TLDR; This tutorial is a complete implementation of CYK Parser in Python1 The Python interpreter is embedded so that you can work through the implementation steps.

A CYK parser is a general context-free parser. It uses dynamic # programming to fill in a chart. Unlike Earley, GLL and GLR parsers, it requires the grammar to be in the Chomsky normal form, which allows at most two symbols on the right hand side of any production. In particular, all the rules have to conform to

\[A -> BC\] \[A -> a\] \[S -> \epsilon\]

Where A,B, and C are nonterminal symbols, a is a terminal symbol, S is the start symbol, and \(\epsilon\) is the empty string.

We previously discussed Earley parser which is a general context-free parser. CYK parser is another general context-free parser that is capable of parsing strings that conform to any given context-free grammar.

Similar to Earley, GLR, GLL, and other general context-free parsers, the worst case for CYK parsing is \(O(n^3)\) . However, unlike those parsers, the best case is also \(O(n^3)\) for all grammars. A peculiarity of CYK parser is that unlike Earley, GLL, and GLR, it is not a left-to-right parser. In this respect, the best known other parser that is not left-to-right is Ungar’s parser. Furthermore, it is a bottom-up parser that builds substrings of fixed length from the bottom up.


import cykparser as P
my_grammar = {'<start>': [['1', '<A>'],
              '<A>'    : [['a']]}
my_parser = P.CYKParser(my_grammar)
assert my_parser.recognize_on(text='1a', start_symbol='<start>'):
for tree in my_parser.parse_on(text='1a', start_symbol='<start>'):


  1. Synopsis
  2. Definitions
    1. Prerequisites
  3. CYKParser
  4. Artifacts

Important: Pyodide takes time to initialize. Initialization completion is indicated by a red border around Run all button.


For this post, we use the following terms:

  • The alphabet is the set all of symbols in the input language. For example, in this post, we use all ASCII characters as alphabet.

  • A terminal is a single alphabet symbol. Note that this is slightly different from usual definitions (done here for ease of parsing). (Usually a terminal is a contiguous sequence of symbols from the alphabet. However, both kinds of grammars have a one to one correspondence, and can be converted easily.)

    For example, x is a terminal symbol.

  • A nonterminal is a symbol outside the alphabet whose expansion is defined in the grammar using rules for expansion.

    For example, <term> is a nonterminal in the below grammar.

  • A rule is a finite sequence of terms (two types of terms: terminals and nonterminals) that describe an expansion of a given terminal. A rule is also called an alternative expansion.

    For example, [<term>+<expr>] is one of the expansion rules of the nonterminal <expr>.

  • A production is a combination of a nonterminal and one of its productions.

    For example <expr>: [<term>+<expr>] is a production.

  • A definition is a set of rules that describe the expansion of a given nonterminal.

    For example, [[<digit>,<digits>],[<digit>]] is the definition of the nonterminal <digits>

  • A context-free grammar is composed of a set of nonterminals and corresponding definitions that define the structure of the nonterminal.

    The grammar given below is an example context-free grammar.

  • A terminal derives a string if the string contains only the symbols in the terminal. A nonterminal derives a string if the corresponding definition derives the string. A definition derives the string if one of the rules in the definition derives the string. A rule derives a string if the sequence of terms that make up the rule can derive the string, deriving one substring after another contiguously (also called parsing).

  • A derivation tree is an ordered tree that describes how an input string is derived by the given start symbol. Also called a parse tree.
  • A derivation tree can be collapsed into its string equivalent. Such a string can be parsed again by the nonterminal at the root node of the derivation tree such that at least one of the resulting derivation trees would be the same as the one we started with.

  • The yield of a tree is the string resulting from collapsing that tree.

  • An epsilon rule matches an empty string.


As before, we start with the prerequisite imports.

Available Packages These are packages that refer either to my previous posts or to pure python packages that I have compiled, and is available in the below locations. As before, install them if you need to run the program directly on the machine. To install, simply download the wheel file (`pkg.whl`) and install using `pip install pkg.whl`.
  1. simplefuzzer-0.0.1-py2.py3-none-any.whl from "The simplest grammar fuzzer in the world".
  2. earleyparser-0.0.1-py2.py3-none-any.whl from "Earley Parser".

We need the fuzzer to generate inputs to parse and also to provide some utilities

We use the display_tree() method in earley parser for displaying trees.

We use the random choice to extract derivation trees from the parse forest.

As before, we use the fuzzingbook grammar style. A terminal symbol has exactly one character (Note that we disallow empty string ('') as a terminal symbol).

Let us start with the following grammar.

We initialize our parser with the grammar, and identify the terminal and nonterminal productions separately. termiinal productions are those that are of the form <A> ::= a where a is a terminal symbol. Nonterminal productions are of the form <A> ::= <B><C> where <B> and <C> are nonterminal symbols.

Next, we define the recognizer. The idea here is that CYK algorithm formulates the recognition problem as a dynamic problem where the parse of a string of length n using a nonterminal <A> which is defined as <A> ::= <B> <C> is defined as a parse of the substring 0..x with the nonterminal <B> and the parse of the substring x..n with nonterminal <C> where 0 < x < n. That is, <A> parses the string if there exists such a parse with <B> and <C> for some x. We first initialize the matrix that holds the results. The cell[i][j] represents the nonterminals that can parse the substring text[i..j]

Let us define a printing routine.

Using it

Next, we define the base case, which is when a single input token matches one of the terminal symbols. In that case, we look at each character in the input, and for each i in the input we identify cell[i][i+1] and add the nonterminal symbol that derives the corresponding token.

Using it

Next, we define the multi-token parse. The idea is to build the table incrementally. We have already indicated in the table which nonterminals parse a single token. Next, we find all nonterminals that parse two tokens, then using that, we find all nonterminals that can parse three tokens etc.

Using it for example, on substrings of length 2

For length 3

For length 4

We combine everything together. At the end, we check if the start_symbol can parse the given tokens by checking (0, n) in the table.

Using it

At this point, we have a recognizer for a grammar in CNF form. However, this is a bit unsatisfying because the CNF form is not very userfriendly. In particluar it lacks two conveniences we are used to in context-free gramamrs

  1. Having more than two tokens in a rule
  2. The ability to add epsilon rules.

The first one is not very difficult to solve. Given an expansion rule for a nonterminal

<nt> ::= <a> <b> <c> <d>

we can always rewrite this as

<nt> ::= <a> <nt_1_1>
<nt_1_1> ::= <b> <nt_1_2>
<nt_1_2> ::= <c> <d>

and so on. We can also recover the structure back from any parse tree by combining the corresponding tokens. The second restriction is more difficult Having to factor out epsilon can change the grammar completely. Turns out, it is not very difficult to incorporate epsilons into this parser.

First, we extract all nullable nonterminals of our grammar. (See Earley parser)

Let us test this out


Once we have the nullable grammar, for each nonterminal in our grammar, we want to identify whether parsinga string with one nonterminal guarantees parse of a parent nonterminal. This is because if parses a string s1, and there exists a rule <A> := <B> <C> and is nullable, then parsing with guarantees that also can parse it. So, this is what we will do.

A grammar. Note that we use <> to represent empty (epsilon) nonterminal


So at this point, all we have to do is, after each cell is computed fully, we just have to extend the cell with the guaranteed parse.


Handling epsilons now allows us to get to the next step in supporting any context free grammar. We require two more steps to support.

  1. Replacing terminal symbols with nonterminal symbols representing tokens
  2. Ensuring that there are exactly two nonterminal symbols in any nonterminal rule. We will handle the first one now.


Next, we want to replace any rule that contains more than two tokens with it decomposition. [] = [] [t1] = [t1] [t1, t2] = [t1, t2] [t1, t2, t3] = [t1, _t2], _t2 = [t2, t3]


decompose grammar

all that remains now is to ensure that each rule is exactly two token nonterminal, or a single token terminal.

connecting everything together

A grammar




Now, all we need to do is to add trees. Unlike GLL, GLR, and Earley, due to restricting epsilons to the start symbol, there are no infinite parse trees.

The parse_1 for terminal symbols

The substring parse


Using it (uses random choice, click run multiple times to get other trees).

assign display_tree


The runnable Python source for this notebook is available here.

The installable python wheel cykparser is available here.

  1. Dick Grune and Ceriel J.H. Jacobs “Parsing Techniques A Practical Guide” 2008