Coinduction in Uniform: Foundations for Corecursive Proof Search with Horn Clauses

Abstract

We establish proof-theoretic, constructive and coalgebraic foundations for proof search in coinductive Horn clause theories. Operational semantics of coinductive Horn clause resolution is cast in terms of coinductive uniform proofs; its constructive content is exposed via soundness relative to an intuitionistic rst-order logic with recursion controlled by the later modality; and soundness of both proof systems is proven relative to a novel coalgebraic description of complete Herbrand models.We can also view Horn clauses coinductively. The greatest complete Herbrand model for a set P of Horn clauses is the largest set of nite and in nite ground atomic formulae coinductively entailed by P. For example, the greatest complete Herbrand model for the above two clauses is the set N ∞ = N ∪ {nat (s (s (· · · )))}, obtained by taking a backward closure of the above two inference rules on the set of all nite and in nite ground atomic formulae. The greatest Herbrand model is the largest set of nite ground atomic formulae coinductively entailed by P. In our example, it would be given by N already. Finally, one can also consider the least complete Hebrand model, which interprets entailment inductively but over potentially in nite terms. In the case of nat, this interpretation does not di er from N . However, nite paths in coinductive structures like transition systems, for example, require such semantics.The need for coinductive semantics of Horn clauses arises in several scenarios: the Horn clause theory may explicitely de ne a coinductive data structure or a coinductive relation. However, it may also happen that a Horn clause theory, which is not explicitly intended as coinductive, nevertheless gives rise to in nite inference by resolution and has an interesting coinductive model. This commonly happens in type inference. We will illustrate all these cases by means of examples.

Flexible coinductive logic programming

“…While coSLD resolution and its proposed extensions are limited by the fact that cycles must be detected in derivations to allow resolution to succeed, a stream of work based on the notion of structural resolution (Komendantskaya et al 2016;) (S-resolution for short) aims to make coinductive resolution more powerful, by allowing to lazily detect infinite derivations which do not have cycles. In particular, recent results (Li 2017;Komendantskaya and Li 2017;Basold et al 2019) investigate how it is possible to integrate coLP cycle detection into Sresolution, by proposing a comprehensive theory. Trying to integrate S-resolution with coclauses is an interesting topic for future work aiming to make coLP even more flexible.…”

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

“…We will use a standard β-and fix-reduction relation on terms, see (Basold et al 2019). The equivalence closure of the reduction relation (convertibility) is denoted by ≡.…”

“…Guarded terms close guarded base terms under operations of the simply typed λ-calculus. Basold et al (2019) provides examples and further discussion of guarded terms. In what follows, we will use the following sets of well-typed terms: the set Λ − Σ of all simple terms, i.e.…”

“…We assume these 6 rules additionally to those in Figure 3. We do not state soundness of CLJ here, as soundness of a very similar proof system CUP relative to the greatest Herbrand models of logic programs was already proven in (Basold et al 2019). Here, our main goal is to prove cut non-elimnation in CLJ.…”

“…CUP is a coinductive extension of uniform proofs, a fragment of the Gentzen's sequent calculus introduced to model the derivations obtained by first-order resolution in Prolog (Miller et al 1991;Miller and Nadathur 2012). As it turns out, CUP is sound with respect to the largest Herbrand models of logic programs (Basold et al 2019).…”