Finite-state automata 2

Day 13

LING 681.02Computational Linguistics

Harry HowardTulane University

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Course organization

http://www.tulane.edu/~ling/NLP/NLTK is installed on the computers in this

room!How would you like to use the Provost's

$150?

SLP §2.2 Finite-state automata

2.2.1 Sheeptalk

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Find your files

>>> import sys>>>sys.path.append("/Users/harryhow/Documents/Work/Research/Sims/NLTK")

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Run program

>>> import fsaproc>>> test = 'baaa!'>>> test = 'baaa!$'>>> fsaproc.machine(test)

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Go over print-out

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Key points

D-recognize is a simple table-driven interpreter.The algorithm is universal for all unambiguous

regular languages.To change the machine, you simply change the table.

Crudely therefore… matching strings with regular expressions (ala Perl, grep, etc.) is a matter of:translating the regular expression into a machine (a

table) and passing the table and the string to an interpreter.

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Recognition as search

You can view this algorithm as a kind of state-space search.

States are pairings of tape positions and state numbers.

The goal state is a pairing with the end of tape position and a final accept state.

SLP §2.2 Finite-state automata

2.2.2 Formal languages

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Generative Formalisms

Formal Languages are sets of strings composed of symbols from a finite set of symbols.

Finite-state automata define formal languages (without having to enumerate all the strings in the language).

The term Generative is based on the view that you can run the machine as a generator to get strings from the language.

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Generative Formalisms

A FSA can be viewed from two perspectives, as:an acceptor that can tell you if a string is in the

language.a generators to produce all and only the strings

in the language.

SLP §2.2 Finite-state automata

2.2.4 Determinism

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Determinism

A deterministic FSA has one unique thing to do at each point in processing.i.e. there are no choices

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Non-determinism

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Non-determinism cont.

Epsilon transitionsAn arc has no symbol on it, represented as .Such a transition does not examine or advance

the tape during recognition:

SLP §2.2 Finite-state automata

2.2.5 Use of a nFSA to accept strings

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Read on your own

pp. 33-5

SLP §2.2 Finite-state automata

2.2.6 Recognition as search

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Non-deterministic recognition: Search

In a ND FSA there is at least one path through the machine for a string that is in the language defined by the machine.

But not all paths directed through the machine for an accept string lead to an accept state.

No paths through the machine lead to an accept state for a string not in the language.

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Non-deterministic recognition

So success in non-deterministic recognition occurs when a path is found through the machine that ends in an accept.

Failure occurs when all of the possible paths for a given string lead to failure.

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

b a a a ! \

q0 q1 q2 q2 q3 q4

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Example

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Key points

States in the search space are pairings of tape positions and states in the machine.

By keeping track of as yet unexplored states, a recognizer can systematically explore all the paths through the machine given an input.

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Ordering of states

But how do you keep track?Depth-first/last in first out (LIFO)/stack

Unexplored states are added to the front of the agenda, and they are explored by going to the most recent.

Breadth-first/first in first out (FIFO)/queueUnexplored states are added to the back of the

agenda, and they are explored by going to the most recent.

SLP §2.2 Finite-state automata

2.2.7 Comparison

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Equivalence

Non-deterministic machines can be converted to deterministic ones with a fairly simple construction.

That means that they have the same power:non-deterministic machines are not more

powerful than deterministic ones in terms of the languages they can accept.

23-Sept-2009 LING 681.02, Prof. Howard, Tulane University

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Why bother?

Non-determinism doesn’t get us more formal power and it causes headaches, so why bother?

More natural (understandable) solutions.

Next time

SLP §2.3 briefly

SLP §3