Abstract
There have been some important methods of combining a recursive path ordering and Tait-Girard’s computability argument to provide an ordering for termination proofs of higher-order rewrite systems. The higher-order recursive path ordering HORPO by Jouannaud and Rubio and the computability path ordering CPO by Blanqui, Jouannaud and Rubio are examples of such an ordering. In this paper, we give a case study of yet another direction of such extension of recursive path ordering, avoiding Tait-Girard’s computability method plugged in the above mentioned works. This motivation comes from Lévy’s question in the RTA open problem 19, which asks for a reasonably straightforward interpretation of simply typed \(\lambda \)-calculus \(\lambda _{\rightarrow }\) in a certain well founded ordering. As in the cases of HORPO and CPO, the addition of \(\lambda \)-abstraction and application into path orderings might be considered as one solution, but the following question still remains; can the termination of \(\lambda _{\rightarrow }\) be proved by an interpretation in a first-order well founded ordering in the sense that \(\lambda \)-abstraction/application are not directly built in the ordering? Reconsidering one of Howard’s works on proof-theoretic studies, we introduce the path ordering with Howard algebra as a case study towards further studies on Lévy’s question.
We thank the readers of the earlier version of this paper and the anonymous reviewers for their valuable comments. The first author is supported by JSPS (Japan Society for the Promotion of Science) KAKENHI Grand Numbers 17H02263, 17H02265 and 19KK0006. The second author is supported by JSPS Overseas Research Fellowship.
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Notes
- 1.
https://www.win.tue.nl/rtaloop/problems/19.html. See also the TLCA open problem 26 (http://tlca.di.unito.it/opltlca/opltlcasu33.html#x38-62000).
- 2.
If one takes the sequent-style formulation of logic, a reasonably straightforward mapping from proofs to a recursive path ordering (for example, a recursive path ordering of size of the ordinal \(\varphi \omega 0\) in Veblen hierarchy) is enough for termination proofs. Lévy’s question is open only when one considers the normalization by means of natural deduction, equivalently of typed \(\lambda \)-calculus, instead of sequent calculus. This fact also suggests that the key issue here, with respect to Lévy’s question, is how to interpret the \(\lambda \)-abstraction/application in a natural way.
- 3.
As stated in Sect. 1, it is not \(\lambda _{\rightarrow }\) but the system \(\mathsf {T}\) of primitive recursive functionals of finite type that Howard actually discussed in [9]. We restrict Howard’s proof to \(\lambda _{\rightarrow }\) because it did not prove the termination of primitive recursive functionals in the sense of strong normalization.
- 4.
Note that \(\mathcal {N}\) is mapped to an ordinal notation system up to \(\varepsilon _0\) if we do not restrict Howard’s proof to \(\lambda _{\rightarrow }\).
- 5.
Though Howard also used the constant 0, we omit it since it is not necessary and this makes the formulation of our path ordering simpler.
- 6.
- 7.
- 8.
Assignment derivations were introduced to formulate Howard-style non-unique assignments of vectors perspicuously. Note that Wilken and Weiermann not only introduced assignment derivations, but also refined Howard’s original non-unique assignments to cope with arbitrary recursor reduction rules of \(\mathsf {T}\). We restrict ourselves to \(\lambda _{\rightarrow }\) here, so we need not use the refined part by Wilken and Weiermann.
- 9.
The rule 2. below is a higher-order version of a usual rule \(\mathsf {ack}_{m+1} ( n ) \rightarrow \mathsf {ack}^{n + 1}_{m} (1)\) with \(f^0 ( k ) := k\) and \(f^{n+1} (k) := f ( f^n ( k ))\).
- 10.
As in the case of Howard’s vector system, we omit 0 from \(\mathcal {O}\).
- 11.
A similar remark can be found in [4, p. 24].
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Okada, M., Takahashi, Y. (2020). A Simplified Application of Howard’s Vector Notation System to Termination Proofs for Typed Lambda-Calculus Systems. In: Escobar, S., Martí-Oliet, N. (eds) Rewriting Logic and Its Applications. WRLA 2020. Lecture Notes in Computer Science(), vol 12328. Springer, Cham. https://doi.org/10.1007/978-3-030-63595-4_8
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