Dynamic Algorithms for the Dyck Languages

We study Dynamic Membership problems for the Dyck languages, the class of strings of properly balanced parentheses. We also study the Dynamic Word problem for the free group. We present deterministic algorithms and data structures which maintain a string under replacements of symbols, insertions, and deletions of symbols, and language membership queries. Updates and queries are handled in polylogarithmic time. We also give both Las Vegas- and Monte Carlo-type randomised algorithms to achieve better running times, and present lower bounds on the complexity for variants of the problems.

[1]  Michael E. Saks,et al.  The cell probe complexity of dynamic data structures , 1989, STOC '89.

[2]  Kim G. Larsen,et al.  Automatic Synthesis of Real Time Systems , 1994, ICALP.

[3]  Mark H. Overmars,et al.  The Design of Dynamic Data Structures , 1987, Lecture Notes in Computer Science.

[4]  L. Aceto.  CPO Models for GSOS Languages - Part I: Compact GSOS Languages , 1994 .

[5]  Paul F. Dietz.  Optimal Algorithms for List Indexing and Subset Rank , 1989, WADS.

[6]  Michael L. Fredman,et al.  The Complexity of Maintaining an Array and Computing Its Partial Sums , 1982, JACM.

[7]  Peter Bro Miltersen.  Lower bounds for union-split-find related problems on random access machines , 1994, STOC '94.

[8]  Kim G. Larsen,et al.  A Constraint Oriented Proof Methodology Based on Modal Transition Systems , 1994, TACAS.

[9]  Leonidas J. Guibas,et al.  A dichromatic framework for balanced trees , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[10]  Peter Bro Miltersen,et al.  Dynamic Algorithms for the Dyck Languages , 1995 .

[11]  Richard M. Karp,et al.  Efficient Randomized Pattern-Matching Algorithms , 1987, IBM J. Res. Dev..

[12]  Luca Aceto,et al.  A Complete Axiomatization of Timed Bisimulation for a Class of Timed Regular Behaviours (Revised Version) , 1994 .

[13]  Richard J. Lipton,et al.  Word Problems Solvable in Logspace , 1977, JACM.

[14]  Michael A. Harrison,et al.  Introduction to formal language theory , 1978 .

[15]  Peter Bro Miltersen,et al.  On data structures and asymmetric communication complexity , 1994, STOC '95.

[16]  Kurt Mehlhorn,et al.  Maintaining dynamic sequences under equality tests in polylogarithmic time , 1994, SODA '94.

[17]  Sten Agerholm,et al.  A HOL Basis for Reasoning about Functional Programs , 1994 .

[18]  Frederick N. Springsteel,et al.  Language Recognition by Marking Automata , 1972, Inf. Control..

[19]  Neil Immerman,et al.  Dyn-FO: A Parallel, Dynamic Complexity Class , 1997, J. Comput. Syst. Sci..

[20]  Dany Breslauer,et al.  Efficient String Matching on Coded Texts , 1994, CPM.

[21]  Kim G. Larsen,et al.  Synthesizing Distinguishing Formulae for Real Time Systems , 1994, Nord. J. Comput..

[22]  W. Magnus,et al.  Combinatorial Group Theory: COMBINATORIAL GROUP THEORY , 1967 .