# Parallel recognition and ranking of context-free languages

## Abstract

We survey the efficiency of the ‘fast’ parallel algorithms for the recognition and ranking of context-free languages on the Parallel Random Access Machine without write conflicts. The efficiency of the algorithm is the total number of operations (the product of time and number of processors). Such efficiency depends heavily on the class of context-free grammars and on the meaning of ‘fast’: log(n), log^{2}n or sublinear time. The slower is the algorithm the better is its total efficiency. Several new results are presented in the paper. A new simpler version of the log(n) time parallel recognition of unambiguous cfl's is presented. The parallel complexity of ranking and max-word problems for several classes of cfl's is related to the complexity of certain (⊕,⊗)-transitive closure problems, where (⊕,⊗)=(+,^{*}) for the ranking problem of unambiguous languages and (⊕,⊗)=(max,concat) for the max-word problem. This simplifies the ranking and max-word algorithms and reduces the number of processors.

## Keywords

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## Bibliography

- [AHU 74]A.Aho, J.Hopcroft, J.Ullman, The design and analysis of computer algorithms, Addison-Wesley 1974Google Scholar
- [AJ 90]C.Alvarez, B.Jenner, A very hard log space counting problem, 5th Structure in Compl. Conf. (1990) 154–168Google Scholar
- [BKR 91]L.Banachowski, A.Kreczmar, W.Rytter, Analysis of algorithms and data structures, Addison-Wesley 1991Google Scholar
- [BGS 87]A. Bertoni, M. Goldwurm, N. Sabatini. Computing the counting function of context-free languages. In Proc. of 4th STACS, LNCS 247, 1987, pages 169–179.zbMATHGoogle Scholar
- [BP 91]D. Bruschi, G. Pighizzini. The complexity of computing maximal word functions. In Proc. of FCT'91, 1991, pages 157–167.Google Scholar
- [BJLR 91]G. Buntrock, B. Jenner, K.-J. Lange, P. Rossmanith. Unambiguity and fewness for logarithmic space. In Proc. of FCT'91, 1991, pages 168–179.Google Scholar
- [BV 85]I. Bar-On, U.Vishkin, Optimal parallel generation of a computation tree form, ACM Trans. on Prog. Lan. and Systems 7:2 (1985) 348–357zbMATHCrossRefGoogle Scholar
- CM 91]M.Chytil, B.Monien, Caterpillars and context-free languages, STACS'90Google Scholar
- [CCMR 91]M.Chytil, M. Crochemore, B.Monien, W.Rytter, On the parallel recognition of unambiguous context-free languages, Theoretical Computer Science 1991Google Scholar
- [CHK 82]J.Cohen, T.Hickey, J.Katcoff, Upper bounds for speedup in parallel parsing, JACM 29:2 (1982) 408–428MathSciNetCrossRefGoogle Scholar
- [CK 85]J.Cohen, S.Kolodner, Estimating the speedup in parallel parsing, IEEE Trans. Software Eng. SE-11 (1985) 114–124Google Scholar
- [DR 87]P. Dymond, W. L. Ruzzo. Parallel RAM's with owned global memory and deterministic language recognition. 13th ICALP, pages 95–104.Google Scholar
- [DR 91]K.Diks, W.Rytter, On optimal parallel computations for sequences of brackets, TCS 87 (1991) 251–262zbMATHMathSciNetCrossRefGoogle Scholar
- [FW 78]S.Fortune, J.Wyllie, Parallelism in random access machines, in Proc. 10th ACM Symposium on Theory of Computation (1978) 114–118Google Scholar
- [GR 88]A.Gibbons, W.Rytter, Efficient parallel algorithms, Cambridge Univ. Press 1988Google Scholar
- [GR 89]A.Gibbons, W.Rytter, Optimal parallel algorithms for dynamic expression evaluation and context-free recognition, Information and Computation 81 (1989) 32–45zbMATHMathSciNetCrossRefGoogle Scholar
- [JoSk79]N.D. Jones, S. Skyum: Complexity of some problems concerning L systems, Math. Systems Theory
**13**, 29–43, 1979.zbMATHMathSciNetCrossRefGoogle Scholar - [Huy 88]Huynh, The complexity of ranking. Math. Systems Theory 23, 1–19, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
- [Kl 84]P.Klein, Parallel recognition of context free languages, B.A. thesis, Harvard University, Cambridge, MA, (1984)Google Scholar
- [Ko 75]R.Kosaraju, Speed of recognition of context free language by array automata, SIAM J.Comp. (1975) 333–340Google Scholar
- [KR 88]Klein, J.Reif, Parallel time O(log n) acceptance of deterministic cfl's on an exclusive-write PRAM, SIAM J. Comp. 17 (1987) 463–485MathSciNetCrossRefGoogle Scholar
- [Lan 90]K-J. Lange, Unambiguity of circuits. To appear in TCS.Google Scholar
- [LP 90]K-J. Lange, P. Rossmanith. Characterizing unambiguous augmented pushdown automata by circuits. MFCS '90, pages 399–406.Google Scholar
- [LR 92]L.Larmore, W.Rytter, Efficient sublinear parallel time algorithms for dynamic programming and context-free recognition, STACS'92Google Scholar
- [LRR91]K-J. Lange, P. Rossmanith, W.Rytter, On the parallel complexity of ranking and max-word problems for simple languages, manuscript, Techn. University of Munich, 1991Google Scholar
- [MF 82]R.Mattheyses, C.Fiduccia, Parsing Dyck languages on parallel machines, 20th Allerton Conf. on Communication, Control and Computing, 1982Google Scholar
- [MSR 91]B.Monien, W.Rytter, H.Schäpers, Fast recognition of cfl's with smaller number of processors. To appear in TCSGoogle Scholar
- [MR 85]G.Miller, J.Reif, Parallel tree contraction and its application, 26th FOCS, 478–489Google Scholar
- [MW 92]E.W.Meyr, R.Werchner, Optimal routing of parentheses on the hypercube, manuscript, J.W.Goethe-University, Frankfurt, 1992Google Scholar
- [PRS 91]W.Plandowski, W.Rytter, T.Szymacha, Exact analysis of three tree contraction algorithms, in FCT'91Google Scholar
- [PS 91]G.Pietsch, E. Schoemer, Optimal parallel recognition of bracket languages on hypercubes, STACS'91, 434–443Google Scholar
- [Re 85]J.Reif, O(log n) time parallel acceptance of deterministic context free languages, FOCS (1985)Google Scholar
- [RG 87]W.Rytter, R.Giancarlo, Optimal parallel parsing of bracket languages, TCS 53 (1987) 295–306zbMATHMathSciNetCrossRefGoogle Scholar
- [RS 92]W.Rytter, A.Saoudi, Parallel recognition of two-dimensional languages, International Conf. on Pattern Recognition, Delft, 1992Google Scholar
- [RS 91]W.Rytter, A.Saoudi, On the complexity of parallel recognition of 2D-images, IPL 38 (1991) 225–229zbMATHMathSciNetCrossRefGoogle Scholar
- [Ry 87a]W.Rytter, On the complexity of parallel parsing of general context free languages, TCS 47 (19870 315–322Google Scholar
- [Ry 85a]W.Rytter, The complexity of two-way pushdown automata and recursive programs, in Combinatorial algorithms on words (ed. A.Apostolico, Z.Galil), Springer-Verlag 1985Google Scholar
- [Ry 87b]W.Rytter, Parallel time O(log n) recognition of unambiguous cfl's. Information and Computation 73, (1987), 75–86zbMATHMathSciNetCrossRefGoogle Scholar
- [Ry 88]W.Rytter, Efficient parallel computations of the cost of paths on a grid graph, Inf.Proc.Letters (1988)Google Scholar
- [Ry 90]W. Rytter:, On parallel evaluation of expressions and straight line programs, Computers and Artificial Intelligence 9 (1990) 427–431zbMATHMathSciNetGoogle Scholar
- [Ry 85b]W.Rytter, On the recognition of context-free languages, in Fifth Symposium on Computation Theory, Lecture Notes in Computer Science 208 (1985) 318–325zbMATHMathSciNetGoogle Scholar
- [Ry 89]W.Rytter, Fast parallel computations for some dynamic programming problems, TCS 1989Google Scholar
- [Ry 88]W.Rytter, A note on parallel transformations of regular expressions to nondeterministic finite automata, IPL (1988)Google Scholar
- [Ru 80]W.Ruzzo, Tree-size bounded alternation, JCSS 21 (1980) 218–235zbMATHMathSciNetGoogle Scholar
- [SB 89]D.Skillicorn, D.Barnard, Parallel parsing on the connection machine, IPL 31 (1989)111–117zbMATHCrossRefGoogle Scholar
- [SS 87]Y.Srikant, P.Shankar, Parallel parsing of programming languages, Inf. Sciences 43 (1987) 55–83zbMATHMathSciNetCrossRefGoogle Scholar
- [Sud77]I.H. Sudborough: The time and tape complexity of developmental languages, In Proc. 4th ICALP, Springer LNCS
**52**, 509–521, 1977.zbMATHMathSciNetGoogle Scholar - [Vi 91]V.Vinay, Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits, 6th Structure in Compl. Conf. (1991) 270–284Google Scholar