## A solution to Curry and Hindley’s problem on combinatory strong reduction.(English)Zbl 1172.03009

From the author’s abstract: “Curry and Hindley’s problem, dating back to 1958, asks for a self-contained proof of the confluence of $$\succ$$ [the combinatory analogue of $$\beta \eta$$-reduction $${\twoheadrightarrow_{\beta\eta}}$$ in $$\lambda$$-calculus], one which makes no detour through $$\lambda$$-calculus. We answer positively to this question, by extending and exploiting the technique of transitivity elimination for ‘analytic’ combinatory proof systems, which has been introduced in previous papers of ours [P. Minari, “Analytic combinatory calculi and the elimination of transitivity”, Arch. Math. Logic 43, No. 2, 159–191 (2004; Zbl 1060.03033); “Proof-theoretical methods in combinatory logic and $$\lambda$$-calculus”, talk presented at CiE 2005: New computational paradigms, Amsterdam, 2005; “Analytic proof systems for $$\lambda$$-calculus: the elimination of transitivity, and why it matters”, Arch. Math. Logic 46, No. 5–6, 385–424 (2007; Zbl 1117.03020)].”

### MSC:

 03B40 Combinatory logic and lambda calculus 03F05 Cut-elimination and normal-form theorems 03F07 Structure of proofs

### Citations:

Zbl 1060.03033; Zbl 1117.03020
Full Text:

### References:

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