From 3f694e020de238ba9027919135236ded3dbabcdc Mon Sep 17 00:00:00 2001 From: quirinecker Date: Wed, 5 Nov 2025 15:49:36 +0100 Subject: [PATCH] finished exercise --- index.pdf | 4711 +++++++++++++++++++++++++++++++++++++++++++++++++++++ index.typ | 184 +++ 2 files changed, 4895 insertions(+) create mode 100644 index.pdf create mode 100644 index.typ diff --git a/index.pdf b/index.pdf new file mode 100644 index 0000000..4447b01 --- /dev/null +++ b/index.pdf @@ -0,0 +1,4711 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 3 + /Kids [349 0 R 351 0 R 353 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 3 0 R + /Count 1 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /First 4 0 R + /Last 12 0 R + /Count -4 + /Title (Exercise 01) + /Dest 348 0 R +>> +endobj + +4 0 obj +<< + /Parent 3 0 R + /Next 8 0 R + /First 5 0 R + /Last 7 0 R + /Count -3 + /Title (01 Eigentschaften einer Gramatik) + /Dest 342 0 R +>> +endobj + +5 0 obj +<< + /Parent 4 0 R + /Next 6 0 R + /Title 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Sollte es mindestens eine +LL(1)-Verletzungen geben, geben Sie bitte alle an. Geben Sie zudem für jede Grammatik, welche die LL(1)- +Eigenschaft nicht besitzt, die theoretisch minimale Anzahl an Vorgriffssymbolen an, die für das Parsen der +jeweiligen Grammatik notwendig wäre. +] + +=== Task a + +#question[ +``` +Temperature = ["-"] ( Celsius | Kelvin ) . +Celsius = { number } "◦C" . +Kelvin = { number } "K" . +``` +] + +Conflicts: + +- $"first"("Celsius") inter "first"("Kelvin") = { "number" } arrow.r.double "LL(1) conflict" $ + - Both productions begin with the same token (number). This is a LL(1) conflict. + - It is not even LL(k) because there can be infinite (number) token before the distiction between Celsius and Kelvin happens. + +=== Task b + + +#question[ +``` +Locals = "class" ident "{" { VarDecl | FuncDecl } "}" . +VarDecl = "int" ident { "," ident } ";" . +FuncDecl = "fn" "int" ident "(" ")" ";" . +``` +] + +$ +"first"("FunDecl") inter "first"("VarDecl") = {"\"int\""} inter {"\"fn\""} = emptyset +$ + +$ +("follow"("VarDecl") union "follow"("FuncDecl")) inter ("first"("FunDecl") union "first"("VarDecl")) \ += {"\"int\"", "\"fn\""} inter { "\"}\"" } = emptyset +$ + +- No LL(1) conflict +- No left recursion +- Grammer is LL(1) + +=== Task c + +#question[ +``` +Var = ident ":" Type . +Type = ( Primitive | FuncType ) . +Primitive = "i32" | "f64" . +FuncType = Type "->" Primitive . +``` +] + +- There is left recurion. Which means that the grammar is not LL(1). Consider the following snippet. The first element of `FuncType` evaluates to `Type`, which then can evaluate to `FuncType` => left recursion + +``` +FuncType = Type "->" Primitive . +Type = ( Primitive | FuncType ) . + +``` + +- `First(Type)` can also not be computed because of the left recursion. So checking the intersection of `First(Type)` and `First(FuncType)` is not possible. +- The amount of lookahead symbols does not matter in this case. The grammar is not LL(k) for any k. + +== 02 Beseitigung von LL(1)-Konflikten mittels Faktorisierung + +Beseitigen Sie in folgenden Grammatiken die enthaltenen LL(1)-Konflikte mittels Einsetzen und Faktorisie- +rung. Geben Sie dazu eine umgeformte, ggf. vereinfachte EBNF Grammatik an. + +=== Task a + +#question[ +``` +StructDecl = ident "=" "{" { Value | NullableValue } "}" . +Value = ident ":" Type . +NullableValue = ident ":" "?" Type . +Type = "i32" | "i64" | "string" . +``` +] + +``` +StructDecl = ident "=" "{" { NullableValue } "}" . +NullableValue = ident ":" ["?"] Type . +Type = "i32" | "i64" | "string" . +``` + +=== Task b + +#question[ +``` +Args = FixArgs ";" VarArgs . +FixArgs = ident ":" Type { ";" ident ":" Type } . +VarArgs = "..." ident . +Type = "i32" | "string" . +``` +] + + +``` +Args = FixArgs VarArgs . +FixArgs = ident ":" Type ";" { ident ":" Type ";" } . +VarArgs = "..." ident . +Type = "i32" | "string" . +``` + +== 03 Beseitigung von Linksrekursion + +#question[ +Beseitigen Sie in folgender Grammatik die enthaltenen LL(1)-Konflikte. Geben Sie dazu eine umgeformte +EBNF Grammatik ohne Rekursion an. Beseitigen Sie mögliche Linksrekursionen durch Umformung in eine +Iteration (nicht durch Umformung in Rechtsrekursion). + +``` +Fields = Type Assigns . +Assigns = ident "=" Expr . +Expr = ident | number | Expr ( "==" | "!=" ) ( ident | number ) . +Type = ident | Type "->" ident . +``` +] + +First lets look at the Type Production: + +``` +Type = ident | Type "->" ident . +``` + +This can be rewritten as: + +``` +Type = ident { "->" ident } . +``` + +Seconnd we need to address the Expr Production: + +``` +Expr = ident | number | Expr ( "==" | "!=" ) ( ident | number ) . +``` + +This can be rewritten as: + +``` +Expr = ( ident | number ) { ( "==" | "!=" ) ( ident | number ) } . +``` + + +== 04 LL(4) Grammatik + +#question[ +Geben Sie eine einfache Grammatik mit mindestens 3 Produktionen an, welche die LL(4)-Eigenschaft, jedoch +nicht die LL(3)-Eigenschaft besitzt. +] + +``` +A = B | C +B = "one" "two" "three" "B" +C = "one" "two" "three" "C" +```