Simplification of Grammars
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Simplification of Grammars
Lecture 17Naveen Z QuazilbashSimplification of GrammarsOverviewAttendanceMotivationSimplification of GrammarsEliminating useless variablesEliminating null productionsEliminating unit productionsQuiz resultMotivation for grammar simplificationParsing ProblemGiven a CFG G and string w, determine if w L(G).Fundamental problem in compiler design and natural language processingIf G is in general form then the procedure maybe very inefficient. So the grammar is transformed into a simpler form to make the parsing problem easier.Simplification of GrammarsIt involves the removal of:
Useless variables-productionsUnit productions
Useless variables:There are two types of useless variables:Variables that cannot be reachedVariables that do not derive any strings-productionsE.g.: A
Note that if we remove these productions, the language no longer includes the empty string.Unit productions:They are of the form ABOrAA1) Unreachable Variables E.g.: SBS|B|EADA|D|SBCB|CCaC|aDbD|bEcE|cTo find unreachable variables, draw a dependency graph
Dependency Graph:Vertices of the graph are variablesThe graph doesnt include alphabet symbols, such as a or bIf there is a production A..B, i.e., the left side is A and the right side includes B, then there is an edge ABA variable is reachable if there is a path from S to this variable
S itself is always reachable
After identifying unreachable variables, remove all productions with unreachable left side.
Drawing its dependency graph:Reachable: S, B, C, ESDAECBGrammar without unreachable variables:SBS|B|EBCB|CCaC|aEcE|c
Ex: Determine its language!!2) Variables that dont terminateA variable A terminates if either:There is a production A. with no variables on the right, e.g. Aaabc, ORThere is a production A where all variables on the right terminate; e.g. AaBbaC, where B and C terminate.
Note: to find all variables that terminate, keep looking for such productions until you cannot find any new ones.TASKExample: SA|BC|DEAaA|bABbB|bCEFDdD|BD|BAEaE|aFcFc|cRemove all productions that include a variable that doesnt terminate. Note: We remove a production if it has such a variable on either side.SolutionxSA|BC|DEXAaA|bAxBbB|bxCEFXDdD|BD|BAxEaE|axFcFc|cSBCBbB|bCEFEaE|aFcFc|c
Ex: Determine its language.3) Eliminating -ProductionsNullable variables:A variable is nullable if either:There is a production A , orThere is a production AB1B2Bn(only variables, no symbols), where all variables on the right side are nullable.Note: to find all nullable variables, keep looking for such productions, until you cannot find any new ones.TASKSSAB|SBC|BCAaA|aBbB|bC|CCcC| First we find variables that can lead to the empty string:C=> B=>C=> S=>BC=>B=>C=>
Thus, S, B, and C can lead to ; they are called nullable variables
For each production that has nullable variables, consider all possible ways to skip some of these variables and add the corresponding productions.E.g. WaWXaYZb, suppose that X, Y and Z are nullable; then there are 8 ways to skip some of them.WaWab|aWXab|aWaYb|aWaZb|aWXaYb|aWXaZb|aWaYZb|aWXaYZb
Back to our grammar where S,B and C are nullable:SA|AB|SA|SAB|S|B|C|SB|BC|SBCAaA|aBb|bB|bC|CCc|cC|Now, we can remove the - productions without changing the language.The only possible change is losing the empty string, if it is in the original language.
So our grammar without null productions becomes:
4) Eliminating Unit ProductionsSAa|BAa|bc|BBA|bb|C|cCCa|CFirst, for every variable, we find all single variables that can be reached from it:For S: S=>B=>A, S=>B=>CFor A: A=>B=>CFor B: B=>A, B=>CFor C: NONE (C itself doesnt count) For finding reachable single variables, what should we do?Use Dependency Graph!Drawing Dependency Graph:Vertices of the graph are variables.If there is a unit production AB, then there is an edge AB. A single variable is reachable from A if there is a pth from A to B.
Dependency Graph:SABCTo construct an equivalent grammar without unit productions:Remove all unit productionsFor each pair A=>*B, where B is a single variable reachable from A, consider all productions Bp1|p2||pn; and add the corresponding productions A p1|p2||pn.for example, since A=>*B and Bbb|cC, add the productions Abb|cCSAa|BAa|bc|BBA|bb|C|cCCa|C
SAaBbb|cCAa|bcCaNote that the variable B has become useless and we need to remove it!Sbb|cC|a|bc|aBa|bc|aAbb|cC|aCaOld non-unit productionsnew productionsSummaryMain steps of simplifying a grammar:Remove useless variables, which cannot be reached or do not terminate.Remove - productions.Remove unit productions.Remove useless variables again!