Abstract
We study languages and formal power series associated to (variants of) the Hammersley process. We show that the ordinary Hammersley process yields a regular language and the Hammersley tree process yields deterministic context-free (but non-regular) languages. For the Hammersley interval process we show that there are two relevant variants of formal languages. One of them leads to the same language as the ordinary Hammersley tree process. The other one yields non-context-free languages.
The results are motivated by the problem of studying the analog of the famous Ulam-Hammersley problem for heapable sequences. Towards this goal we also give an algorithm for computing formal power series associated to the Hammersley process. We employ this algorithm to settle the nature of the scaling constant, conjectured in previous work to be the golden ratio. Our results provide experimental support to this conjecture.
This work was supported by a grant of Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0842, within PNCDI III.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Nonrigorous computations predict that \(c_{0}=c_{2}=\frac{\sqrt{5}-1}{2}, c_{1}=\frac{3+\sqrt{5}}{2}.\).
References
Aldous, D., Diaconis, P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theor. Relat. Fields 103(2), 199–213 (1995)
Aldous, D., Diaconis, P.: Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. A.M.S. 36(4), 413–432 (1999)
Balogh, J., Bonchiş, C., Diniş, D., Istrate, G., Todincã, I.: Heapability of partial orders. arXiv preprint arXiv:1706.01230 (2017)
Basdevant, A.-L., Gerin, L., Gouéré, J.-B., Singh, A.: From Hammersley’s lines to Hammersley’s trees. Prob. Theory Related Fields 171(1–2), 1–51 (2018). https://doi.org/10.1007/s00440-017-0772-2
Basdevant, A.-L., Singh, A.: Almost-sure asymptotic for the number of heaps inside a random sequence. arXiv preprint arXiv:1702.06444 (2017)
Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications, vol. 137. Cambridge University Press, Cambridge (2011)
Byers, J., Heeringa, B., Mitzenmacher, M., Zervas, G.: Heapable sequences and subseqeuences. In: Proceedings of ANALCO 2011, pp. 33–44. SIAM Press (2011)
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)
Istrate, G., Bonchiş, C.: Partition into heapable sequences, heap tableaux and a multiset extension of Hammersley’s process. In: Cicalese, F., Porat, E., Vaccaro, U. (eds.) CPM 2015. LNCS, vol. 9133, pp. 261–271. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19929-0_22
Istrate, G., Bonchiş, C.: Heapability, interactive particle systems, partial orders: results and open problems. In: Câmpeanu, C., Manea, F., Shallit, J. (eds.) DCFS 2016. LNCS, vol. 9777, pp. 18–28. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41114-9_2
Justicz, J., Scheinerman, E.R., Winkler, P.M.: Random intervals. Am. Math. Mon. 97(10), 881–889 (1990)
Liggett, T.: Interacting Particle Systems. Springer, Heidelberg (2005). https://doi.org/10.1007/b138374
Moore, C., Lakdawala, P.: Queues, stacks and transcendentality at the transition to chaos. Physica D 135(1–2), 24–40 (2000)
Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp. 257–289. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01492-5_7
Porfilio, J.: A combinatorial characterization of heapability. Master’s thesis, Williams College, May 2015. https://unbound.williams.edu/theses/islandora/object/studenttheses%3A907. Accessed Dec 2017
Reutenauer, C.: Séries formelles et algebres syntactiques. J. Algebra 66(2), 448–483 (1980)
Romik, D.: The Surprising Mathematics of Longest Increasing Subsequences. Cambridge University Press, Cambridge (2015)
Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Springer, New York (1978). https://doi.org/10.1007/978-1-4612-6264-0
Szpankowski, W.: Average Case of Algorithms on Sequences. Wiley, New York (2001)
Welsh, D.: Complexity: Knots, Colourings and Counting. Cambridge University Press, Cambridge (1994)
Xie, H.: Grammatical Complexity and One-Dimensional Dynamical Systems. Directions in Chaos, vol. 6. World Scientific, Singapore (1996)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Bonchiş, C., Istrate, G., Rochian, V. (2018). The Language (and Series) of Hammersley-Type Processes. In: Durand-Lose, J., Verlan, S. (eds) Machines, Computations, and Universality. MCU 2018. Lecture Notes in Computer Science(), vol 10881. Springer, Cham. https://doi.org/10.1007/978-3-319-92402-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-92402-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92401-4
Online ISBN: 978-3-319-92402-1
eBook Packages: Computer ScienceComputer Science (R0)