CS 208: Computing Theory

Assoc. Prof. Dr. Brahim HnichFaculty of Computer SciencesIzmir University of Economics

Context Free LanguagesContext-Free Languages

So far Methods for describing regular languagesFinite AutomataDeterministicNon-deterministicRegular ExpressionsThey are all equivalent, and limitedCannot some simple languages like {0n1n | n is positive}Now, we introduce a more powerful method for describing languagesContext-free Grammars (CFG)

Are CFGs any useful?Extremely useful!

Artificial IntelligenceNatural language Processing

Programming Languagesspecificationcompilation

ExampleThis is a CFG which we call G1A0A1ABB#

Example: production rulesThis is a CFG which we call G1A0A1ABB#Each line is a substitution rules or production rules

Example: variablesThis is a CFG which we call G1A0A1ABB#A and B are called variables or non-terminals

Example: variablesThis is a CFG which we call G1A0A1ABB#0,1, and # are called terminals

Example: variablesThis is a CFG which we call G1A0A1ABB#A is the start variable

RulesWe use a CFG to describe a language by generating each string of that languageWrite down the start variablePick a variable written down and a production rule that starts with that variableReplace that variable with right-hand side of the production ruleRepeat until no variable remain

DerivationsThis is a CFG which we call G1A0A1ABB#Derivations with G1A0A10B10#1A0A100A1100B1100#11A0A100A11000A111000B111000#111

Parse treeParse tree for 0#1 in G1A0A10B10#1

AB0#1A

Parse treeParse tree for 00#11 in G1 A0A100A1100B1100#11AB0#1AA01

Context-free languages

All strings generated by a CFG constitute the language of the grammarExample: L(G1)={0n#1n | n is positive}

Any language generated by a context-free grammar is a context-free language

A useful abbreviationProduction rulesA 0A1A BB #Can be written asA 0A1 | BB #

Another exampleCFG G2 describing a fragment of English ||| a | the boy | girl | flower touches | likes | sees with

Another exampleExamples of strings belonging to L(G2)

a boy sees

the boy sees a flower

a girl with a flower likes the boy with a flower

Another exampleDerivation of a boy sees a a boy a boy a boy a boy sees

Formal definitionsA context-free grammar is a 4-tuple whereV is a finite set of variablesis a finite set of terminalsR is a finite set of rules: each rule is a variable and a finite string of variable and terminalsS is the start symbol

Formal definitionsIf u and v are strings of variable and terminals, andA w is a rule of the grammar,Then uAv yields uwv, written uAv uwv

We write u * v if u = v oru u1 . uk v

Formal definitionsThe language of grammar G is

L(G) = {w | S * w}

ExampleConsider G4 = where R isS (S) | SS | What is the language of G4?Examples: (), (()((())),

ExampleConsider G4 = where R isS (S) | SS | What is the language of G4?L(G4) is the set of strings of properly nested parenthesis

ExampleConsider G4 = where R isE E + T | TT T X F | FF (E) | a

What is the language of G4?Examples: a+a+a, (a+a) x a

ExampleConsider G4 = where R isE E + T | TT T x F | FF (E) | a

What is the language of G4?E stands for expression, T for Term, and F for Factor: so this grammar describes some arithmetic expressions

AmbiguitySometimes a grammar can generate the same string in several different ways!This string will have several parse treesThis is a very serious problemThink if a C program can have multiple interpretations?If a language has this problem, we say that it is ambiguous

ExampleConsider G5:+|x |() | aG5 is ambiguous because a+axa has two parse tress!

ExampleConsider G5:+|x |() | aG5 is ambiguous because a+axa has two parse tress!

a+ a xa

ExampleConsider G5:+|x |() | aG5 is ambiguous because a+axa has two parse tress!

a+ a xa

a+ a xa

Formal definition: ambiguityA string w is generated ambiguously in CFG G if it has two or more different leftmost derivations!A derivation is leftmost if at every step the variable being replaced is the leftmost one

Grammar G is ambiguous if it generates some string ambiguously

Chomsky Normal Form (CNF)Every rule has the formA BCA aS Where S is the start symbol, A, B, and C are any variables except that B and C may not be the start symbol

TheoremTheorem: Any context-free language is generated by a context-free grammar in Chomsky normal formHow?Add new start symbol S0Eliminate all rules of the form A Eliminate all unit rules of the form A BPatch up rules so that grammar denotes the same languageConvert remaining rules to proper form

Steps to convert any grammar into CNFStep1Add a new start symbol S0Add the rule S0S

Steps to convert any grammar into CNFStep2: RepeatRemove some rule of the form A where A is not the start symbolThen, for each occurrence of A on the right-hand side of a rule, we add a new rule with that occurrence deletedE.g., if R uAvAu where u and v are strings of variables and terminalsWe add rules: R uvAu, RuAvu, and RuvuFor RA add R, except if R has already been removedUntil all -rules not involving the start symbol have been removed

Steps to convert any grammar into CNFStep3: eliminate unit rulesRepeatRemove some rule of the form A BFor each Bu, add Au, except if Au has already been removedUntil all unit rules have been removed

Steps to convert any grammar into CNFStep4: convert remaining rulesReplace each rule A u1u2uk, where k >2 and each ui is a terminal or a variable with the rulesAu1A1A1u2A2A2u3A3.Ak-2uk-1ukIf k=2, we replace any terminal ui in the preceding rules with the new variable Ui and add the rule Uiui

ExampleStart withS ASA | aBA B | SB b |

ExampleStep 1: add new start symbol and new ruleS0 SS ASA | aBA B | SB b |

ExampleStep 2: remove -rule B S0 SS ASA | aB | aA B | S | B b

ExampleStep 2: remove -rule A S0 SS ASA | aB | a | SA | AS | SA B | S B b

ExampleStep 3: remove unit rule S SS0 SS ASA | aB | a | SA | AS | SA B | S B b

ExampleStep 3: remove unit rule S0 SS0 S | ASA | aB | a | SA | ASS ASA | aB | a | SA | AS A B | S B b

ExampleStep 3: remove unit rule A BS0 ASA | aB | a | SA | ASS ASA | aB | a | SA | AS A B | S | bB b

ExampleStep 3: remove unit rule A SS0 ASA | aB | a | SA | ASS ASA | aB | a | SA | AS A S | b | ASA | aB | a | SA | ASB b

ExampleStep 3: remove unit rule A SS0 ASA | aB | a | SA | ASS ASA | aB | a | SA | AS A b | ASA | aB | a | SA | ASB b

ExampleStep 4: convert remaining rulesS0 AA1|UB| a| SA | ASS AA1|UB | a | SA | AS A b | AA1 | UB | a | SA | ASB bUaA1SA

Pushdown Automata

Pushdown automataPushdown automat (PDA) are like nondeterministic finite automat but have an extra component called a stackCan push symbols onto the stackCan pop them (read them back) laterStack is potentially unbounded

State controlaabaxyzstackinput

Formal DefinitionA pushdown automaton is a 6-tuple (Q,,S, ,q0,F), whereQ is a finite set of states is a finite set of symbols called the alphabetS is the stack alphabet : Q x x S P(Q x S) is the transition functionq0 Q is the start stateF Q is the set of accept states or final states

ConventionsQuestion: when is the stack empty?Start by pushing a $ onto the stackIf you see it again, stack is empty

Question: when is input string emptyDoesnt matterAccepting states accept only if inputs exhausted

NotationTransition a,bc means Read a from the inputPop b from stackPush c onto stackMeaning of transitionIf a = , dont read inputIf b= , dont pop any symbolIf c= , dont push any symbols

ExampleRecall 0n1n which is not regularConsider the following PDARead input symbolsFor each 0, push it on the stackAs soon as a 1 is seen, pop a 0 for each 1 readAccept if stack is empty when last symbol readReject if stack non-empty, or if input symbol exist, or if 0 read after a 1, etc

Example,$0, 01,0 1,0 ,$ {0n1n| n is positive}

Example,${aibjck| i=j or i=k}a, a, , , , $ b, c, a b, a ,$ c,

TheoremTheorem: A language is context-free if and only some pushdown automaton accepts itProof: we will skip it! (Those interested may read the book)Corollary: Every regular language is a context-free languageRegular languagesContext-free languages

Conclusions

Context-free grammarsdefinitionambiguityChomsky normal form

Pushdown automatadefinition

Next: Part C; Computability Theory