Skip to main content Accessibility help
×
Home
Hostname: page-component-768dbb666b-k9l4c Total loading time: 0.41 Render date: 2023-02-05T12:28:21.714Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Clocked lambda calculus

Published online by Cambridge University Press:  15 October 2015

JÖRG ENDRULLIS
Affiliation:
Department of Computer Science, VU University Amsterdam Email: j.endrullis@vu.nl, r.d.a.hendriks@vu.nl, j.w.klop@vu.nl, a.polonsky@vu.nl
DIMITRI HENDRIKS
Affiliation:
Department of Computer Science, VU University Amsterdam Email: j.endrullis@vu.nl, r.d.a.hendriks@vu.nl, j.w.klop@vu.nl, a.polonsky@vu.nl
JAN WILLEM KLOP
Affiliation:
Department of Computer Science, VU University Amsterdam Email: j.endrullis@vu.nl, r.d.a.hendriks@vu.nl, j.w.klop@vu.nl, a.polonsky@vu.nl
ANDREW POLONSKY
Affiliation:
Department of Computer Science, VU University Amsterdam Email: j.endrullis@vu.nl, r.d.a.hendriks@vu.nl, j.w.klop@vu.nl, a.polonsky@vu.nl

Abstract

One of the best-known methods for discriminating λ-terms with respect to β-convertibility is due to Corrado Böhm. The idea is to compute the infinitary normal form of a λ-term M, the Böhm Tree (BT) of M. If λ-terms M, N have distinct BTs, then MβN, that is, M and N are not β-convertible. But what if their BTs coincide? For example, all fixed point combinators (FPCs) have the same BT, namely λx.x(x(x(. . .))).

We introduce a clocked λ-calculus, an extension of the classical λ-calculus with a unary symbol τ used to witness the β-steps needed in the normalization to the BT. This extension is infinitary strongly normalizing, infinitary confluent and the unique infinitary normal forms constitute enriched BTs, which we call clocked BTs. These are suitable for discriminating a rich class of λ-terms having the same BTs, including the well-known sequence of Böhm's FPCs.

We further increase the discrimination power in two directions. First, by a refinement of the calculus: the atomic clocked λ-calculus, where we employ symbols τp that also witness the (relative) positions p of the β-steps. Second, by employing a localized version of the (atomic) clocked BTs that has even more discriminating power.

Type
Paper
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We dedicate our paper to Corrado Böhm in honour of his 90th birthday, in gratitude and admiration.

References

Abramsky, S. and Ong, C.-H. (1993). Full abstraction in the lazy lambda calculus. Information and Computation 105 (2) 159267.CrossRefGoogle Scholar
Aehlig, K. and Joachimski, F. (2002). On continuous normalization. In: Proceedings Workshop on Computer Science Logic (CSL 2002). Springer Lecture Notes in Computer Science 2471 5973.CrossRefGoogle Scholar
Barendregt, H. (1984). Studies in Logic and the Foundations of Mathematics. The Lambda Calculus. Its Syntax and Semantics, vol 103, North-Holland.Google Scholar
Barendregt, H. and Klop, J. (2009). Applications of infinitary lambda calculus. Information and Computation 207 (5) 559582.CrossRefGoogle Scholar
Berarducci, A. (1996). Infinite λ-calculus and non-sensible models. In: Logic and Algebra (Pontignano, 1994), Dekker, New York, 339377.Google Scholar
Bethke, I. (2003). Lambda calculus. Chapter 10 in Terese (2003).Google Scholar
Bethke, I., Klop, J. and de Vrijer, R. (2000). Descendants and origins in term rewriting. Information and Computation 159 (1–2) 59124.CrossRefGoogle Scholar
Coquand, T. (1994). Infinite objects in type theory. In: Barendregt, H. and Nipkow, T. (eds.) TYPES, vol 806, Springer–Verlag, Berlin, 6278.Google Scholar
Coquand, T. and Herbelin, H. (1994). A-translation and looping combinators in pure type systems. Journal of Functional Programming 4 (1) 7788.CrossRefGoogle Scholar
Endrullis, J., Hendriks, D. and Klop, J. (2010). Modular construction of fixed point combinators and clocked Böhm trees. In: Proceedings Symposium on Logic in Computer Science (LICS 2010) 111–119.Google Scholar
Endrullis, J., Hendriks, D., Klop, J. W. and Polonsky, A. (2014). Discriminating Lambda-terms using clocked Böhm trees. Logical Methods in Computer Science 10 (2). doi: 10.2168/LMCS-10(2:4)2014.CrossRefGoogle Scholar
Faustini, A. (1982). The Equivalence of an Operational and a Denotational Semantics for Pure Dataflow, Ph.D. thesis, University of Warwick.Google Scholar
Geuvers, H. and Werner, B. (1994). On the Church–Rosser property for expressive type systems and its consequences for their metatheoretic study. In: Proceedings Symposium on Logic in Computer Science (LICS 1994) 320–329.Google Scholar
Intrigila, B. (1997). Non-existent Statman's double fixed point combinator does not exist, indeed. Information and Computation 137 (1) 3540.CrossRefGoogle Scholar
Kennaway, R., Klop, J., Sleep, M. and de Vries, F.-J. (1997). Infinitary lambda calculus. Theoretic Computer Science 175 (1) 93125.CrossRefGoogle Scholar
Ketema, J. and Simonsen, J. (2009). Infinitary combinatory reduction systems: Confluence. Logical Methods in Computer Science 5 (4) 129.CrossRefGoogle Scholar
Klop, J. (2007). New fixed point combinators from old. In: Reflections on Type Theory, λ-Calculus, and the Mind. Essays Dedicated to Henk Barendregt on the Occasion of his 60th Birthday 197–210. Online version: http://www.cs.ru.nl/barendregt60.Google Scholar
Klop, J. and de Vrijer, R. (2005). Infinitary normalization. In: We Will Show Them: Essays in Honour of Dov Gabbay vol 2, 169192. College Publ. Techn. report: http://www.cwi.nl/ftp/CWIreports/SEN/SEN-R0516.pdf.Google Scholar
Matthews, S. (1985). Metric Domains for Completeness, Ph.D. thesis, University of Warwick.Google Scholar
McCune, W. and Wos, L. (1991). The absence and the presence of fixed point combinators. Theoretic Compututer Science 87 (1) 221228.CrossRefGoogle Scholar
Naoi, T. and Inagaki, Y. (1989). Algebraic semantics and complexity of term rewriting systems. In: Proceedings of the Conference on Rewriting Techniques and Applications (RTA 1989). Springer Lecture Notes in Computer Science 355 311325.CrossRefGoogle Scholar
Park, D. (1983). The fairness problem and nondeterministic computing networks. Foundations of Computer Science IV, Distributed Systems: Part 2 (159) 133161.Google Scholar
Plotkin, G. (1977). Lcf considered as a programming language. Theoretical Computer Science 5 (3) 223255.CrossRefGoogle Scholar
Plotkin, G. (2007). Personal communication at the symposium for H. Barendregt's 60th birthday.Google Scholar
Sangiorgi, D. and Rutten, J. (2012). Advanced Topics in Bisimulation and Coinduction, Cambridge Tracts in Theoretical Computer Science vol 52, Cambridge University Press.Google Scholar
Smullyan, R. (1985). To Mock a Mockingbird, and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic, Knopf, New York.Google Scholar
Terese (2003). Term Rewriting Systems, Cambridge Tracts in Theoretical Computer Science vol 55, Cambridge University Press.Google Scholar
Wadge, W. (1981). An extensional treatment of dataflow deadlock. Theoretical Computer Science 13 315.CrossRefGoogle Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Clocked lambda calculus
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Clocked lambda calculus
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Clocked lambda calculus
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *