Abstract
We investigate the notion of ‘infinitary strong normalization’ (SN ∞ ), introduced in [6], the analogue of termination when rewriting infinite terms. A (possibly infinite) term is SN ∞ if along every rewrite sequence each fixed position is rewritten only finitely often. In [9], SN ∞ has been investigated as a system-wide property, i.e. SN ∞ for all terms of a given rewrite system. This global property frequently fails for trivial reasons. For example, in the presence of the collapsing rule tail(x:σ)→σ, the infinite term t =tail(0:t) rewrites to itself only. Moreover, in practice one usually is interested in SN ∞ of a certain set of initial terms. We give a complete characterization of this (more general) ‘local version’ of SN ∞ using interpretations into weakly monotone algebras (as employed in [9]). Actually, we strengthen this notion to continuous weakly monotone algebras (somewhat akin to [5]). We show that tree automata can be used as an automatable instance of our framework; an actual implementation is made available along with this paper.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 133–178 (2000)
Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications (2007), http://www.grappa.univ-lille3.fr/tata
Endrullis, J., Grabmayer, C., Hendriks, D., Isihara, A., Klop, J.W.: Productivity of Stream Definitions. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) FCT 2007. LNCS, vol. 4639, pp. 274–287. Springer, Heidelberg (2007)
Endrullis, J., Waldmann, J., Zantema, H.: Matrix Interpretations for Proving Termination of Term Rewriting. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 574–588. Springer, Heidelberg (2006)
Goguen, J.A., Thatcher, J.W., Wagner, E.G., Wright, J.B.: Initial Algebra Semantics and Continuous Algebras. JACM 24(1), 68–95 (1977)
Klop, J.W., de Vrijer, R.C.: Infinitary Normalization. In: Artemov, S., Barringer, H., d’Avila Garcez, A.S., Lamb, L.C., Woods, J. (eds.) We Will Show Them: Essays in Honour of Dov Gabbay, vol. 2, pp. 169–192. College Publ. (2005)
Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)
Zantema, H.: Termination of Term Rewriting: Interpretation and Type Elimination. Journal of Symbolic Computation 17, 23–50 (1994)
Zantema, H.: Normalization of Infinite Terms. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 441–455. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Endrullis, J., Grabmayer, C., Hendriks, D., Klop, J.W., de Vrijer, R. (2009). Proving Infinitary Normalization. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds) Types for Proofs and Programs. TYPES 2008. Lecture Notes in Computer Science, vol 5497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02444-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-02444-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02443-6
Online ISBN: 978-3-642-02444-3
eBook Packages: Computer ScienceComputer Science (R0)