FSM/regex conversion library
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greenery
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LICENCE
README.md
main.py
module map.txt
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README.md

greenery

Tools for parsing and manipulating regular expressions (greenery.lego), for producing finite-state machines (greenery.fsm), and for freely converting between the two. Python 3 only.

This project was undertaken because I wanted to be able to compute the intersection between two regular expressions. The "intersection" is the set of strings which both regexes will accept, represented as a third regular expression.

Example

>>> from greenery.lego import parse
>>> print(parse("abc...") & parse("...def"))
abcdef
>>> print(parse("\d{4}-\d{2}-\d{2}") & parse("19.*"))
19\d{2}-\d{2}-\d{2}
>>> print(parse("\W*") & parse("[a-g0-8$%\^]+") & parse("[^d]{2,8}"))
[$%\^]{2,8}
>>> print(parse("[bc]*[ab]*") & parse("[ab]*[bc]*"))
([ab]*a|[bc]*c)?b*
>>> print(parse("a*") & parse("b*"))

>>> print(parse("a") & parse("b"))
[]

In the penultimate example, the empty string is returned, because only the empty string is in both of the regular languages a* and b*. In the final example, an empty character class has been returned. An empty character class can never match anything, which means that this is the smallest representation of a regular expression which matches no strings at all. (Note that this is different from only matching the empty string.)

greenery works by converting both regexes to finite state machines, computing the intersection of the two FSMs as a third FSM, and converting the third FSM back to a regex.

As such, greenery is divided into two libraries:

greenery.fsm

This module provides for the creation and manipulation of deterministic finite state machines.

Example

To do: a slightly more impressive example.

>>> from greenery import fsm
>>> a = fsm.fsm(
...     alphabet = {"a", "b"},
...     states   = {0, 1},
...     initial  = 0,
...     finals   = {1},
...     map      = {
...             0 : {"a" : 1},
...     },
... )
>>> print(a)
  name final? a b
------------------
* 0    False  1
  1    True
>>> a.accepts([])
False
>>> a.accepts(["a"])
True
>>> a.accepts(["b"])
False
>>> print(a.accepts(["c"]))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "fsm.py", line 68, in accepts
    state = self.map[state][symbol]
KeyError: 'c'

Functions in this module

fsm(alphabet, states, initial, finals, map)

Constructor for an fsm object, as demonstrated above. fsm objects are intended to be immutable.

map may be sparse. If a transition is missing from map, then it is assumed that this transition leads to an undocumented "oblivion state" which is not final. This oblivion state does not appear when the FSM is printed out.

Ordinarily, you may only feed known alphabet symbols into the FSM. Any other symbol will result in an exception, as seen above. However, if you add the special symbol fsm.anything_else to your alphabet, then any unrecognised symbol will be automatically converted into fsm.anything_else before following whatever transition you have specified for this symbol.

crawl(alphabet, initial, final, follow)

Crawl what is assumed to be an FSM and return a new fsm object representing it. Starts at state initial. At any given state, crawl calls final(state) to determine whether it is final. Then, for each symbol in alphabet, it calls follow(state, symbol) to try to discover new states. Obviously this procedure could go on for ever if your implementation of follow is faulty. follow may also throw an OblivionError to indicate that you have reached an inescapable, non-final "oblivion state"; in this case, the transition will be omitted from the resulting FSM.

null(alphabet)

Returns an FSM over the supplied alphabet which accepts no strings at all.

epsilon(alphabet)

Returns an FSM over the supplied alphabet which accepts only the empty string, "".

Methods on class fsm

An FSM accepts a possibly-infinite set of strings. With this in mind, fsm implements numerous methods like those on frozenset, as well as many FSM-specific methods. FSMs are immutable.

Method Behaviour
fsm1.accepts("a")
"a" in fsm1
Returns True or False or throws an exception if the string contains a symbol which is not in the FSM's alphabet. The string should be an iterable of symbols.
fsm1.strings()
for string in fsm1
Returns a generator of all the strings that this FSM accepts.
fsm1.empty() Returns True if this FSM accepts no strings, otherwise False.
fsm1.cardinality()
len(fsm1)
Returns the number of strings which the FSM accepts. Throws an OverflowError if this number is infinite.
fsm1.equivalent(fsm2)
fsm1 == fsm2
Returns True if the two FSMs accept exactly the same strings, otherwise False.
fsm1.different(fsm2)
fsm1 != fsm2
Returns True if the FSMs accept different strings, otherwise False.
fsm1.issubset(fsm2)
fsm1 <= fsm2
Returns True if the set of strings accepted by fsm1 is a subset of those accepted by fsm2, otherwise False.
fsm1.ispropersubset(fsm2)
fsm1 < fsm2
Returns True if the set of strings accepted by fsm1 is a proper subset of those accepted by fsm2, otherwise False.
fsm1.issuperset(fsm2)
fsm1 >= fsm2
Returns True if the set of strings accepted by fsm1 is a superset of those accepted by fsm2, otherwise False.
fsm1.ispropersuperset(fsm2)
fsm1 > fsm2
Returns True if the set of strings accepted by fsm1 is a proper superset of those accepted by fsm2, otherwise False.
fsm1.isdisjoint(fsm2) Returns True if the set of strings accepted by fsm1 is disjoint from those accepted by fsm2, otherwise False.
fsm1.copy() Returns a copy of fsm1.
fsm1.reduce() Returns an FSM which accepts exactly the same strings as fsm1 but has a minimal number of states.
fsm1.star() Returns a new FSM which is the Kleene star closure of the original. For example, if fsm1 accepts only "asdf", fsm1.star() accepts "", "asdf", "asdfasdf", "asdfasdfasdf", and so on.
fsm1.everythingbut() Returns an FSM which accepts every string not accepted by the original. x.everythingbut().everythingbut() accepts the same strings as x for all fsm objects x, but is not necessarily mechanically identical.
fsm1.reversed()
reversed(fsm1)
Returns a reversed FSM. For each string that fsm1 accepted, reversed(fsm1) will accept the reversed string. reversed(reversed(x)) accepts the same strings as x for all fsm objects x, but is not necessarily mechanically identical.
fsm1.times(7)
fsm1 * 7
Essentially, this is repeated self-concatenation. If fsm1 only accepts "z", fsm2 only accepts "zzzzzzz".
fsm1.concatenate(fsm2, ...)
fsm1 + fsm2 + ...
Returns the concatenation of the FSMs. If fsm1 accepts all strings in A and fsm2 accepts all strings in B, then fsm1 + fsm2 accepts all strings of the form a·b where a is in A and b is in B.
fsm1.union(fsm2, ...)
`fsm1
fsm2
fsm1.intersection(fsm2, ...)
fsm1 & fsm2 & ...
Returns an FSM accepting any string accepted by all input FSMs.
fsm1.difference(fsm2, ...)
fsm1 - fsm2 - ...
Subtract the set of strings accepted by fsm2 onwards from those accepted by fsm1 and return the resulting new FSM.
fsm1.symmetric_difference(fsm2, ...)
fsm1 ^ fsm2 ^ ...
Returns an FSM accepting any string accepted by fsm1 or fsm2 but not both.
fsm1.derive("a") Return the Brzozowski derivative of the original FSM with respect to the input string. E.g. if fsm1 only accepts "ab" or "ac+", returns an FSM only accepting "b" or "c+".

greenery.lego

This module provides methods for parsing a regular expression (i.e. a string) into a manipulable nested data structure, and for manipulating that data structure.

Note that this is an entirely different concept from that of simply creating and using those regexes, functionality which is present in basically every programming language in the world, Python included.

This module requires greenery.fsm in order to carry out many of its most important functions. (greenery.fsm, in comparison, is completely standalone.)

Classes in this module

lego.bound

A non-negative integer, or inf, plus a bunch of arithmetic methods which make it possible to compare, add and multiply them.

Special bounds
  • inf

lego.multiplier

A combination of a finite lower bound and a possibly-infinite upper bound, plus a bunch of methods which make it possible to compare, add and multiply them.

Special multipliers
  • zero (multiplier(bound(0), bound(0)) (has some occasional uses internally)
  • qm (multiplier(bound(0), bound(1)))
  • star (multiplier(bound(0), inf))
  • one (multiplier(bound(1), bound(1)))
  • plus (multiplier(bound(1), inf))

lego.lego

Parent class for charclass, mult, conc and pattern. In general, this represents a regular expression object.

lego.charclass

Represents a character class, e.g a, [abc], [^xyz], \d.

Special character classes
  • w (charclass("0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ_abcdefghijklmnopqrstuvwxyz"))
  • d (charclass("0123456789"))
  • s (charclass("\t\n\v\f\r "))
  • W (any character except those matched by w)
  • D (any character except those matched by d)
  • S (any character except those matched by s)
  • dot (any character)
  • nothing (empty character class, no matches possible)

lego.mult

Represents a charclass or pattern combined with a multiplier, e.g. [abc]* or (a|bc)*.

lego.conc

Represents a sequence of zero or more mults, e.g. ab, [abc]*d.

Special concatenations
  • emptystring, the regular expression which only matches the empty string (conc())

lego.pattern

Represents an alternation between one or more concs, e.g. [abc]*d|e.

Methods in this module

lego.from_fsm()

Uses the Brzozowski algebraic method to convert a greenery.fsm object into a lego object, which is a regular expression.

lego.parse(string)

Returns a lego object representing the regular expression in the string.

The following metacharacters and formations have their usual meanings: ., *, +, ?, {m}, {m,}, {m,n}, (), |, [], ^ within [] character ranges only, - within [] character ranges only, and \ to escape any of the preceding characters or itself.

These character escapes are possible: \t, \r, \n, \f, \v.

These predefined character sets also have their usual meanings: \w, \d, \s and their negations \W, \D, \S. . matches any character, including new line characters and carriage returns.

An empty charclass [] is legal and matches no characters: when used in a regex, the regex may match no strings.

Unsupported constructs
  • This method is intentionally rigorously simple, and tolerates no ambiguity. For example, a hyphen must be escaped in a character class even if it appears first or last. [-abc] is a syntax error, write [\-abc]. Escaping something which doesn't need it is a syntax error too: [\ab] resolves to neither [\\ab] nor [ab].

  • The ^ and $ metacharacters are not supported. By default, greenery assumes that all regexes are anchored at the start and end of any input string. Carets and dollar signs will be parsed as themselves. If you want to not anchor at the start or end of the string, put .* at the start or end of your regex respectively.

This is because computing the intersection between .*a.* and .*b.* (1) is largely pointless and (2) usually results in gibberish coming out of the program.

  • The greedy operators *?, +?, ?? and {m,n}? are not supported, since they do not alter the regular language.

  • Parentheses are used to alternate between multiple possibilities e.g. (a|bc) only, not for capture grouping. Here's why:

      >>> print(parse("(ab)c") & parse("a(bc)"))
      abc
    

The (?:...) syntax for non-capturing groups is permitted, but does nothing.

Methods on the lego class

All objects of class lego (charclass, mult, conc and pattern) share these methods.

Method Behaviour
lego1.to_fsm() Returns an fsm object, a finite state machine which recognises exactly the strings that the original regular expression can match. The majority of the other methods employ this one.
lego1.matches("a")
"a" in lego1
Returns True if the regular expression matches the string or False if not.
lego1.strings()
for string in lego1
Returns a generator of all the strings that this regular expression matches.
lego1.empty() Returns True if this regular expression matches no strings, otherwise False.
lego1.cardinality()
len(lego1)
Returns the number of strings which the regular expression matches. Throws an OverflowError if this number is infinite.
lego1.equivalent(lego2) Returns True if the two regular expressions match exactly the same strings, otherwise False.
lego1.copy() Returns a copy of lego1.
lego1.everythingbut() Returns a regular expression which matches every string not matched by the original. x.everythingbut().everythingbut() matches the same strings as x for all lego objects x, but is not necessarily identical.
lego1.reversed()
reversed(lego1)
Returns a reversed regular expression. For each string that lego1 matched, reversed(lego1) will match the reversed string. reversed(reversed(x)) matches the same strings as x for all lego objects x, but is not necessarily identical.
lego1.times(star)
lego1 * star
Returns the input regular expression multiplied by any multiplier.
lego1.concatenate(lego2, ...)
lego1 + lego2 + ...
Returns the concatenation of the regular expressions.
lego1.union(lego2, ...)
`lego1
lego2
lego1.intersection(lego2, ...)
lego1 & lego2 & ...
Returns a regular expression matching any string matched by all input regular expressions. The successful implementation of this method was the ultimate goal of this entire project.
lego1.difference(lego2, ...)
lego1 - lego2 - ...
Subtract the set of strings matched by lego2 onwards from those matched by lego1 and return the resulting regular expression.
lego1.symmetric_difference(lego2, ...)
lego1 ^ lego2 ^ ...
Returns a regular expression matching any string accepted by lego1 or lego2 but not both.
lego1.reduce() Returns a regular expression which matches exactly the same strings as lego1 but is simplified as far as possible. See dedicated section below.
lego1.derive("a") Return the Brzozowski derivative of the input regular expression with respect to "a".

reduce()

Call this method to try to simplify the regular expression object, according to the following patterns:

  • (ab|cd|ef|)g to (ab|cd|ef)?g
  • ([ab])* to [ab]*
  • ab?b?c to ab{0,2}c
  • a(d(ab|a*c)) to ad(ab|a*c)
  • 0|[2-9] to [02-9]
  • abc|ade to a(bc|de)
  • xyz|stz to (xy|st)z
  • abc()def to abcdef
  • a{1,2}|a{3,4} to a{1,4}

The various reduce() methods are extensible.

Note that in a few cases this did not result in a shorter regular expression.

Name

I spent a long time trying to find an appropriate metaphor for what I was trying to do: "I need an X such that lots of Xs go together to make a Y, but lots of Ys go together to make an X". Unfortunately the real world doesn't seem to be recursive in this way so I plumped for "lego" as a basic catchall term for the various components that go together to make up a data structure.

This was a dumb idea in retrospect and it will be changed to greenery.re or greenery.rx in the near future. Vote now if you have an opinion.