The Membership Problem for Regular Expressions with Intersection Is Complete in LOGCFL

We show that the recognition problem of context-free languages can be reduced to membership in the language defined by a regular expression with intersection by a log space reduction with linear output length. We also show a matching upper bound improving the known fact that the membership problem for these regular expressions is in NC2. Together these results establish that the membership problem is complete in LOGCFL. For unary expressions we show hardness for the class NL and some related results.

[1]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[2]  Róbert Szelepcsényi,et al.  The method of forced enumeration for nondeterministic automata , 1988, Acta Informatica.

[3]  Hiroaki Yamamoto.  An Automata-Based Recognition Algorithm for Semi-extended Regular Expressions , 2000, MFCS.

[4]  Ivan Hal Sudborough,et al.  On the Tape Complexity of Deterministic Context-Free Languages , 1978, JACM.

[5]  Dexter Kozen,et al.  Lower bounds for natural proof systems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[6]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[7]  Neil Immerman.  Nondeterministic Space is Closed Under Complementation , 1988, SIAM J. Comput..

[8]  Markus Lohrey.  On the Parallel Complexity of Tree Automata , 2001, RTA.

[9]  Martin Fürer.  The Complexity of the Inequivalence Problem for Regular Expressions with Intersection , 1980, ICALP.

[10]  John Michael Robson.  The Emptiness of Complement Problem for Semi Extended Regular Expressions Requires cn Space , 1979, Inf. Process. Lett..

[11]  Sheila A. Greibach,et al.  The Hardest Context-Free Language , 1973, SIAM J. Comput..

[12]  Alfred V. Aho,et al.  Algorithms for Finding Patterns in Strings , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[13]  H. James Hoover,et al.  Limits to Parallel Computation: P-Completeness Theory , 1995 .

[14]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[15]  Eugene W. Myers,et al.  A Four Russians algorithm for regular expression pattern matching , 1992, JACM.

[16]  Tao Jiang,et al.  A Note on the Space Complexity of Some Decision Problems for Finite Automata , 1991, Inf. Process. Lett..

[17]  Harry B. Hunt,et al.  The Equivalence Problem for Regular Expressions with Intersection is Not Polynomial in Tape , 1973 .

[18]  Albert R. Meyer,et al.  Word problems requiring exponential time(Preliminary Report) , 1973, STOC.

[19]  Ken Thompson,et al.  Programming Techniques: Regular expression search algorithm , 1968, Commun. ACM.

[20]  Allan Borodin,et al.  Two Applications of Inductive Counting for Complementation Problems , 1989, SIAM J. Comput..