined predicates based on sequent calculus including the cyclic proof system CLKID-omega and the infinite-descent proof system LKID-omega have attracted much attention. Although the relation among their provabilities has been clarified so far, the logical complexity of these systems has not been much studied. The infinite-descent proof system LKID-omega is an infinite proof system for inductive definitions and allows infinite paths in proof figures. It serves as a basis for the cyclic proof system. This paper shows that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete. To show this, first it is shown that the validity for inductive definitions in standard models is equivalent to the validity for inductive definitions in standard term models. Next, using this equivalence, this paper extends the truth predicate of omega-languages, as given in Girard’s textbook, to inductive definitions by employing arithmetical coding of inductive definitions. This shows that the validity of inductive definitions in standard models is a (Pi-1-1) relation. Then, using the completeness of LKID-omega for standard models, it is shown that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete.

💡 Research Summary

This paper investigates the logical complexity of the infinite‑descent proof system LKID‑ω, which is designed for reasoning about inductively defined predicates in first‑order logic (FO​L​ID). The authors establish that provability in LKID‑ω is Π¹₁‑complete, thereby locating the proof‑search problem for this system at the first level of the analytical hierarchy.

The work proceeds in several stages. First, the authors formalize FO​L​ID, a first‑order language enriched with a finite collection of inductive predicate symbols P₁,…,Pₙ. Each inductive predicate is given by production rules of the form
 Q₁(ū₁)…Qₕ(ūₕ) Pⱼ₁(t̄₁)…Pⱼₘ(t̄ₘ) → Pᵢ(t̄),
where the Q’s are ordinary predicates. From these rules a monotone operator ϕᵢ on subsets of the universe is derived, and the whole operator ϕ = (ϕ₁,…,ϕₙ) determines the least fixed point (lfp ϕ)ᵢ that interprets each inductive predicate in a standard model.

Next, the paper introduces a “name‑extension” technique. By adding countably many fresh constants c₁,c₂,… to the signature (forming Σᶜ) and interpreting each element of a countable model M by a distinct constant, a name‑extended model Mᶜ is obtained. Lemma 3.4 shows that if M is standard then Mᶜ is also standard.

From a name‑extended model the authors construct a term model Mᵀ whose domain consists of equivalence classes of closed terms modulo the equality provable in the original model. The interpretation of function symbols and constants is term‑wise. Crucially, Lemma 3.12 proves that if the original model is standard and name‑extended, then its term model is again a standard model. This equivalence between validity in ordinary standard models and in term models is the cornerstone for the subsequent truth‑predicate construction.

To analyse complexity, the authors adapt Girard’s truth predicate for ω‑languages. In Girard’s setting, the set of infinite words over a finite alphabet can be encoded as a Π¹₁ set, and the truth of an ω‑sentence is a Π¹₁ property. The paper shows that inductive definitions can be arithmetically coded in a similar way: each inductive predicate Pᵢ is approximated by a family of formulas Pᵢ(k, x̄) that converge to the true predicate as k grows. Proposition 3.10 and Corollary 3.11 establish that “M ⊨ Pᵢ(t̄)” is equivalent to “M ⊨ Pᵢ(k, t̄)” for some k, and that this equivalence can be expressed by a Π¹₁ formula. Consequently, the validity problem for FO​L​ID in standard models is a Π¹₁ relation.

The infinite‑descent system LKID‑ω is then defined: it is a sequent calculus that permits infinite proof trees, provided every infinite branch descends with respect to a well‑founded measure (the usual infinite‑descent condition). Prior work