12.1 Languages and Grammars

12.1 Languages and Grammars

• Define– Vocabulary V (alphabet)

– Word (string)

– Empty string λ

– Free language V*

– A language over V

– an, for a in V, n in N.

Phrase-Structure Grammars

• A phrase-structure grammar is a 4-tuple

(V,T,S,P)

Example

The Language Generated by a Grammar

• The language L(G) generated by a grammar G is the subset of T* consisting of all the strings of terminal symbols which can be generated from the start symbol S by applying a series of productions.

ExampleGenerate the string aaabb from the start symbol of the grammar at right.

Solution

• Note that the language L generated by a grammar consists of all strings of terminal symbols that are derivable from the start symbol S

• In the above example,

Example

• A grammar that generates the set

is given at right• Show how the strings 012 and 001122 can be

derived from the start symbol S

}0|210{ nL nnn

Example

• Find a grammar that generates the set

}1|10{ 2 nL nn

Language Types

• Type 0 – No restrictions• Type 1 – All productions are of the form

w1w2, where length(w1) length(w2) and w1 contains a non-terminal

• Type 2 – All productions are of the form Nw

• Type 3 – All productions are of the form Aa or AaB

Context-Free Grammars

• Type 2 languages (where all productions are of the form Nw) are called context-free grammars.

• If a grammar is context-free, then all the words in the language have a parse tree, also called a derivation tree.

Example

• Construct a parse tree for aaabb in the grammar at right.

Parsing

• To parse is to discover the form of the derivation tree

• Top-down parsing starts at the root, with the start symbol, and constructs the tree from there

• Bottom-up parsing builds the tree up from the leaves, by grouping symbols into valid right-hand sides of productions

Backus-Naur Form (BNF)

• Backus-Naur Form, or BNF, is a notation for describing grammars, first used in the description of the syntax of the ALGOL 60 language

• Non-terminal symbols are descriptive terms enclosed in angular brackets (e.g. <sentence> instead of S)

• The symbol used for is ::=• Alternative right-hand sides can be combined

using a vertical slash (|) to separate them

12.2 Finite State Machines with Output

• A finite state machine with output is a 6-tuple (S, I, O, f, g, s0), where S is a finite set of states, I is a finite set of input symbols, O is a finite set of output symbols, f is a state transition function, g is an output function, and s0 is a designated state from the set S, called the start state.

Example:

Extending the Output Function

• Recall that the output function g maps a (state, input symbol) pair to a single output symbol

• We can extend that function so that it maps an input string to a string of output symbols.

• For instance, in the above example g(01100100) =

Example

• Input alphabet is {a, b}. Output a 1 if the input string began with bab, output 0 otherwise.

More Examples

• Machine to reverse bits

• Machine to output 1 for an odd number of 1’s, 0 otherwise

Vending Machine Example• Machine takes quarters only. • Outputs a coke if 75 cents has been entered and

the vend button has been pressed (use output symbol c)

• Outputs a quarter if 75 cents has already been entered (use output symbol 25)

• No output otherwise (use output symbol n)

12.3 Finite State Machines with No Output

• Concatenation of sets of strings

• Powers An of a set of symbols A

• Kleene Closure A* of a set of strings A

Examples:

Let

Find:

Deterministic Finite State Automata

• A deterministic finite state automaton (DFSA) is a 5-tuple M = (S, I, f, s0 , F), where S is a finite set of states, I is an input alphabet, f:SIS is a state transition function, and F is a subset of S called the set of final states.

• The language accepted by M is the set of strings in I* which, if given as input to M, will leave the machine in a final state.

Examples

• Construct a machine with input alphabet {0, 1} which accepts any string with an even number of 1’s

• Construct a machine which accepts {0,1}*.

Example

• What language is accepted by the following machine?

Non-Deterministic Finite-State Automata

• A non-deterministic finite-state automaton (NFSA) is defined the same as a dfsa, except that the state transition function maps each (state, input symbol) pair to a set of states; i.e. f:SIP(S).

• If f(s,a) is {s',s"}, then the machine in state s could, if its input is a, either transition to state s' or to state s".

• If f(s,a) is {} (the empty set), then the machine in state s would consider a an invalid input symbol.

• The language accepted by an NFSA is the set of all input strings which could put the machine in a final state if the right choices were made at each step in the execution of the machine.

Example

• What language is accepted by the following NFSA?

Constructing a DFSA from an NFSA

• Any NFSA can be transformed into a DFSA by using sets of states as the states in the new machine.

• The start state of the new machine is {s0}.

• The transition function g of the new machine can be defined in terms of the original function f and sets of states.

• A set representing a state in the new machine is considered final if any state in the set is final.

• Example: Transform the machine

on the previous slide into a DFSA.

• Note that the language accepted by the original NFSA is the same as that accepted by the DFSA constructed from it.

• This can be expressed as a theorem about languages…

12.4 Language Recognition

• Let I be a finite set of input symbols. The “regular expressions over I” are the symbols in I along with the empty string λ, the empty set , and any expression that can be formed from those elements using concatenation, union, and Kleene closure

• Regular expressions are used to generate the category of languages called “regular languages”

Regular Languages

• Regular expressions always describe sets, and when a symbol, say a, is used in a regular expression it stands for {a}.

• Thus (a ab)* is the Kleene closure of the set {a, ab}.• A language generated in this way by a regular expression is

called a regular language.• Examples: Give a regular expression describing…

– The set {α | α is λ or α is all zeros or α is all ones}

– the set {α | α is λ or α is an alternating sequence of 0’s and 1’s}

What are the Regular Languages? • Kleene’s Theorem: The regular languages are precisely

those languages which can be recognized by FSA’s.• Draw a DFSA which recognizes the last example of a

regular language.

Regular Grammars

• Recall that in a regular grammar, all the productions are of the form Aa or AaB, where A and B are non-terminals and a is a terminal symbol.

• A regular language has the property that all of its non-empty strings can be generated by the productions of a regular grammar

Example:

• Find a regular grammar capable of generating the nonempty strings recognized by the FSA shown above

Which Languages are not Regular?

• Example: {0n1n | n 0} is not regular

12.5 Turing Machines

• A Turing Machine (TM) is a quadruple (S, I, f, s0), where S is a set of states, I is a set of input/output symbols, f is a transition function, and s0 is an initial state. A special symbol B, called the blank symbol, is always an element of I.

• The transition function is actually a partial function, since its domain is a subset of SI, not the complete set. But for every pair (s,i) for which f(s,i) is defined, f(s,i) will be a triple (s′,i′,D), where s′ is a state, i′ is an ouput symbol, and D is either the symbol R (for right) or the symbol L (for left).

PictureA Turing Machine is conceptualized as taking its input from and writing its output to a discrete tape (actually a sequence of storage cells) that is infinite in both directions. All but a finite number of cells contain the blank symbol B.

By convention, the initial position of a TM is with the read head of the machine positioned over the leftmost non-blank symbol.

At each step of the machine’s operation, it reads the input symbol from the tape, changes its state, writes a symbol back into the same cell, then moves either left or right.

Example• States:

– s0, start state – moving right, looking for 1’s

– s1, moving right, looking for 1’s, found 1



– s4, found four 1’s.

Transition Function

• The transition function for this TM is given below:

Final States

• A state si is said to be a final state if there are no actions defined for that state, i.e. if there are no pairs of the form (si, x) for which the transition function is defined.

• The language recognized by a Turing Machine T is the set of all strings of input symbols which, if placed on the tape with the read head at the initial position, will cause the machine to eventually transition into a final state.

Example

• A TM that recognizes {0n1n| n 1}.

0 0 1 M R1 0 1 0 R1 1 1 1 R1 M 2 M L1 _ 2 _ L2 1 3 M L3 M 5 M R3 0 4 0 L3 1 4 1 L4 0 4 0 L4 1 4 1 L4 M 0 M R

Example:

What does the Turing machine described by the five-tuples and do when given

a) 11 as input?

b) An arbitrary bit string as input?

Example: Construct a Turing machine with tape symbols 0, 1 and B that, when given a bit string as input, replaces the first 0 with a 1 and does not change any of the other symbols on the tape.

Example: Construct a Turing machine that recognizes the set of all bit strings that end with a 0.

Theorem

• The languages that can be recognized by a TM are precisely the type 0 languages (languages for which a grammar exists)

Computing Functions with a TM

• Unary Representation– 1 for 0– 11 for 1– 111 for 2– Etc.

• Example: f(n) = n mod 3. – One way is to write an * after n, then place f(n) to the

right of that.– Use the following 5-tuples:

(s0,1,s1,1,R), (s1,B,s5,*,R), (s1,1,s2,1,R), (s2,B,s6,*,R), (s5,B,s4,1,R),(s2,1,s3,1,R), (s3,B,s7,*,R), (s7,B,s6,1,R), (s6,B,s5,1,R), (s3,1,s1,1,R)

(s0,1,s1,1,R), (s1,B,s5,*,R), (s1,1,s2,1,R), (s2,B,s6,*,R), (s5,B,s4,1,R)

(s2,1,s3,1,R), (s3,B,s7,*,R), (s7,B,s6,1,R), (s6,B,s5,1,R), (s3,1,s1,1,R)

Example: Another way is to replace the input with the answer. For example, suppose we want a Turing machine for adding two nonnegative integers:

The input is represented by a string of 1s followed by an asterisk followed by 1’s. The five-tuples are: and

Computable FunctionsA function is computable if and only if

Example of a non-computable function:

The Church-Turing Thesis

Decision Problems• A decision problem asks

• The Halting problem is an unsolvable decision problem.

• A decision problem is P if

• A decision problem is NP if

12.1 Languages and Grammars

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