Regular Expressions

Formal Languages, Regular Expressions, Automata and Transducers

Useful links:

RegEx online validation tool (Chinese)

Formal Language = Set of Strings of Symbols

A Formal Language can model a phenomenon, e.g. written English

Examples: All combinations of the letters, any number of As, followed by any number of Bs, mathematical equations, all the sentences of a simplified version of written English, a sequence of musical notation (e.g., the notes in Beethoven's 9th Symphony), etc.

What is a Formal Grammar for?

A formal grammar: a set of rules that match all and only instances of a formal language. A formal grammar defines a formal language.

In Computer Science, Formal grammars are used to generate and recognize formal languages (e.g., programming languages)

Parsing a string of a language involves:
- Recognizing the string and
- Recording the analysis showing it is part of the language
A compiler translates from language X to language Y, e.g.,This may include parsing language X and generating language Y
If all natural languages were formal languages, then Machine Translation systems would just be compilers

A Formal Grammar Consists of

N: a Finite set of non-terminal symbols

Symbols that can be replaced by other symbols

T: a Finite set of terminal symbols

Symbols that cannot be replaced by other symbols

R: a set of rewrite rules

Replace the symbol sequence XYZ with abXzY: XYZ → abXzY

S: A special non-terminal that is the start symbol

Marks the start of the language

The Chomsky Hierarchy

Type0 ⊇ Type1 ⊇ Type2 ⊇ Type3

Type 0: No restrictions on rules

Equivalent to Turing Machine, general system capable of simulating any algorithm.

Type 1: Context-sensitive rules

αAβ → αγβ

Greek letters = 0 or more non-terms/terms.
A = non-terminal
Rule means: replace A with γ, when A is between α and β

Type 2: Context-free rules

A → αγβ

Like context-sensitive, except left-hand side can only contain exactly one non-terminal

Example Rule from linguistics:

Type 3: Context-free rules with restrictions

Regular (finite state) grammars

A → βa or A → ϵ (left regular)
A → aβ, or A → ϵ (right regular)

Like Type 2, except:

Non-terminals can precede terminals in left regular grammar
Non-terminals can follow terminals in right regular grammar
Null string is allowed

Type-3 grammars generate the regular languages

Further Simplifications

Type-3 grammars must have a single non-terminal on the left-hand side and a right-hand side consisting of a single terminal or single terminal followed by a single non-terminal.

The productions must be in the form X → a or X → aY

where X, Y ∈ N (Non terminal)

and a ∈ T (Terminal)

The rule S → ε is allowed if S does not appear on the right side of any rule.

Comparisons

Type 3 grammars: Least expressive, Most efficient processors

Type 0 grammars: Most expressive, Least efficient processors

Complexity of recognizer for languages:

Type 0: exponential
Type 1: polynomial
Type 2: $O(n^3)$
Type 3: $O(n logn)$

CL mainly features Type 2 & 3 Grammars

Type 3 grammars:

Include regular expressions and finite state automata (aka, finite state machines)
The focal point of the rest of this talk
Also see Nooj platform for NLP

Type 2 grammars:

Commonly used for natural language parsers
Used to model syntactic structure in many linguistics theories (often supplemented by other mechanisms)
Important for later talks on constituent structure & parsing

Type 1.5 Grammars

Human Language believed to be “mildly context sensitive”

Less expressive than type 1 (context sensitive)
More expressive than type 2 (context-free)

Some complex dependencies cannot be expressed in context free rules, e.g. see this

Three Adjoining Grammars

https://repository.upenn.edu/cgi/viewcontent.cgi?article=1706&context=cis_reports
https://www.aclweb.org/anthology/H86-1020.pdf
Formalism by A. Joshi & others
May be able to handle these cases

Regular Expressions

Concatenation

If X is a regexp and Y is a regexp, then XY is a regexp
Examples:
- If ABC and DEF are regexps, then ABCDEF is a regexp
- If AB* and BC* are regexps, then AB*BC* is a regexp Note: Kleene _ is explained below

Disjunction

If X is a regexp and Y is a regexp, then X | Y is a regexp
Example: ABC|DEF will match either ABC or DEF

Repetition

If X is a regexp then a repetition of X will also be a regexp
- The Kleene Star: A* means 0 or more instances of A
- Regexp{number}: A{2} means exactly 2 instances of A

Disjunction of characters

[ABC] – means the same thing as A | B | C
[a-zA-Z0-9] – character ranges are equivalent to lists: a|b|c|...|A|B|...|0|1|...|9

Negation of character lists/sequences

^ inside bracket means complement of disjunction, e.g., [^a-z] means a character that is neither a nor b nor c … nor z

Parentheses

Disambiguate scope of operators
- A(BC)|(DEF) means ABC or ADEF
- Otherwise defaults apply, e.g., ABC|D means ABC or ABD

? signifies optionality

ABC? is equivalent to (ABC)|(AB)

* indiates 1 or more

A(BC)* is equivalent to A|(A(BC)+)

Special Symbols:

Period means any character, e.g., A.*B – matches A and B and any characters between
Carrot (^) means the beginning of a line, e.g., ^ABC matches ABC at the beginning of a line [*Note dual usage of ^ as negation operator]
Dollar sign ($) means the end of a line, .e.g., [\.?!] *$ matches final punctuation, zero or more spaces and the end of a line

Sets of characters:

\w = [A-Za-z0-9_]
\W = [^A-Za-z0-9_]

Generator

Finite State Automata

Devices for recognizing finite state grammars (include regexps)

Two types

Deterministic Finite State Automata (DFSA): Rules are unambiguous
NonDeterministic FSA (NDFSA): Rules are ambiguous. Sometimes more than one sequence of rules must be attempted to determine if a string matches the grammar. Ways to solve this: Backtracking, Parallel Processing and Look Ahead.

Any NDFSA can be mapped into an equivalent (but larger) DFSA

DFSA

Algorithm:

D-Recognize(tape, machine)
    pointer ← beginning of tape
    current state ← initial state Q0
    repeat until the end of the input is reached
        look up (current state,input symbol) in transition table
        if found: set current state as per table look up
                  advance pointer to next position on tape
        else: reject string and exit function
    if current state is a final state: accept the string
    else: reject the string

NDFSA

Algorithm:

ND-Recognize(tape, machine)
    agenda ← {(initial state, start of tape)}
    current state ← next(agenda)
    repeat until accept(current state) or agenda is empty
        agenda ← Union(agenda,look_up_in_table(current state,next_symbol))
        current state ← next(agenda)
    if accept(current state): return(True)
    else: false

Accept if at the end of the tape and current state is a final state

Next defined differently for different types of search

Choose most recently added state first (depth first)
Chose least recently added state first (breadth first)
Etc.

PreviousIntroduction NextHMM POS Tagging

Last updated 2 years ago