Regular Expressions and Automata Chapter 2

Regular Expressions Standard notation for characterizing text sequences Used in all kinds of text processing and information extraction tasks As things have progressed, the RE languages used in various tools and languages (grep, Emacs, Python, Ruby, Java, …) are very similar 7/3/2015 Speech and Language Processing - Jurafsky and Martin 2

Regular Expressions We’ll look at a few examples [in lecture], make a note about types of errors, and then move toward automata 7/3/2015 Speech and Language Processing - Jurafsky and Martin 3

7/3/2015 Speech and Language Processing - Jurafsky and Martin 4 Example Find all the instances of the word “the” in a text.  /the/  /[tT]he/  /\b[tT]he\b/

7/3/2015 Speech and Language Processing - Jurafsky and Martin 5 Errors We fixed two kinds of errors  Matching strings that we should not have matched  False positives (Type I)  Not matching things that we should have matched  False negatives (Type II)

7/3/2015 Speech and Language Processing - Jurafsky and Martin 6 Errors We’ll see the same story for many tasks, all semester. Reducing the error rate for an application often involves two antagonistic efforts:  Increasing precision, (minimizing false positives)  Increasing coverage, or recall, (minimizing false negatives)

Formal Languages and Models Language: a (possibly infinite) set of strings made up of symbols from a finite alphabet Model of a language: can recognize and generate all and only the strings from the language  Serves as a definition of the formal language 7/3/2015 Speech and Language Processing - Jurafsky and Martin 7

Chomsky Hierarchy Regular language  Model: regular expressions, finite state automata Context free language Context sensitive language Unrestricted language  Model: Turning Machine 7/3/2015 Speech and Language Processing - Jurafsky and Martin 8

Regular Expressions and Languages A regular expression pattern can be mapped to a set of strings A regular expression pattern defines a language (in the formal sense) – the class of this type of languages is called a regular language 7/3/2015 Speech and Language Processing - Jurafsky and Martin 9

7/3/2015 Speech and Language Processing - Jurafsky and Martin 10 Finite State Automata FSAs and their probabilistic relatives are at the core of much of what we’ll be doing all semester. They also capture significant aspects of what linguists say we need for morphology and parts of syntax. They are formally equivalent to regular expressions

Formal Definition of a Finite Automaton 1.Finite set of states, typically Q. 2.Alphabet of input symbols, typically  3.One state is the start/initial state, typically q 0 // q 0  Q 4.Zero or more final/accepting states; the set is typically F.// F  Q 5.A transition function, typically δ. This function Takes a state and input symbol as arguments. Returns a state. One “rule” would be written δ(q, a) = p, where q and p are states, and a is an input symbol. Intuitively: if the FA is in state q, and input a is received, then the FA goes to state p (note: q = p OK). 6.A FA is represented as the five-tuple: A = (Q, , δ,q 0, F). Here, F is a set of accepting states.

A Simple Example Language: “Sheepish” Any string that starts with the letter b, followed by two or more a’s, and ending in ! {“baa!”,”baaa!”,”baaaa!”,”baaaaa!”,…} Regular expression for this? 7/3/2015 Speech and Language Processing - Jurafsky and Martin 12

7/3/2015 Speech and Language Processing - Jurafsky and Martin 13 One Possible Sheepish FSA Formal definition of this FSA?

FSA as a Recognizer Does a string belong to its language? 1.Place the input string on a tape, point at start 2.Initialize current state to q 0 3.Iteratively check the next letter on tape. 1.From the current state, if an outgoing arc label matches new letter, move to new state 2.If stuck, REJECT 4.If reach the end of the tape and in a final state, then ACCEPT; else, REJECT 7/3/2015 Speech and Language Processing - Jurafsky and Martin 14

7/3/2015 Speech and Language Processing - Jurafsky and Martin 15 Recognition Traditionally, (Turing’s notion) this process is depicted with a tape.

FSA as Generator FSA can also produce strings in the language it represents 1.Start from q 0 2.Pick an out-going arc to a new state (for now, assume picking randomly) and print the symbol on the arc 3.Follow the arc to the new state 4.Repeat from step 2 until reaching a final state 7/3/2015 Speech and Language Processing - Jurafsky and Martin 16

FSAs and Regular Expressions These are formally equivalent. Both of these classes of models recognize/generate exactly the class of regular languages Interesting proofs: constructive! Given any regular expression, create an equivalent FSA; given any FSA, create an equivalent regular expression 7/3/2015 Speech and Language Processing - Jurafsky and Martin 17

Note on Practical Regular Expression Utilities NOTE: additional features added to regular expression processing can make them more powerful; think of memory 7/3/2015 Speech and Language Processing - Jurafsky and Martin 18

7/3/2015 Speech and Language Processing - Jurafsky and Martin 19 About Alphabets Don’t take term alphabet word too narrowly; it just means we need a finite set of symbols in the input. These symbols can and will stand for bigger objects that can have internal structure.

7/3/2015 Speech and Language Processing - Jurafsky and Martin 20 Often there is more than one FSA for a given language E.g., here is another FSA for “Sheepish”

7/3/2015 Speech and Language Processing - Jurafsky and Martin 21 Yet Another View The guts of FSAs can ultimately be represented as tables ba!e , If you’re in state 1 and you’re looking at an a, go to state 2

7/3/2015 Speech and Language Processing - Jurafsky and Martin 22 Deterministic versus Non- Deterministic FSAs Deterministic means that at each point in processing there is always one unique thing to do (no choices). Non-deterministic means there are choices Go back and look at previous DFA How do deterministic and non- deterministic FSAs compare?

7/3/2015 Speech and Language Processing - Jurafsky and Martin 23 Non-Deterministic FSAs May include  Epsilon transitions  Key point: these transitions do not examine or advance the tape during recognition

7/3/2015 Speech and Language Processing - Jurafsky and Martin 24 ND Recognition Two basic approaches 1.Either take a ND machine and convert it to a D machine and then do recognition with that. 2.Or explicitly manage the process of recognition as a state-space search (leaving the machine as is).

7/3/2015 Speech and Language Processing - Jurafsky and Martin 25 Non-Deterministic Recognition: Search In a ND FSA there exists at least one path through the machine for a string that is in the language defined by the machine. But not all paths through the machine for an accept string lead to an accept state. If a string is not in the language, there are no paths through the machine that lead to an accept state

7/3/2015 Speech and Language Processing - Jurafsky and Martin 26 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.

7/3/2015 Speech and Language Processing - Jurafsky and Martin 27 Example

7/3/2015 Speech and Language Processing - Jurafsky and Martin 28 Why Non-Determinism? Non-determinism doesn’t get us more formal power and it causes headaches so why bother?  More natural (understandable) solutions

7/3/2015 Speech and Language Processing - Jurafsky and Martin 29 Compositional Machines Formal languages are just sets of strings Therefore, we can talk about various set operations (intersection, union, concatenation) We’ll just do a couple

7/3/2015 Speech and Language Processing - Jurafsky and Martin 30 Union

7/3/2015 Speech and Language Processing - Jurafsky and Martin 31 Concatenation