Implementation of algorithm from "General Context-Free Recognition in Less than Cubic Time" by Leslie G. Valiant (1975.)
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Valiant's parsing algorithm

... is the asymptotically fastest known parsing algorithm.

Algorithm goes as follows:

  1. Convert grammar to Chomsky normal form.
  2. Build a Cocke–Younger–Kasami-like (CYK) parsing matrix. Define semiring operations on sets of nonterminals (clauses).
  3. Unlike CYK, compute closure using Valiant's divide-and-conquer algorithm.
  4. Use binary (boolean) matrix multiplication.

Things inside

  • matrix and matrix_view generic structures. Both expect certain operations to be defined on used element type. Inside: matrix.h.
  • grammar structure, representing a context-free grammar. Can convert itself to CNF. Doesn't support epsilons/blanks. Can be loaded from a file (see sample_input.txt), Inside: grammar.h, grammar.cpp.
  • bunch of functions implementing Valiant's algorithm. See main() for usage. Inside: main.cpp.

Current implementation isn't using the fastest known matrix multiplication, so the graph looks like this (loglog). Graph valiant


On Unix-based systems:

$ git clone
$ cd valiant-parsing
$ g++ *.cpp -std=c++11 -o valiant

On Windows, use any modern C++ compiler with "-std=c++11" or equivalent compiler flag.



Evaluating (0+1) and (1*(0+0)) in BA:

$. echo "(0+1) (1*(0+0))" | ./valiant -g boolean_algebra.txt

Search for closed algebraic expressions of BA evaluating to 1, up to length 5, by iterating all words over alphabet:

$ ./valiant -g boolean_algebra.txt -language 5 -skip_interactive

Full syntax

$ ./valiant [-g <grammar_file_address>] [-language <inclusive_maximal_length>] [-skip_interactive] [-bmm] [-show_grammar] [-show_table]


  1. -g <grammar_file_address> specifies grammar. See below for details on this file's format. Default value: boolean_algebra.txt
  2. -language <inclusive_maximal_length> will enumerate all words over alphabet of up to given length (inclusive).
  3. -skip_interactive will skip waiting for STDIN. Without using this parameter, waiting can still be skipped later by pressing Ctrl + D while in interactive mode.
  4. -bmm will use binary matrix multiplication. This won't improve the algorithm in any way, but won't make it asymptotically worse either. It will make the calculation slower though. Implemented only because Valiant used it in the article.
  5. -show_grammar will display loaded grammar before and after conversion to Chomsky normal form.
  6. -show_table will display final (CYK-based) parsing matrix.

Only the first argument is mandatory.


Strings to test can be entered at runtime. Use Ctrl + D to skip interactive mode. Use Ctrl + C to exit. Otherwise, write all the strings in a file, separated by whitespace.

Output is a list of 0's and 1's, ith number representing whether ith input string belongs to language. Output is given one character per line.

Grammar file


See boolean_algebra.txt for a larger example. New lines are optional.

Grammar (V, T, P, S) = ({S, A, B}, {a, b}, {S -> AB, A -> AB | aB, B .> b}, S) is given by

a b
S -> AB
A -> AB | aB
B -> b

Full syntax

Grammar file consists of:

  1. Whitespace-separated list of terminal symbols. Terminal symbol is any non-whitespace ASCII byte, followed by
  2. the symbol #, a whitespace and then:
  3. Whitespace-separated list of nonterminal symbols. Nonterminal symbol is any non-whitespace ASCII byte that wasn't already used. The list should probably include S, the root nonterminal symbol. Then:
  4. the symbol #, a whitespace and then:
  5. Whitespace-separated list of productions, where production is:
    1. A left-hand-side, which is any previosly declared nonterminal symbol.
    2. A right-hand-side, which is one of the following:
      1. A chain of symbols, each of which is either a terminal or a nonterminal.
      2. A right-hand-side, followed by whitespace, symbol |, whitespace, then another right-hand-side.