1. The grammar G: SSS alb is ambiguous. That means at least some of the strings in its language have more than one leftmost derivation. However, it may be that some strings in the language have only one derivation. Identify from the list below a string that has exactly TWO leftmost derivations in G. a) abbab b) aa c) bb d) aab 2. Programming languages are often described using an extended form of context-free grammar, where square brackets are used to denote an optional construct. For example, A → B[C]D says that an A can be replaced by a B and a D, with an optional C between them. This notation does not allow us to describe anything but context-free languages, since an extended production can always be replaced by several conventional productions. Suppose a grammar has the extended productions: A → B[CD] EF | BC [DE] F Convert this pair of extended productions to conventional productions. Identify, from the list below, the conventional productions that are equivalent to the extended productions above. a) A - → BCDEF BEF | BCF BA1F CD | DE | E b) A A1 →> c) A BCDEF | BEF d) A →> BA1F A1 CD | DE 3. Consider the grammars: G₁: S → SaS | aa | a G2: S→ SS & G3: S→ SS a G4: S→ SS | aa G5: S→ Sala G6: SaSalaa|a G7: S→ SAS | ε Describe the language of each of these grammars. Then, identify from the list below a pair of grammars that define the same language. a) G5, G4 b) G6, G2 c) G1, G2 d) G1, G3 b) A → BA1F A1 → CD | DE | ɛ BCDEF ❘ BEF c) A -> d) A A1 -> CD | DE → BA1F 3. Consider the grammars: G₁: S SaS aaa G2: S→ SS & G3: S→ SS a G4: S SS aa G5: S→ Sala G6: SaSa aala G7: S SAS | E →> Describe the language of each of these grammars. Then, identify from the list below a pair of grammars that define the same language. a) G5, G4 b) G6, G2 c) G1, G2 d) G1, G3 4. Here is a context-free grammar G: S →> AB A → 0A1 | 2 B1B3A Which of the following strings is in L(G)? a) 00213021 b) 0021113002111 c) 21113021 d) 021131021 5. Consider the languge L={a}. Which grammar defines L? a) G₁:SACIC, A → b b) G₁:S ACla, A → b →> c) G₁:Sacla, A → clblɛ d) G₁:S →dla, A → clblɛ Submit Homework

1. The grammar G: SSS alb is ambiguous. That means at least some of the strings in its language have more than one leftmost derivation. However, it may be that some strings in the language have only one derivation. Identify from the list below a string that has exactly TWO leftmost derivations in G. a) abbab b) aa c) bb d) aab 2. Programming languages are often described using an extended form of context-free grammar, where square brackets are used to denote an optional construct. For example, A → B[C]D says that an A can be replaced by a B and a D, with an optional C between them. This notation does not allow us to describe anything but context-free languages, since an extended production can always be replaced by several conventional productions. Suppose a grammar has the extended productions: A → B[CD] EF | BC [DE] F Convert this pair of extended productions to conventional productions. Identify, from the list below, the conventional productions that are equivalent to the extended productions above. a) A - → BCDEF BEF | BCF BA1F CD | DE | E b) A A1 →> c) A BCDEF | BEF d) A →> BA1F A1 CD | DE 3. Consider the grammars: G₁: S → SaS | aa | a G2: S→ SS & G3: S→ SS a G4: S→ SS | aa G5: S→ Sala G6: SaSalaa|a G7: S→ SAS | ε Describe the language of each of these grammars. Then, identify from the list below a pair of grammars that define the same language. a) G5, G4 b) G6, G2 c) G1, G2 d) G1, G3 b) A → BA1F A1 → CD | DE | ɛ BCDEF ❘ BEF c) A -> d) A A1 -> CD | DE → BA1F 3. Consider the grammars: G₁: S SaS aaa G2: S→ SS & G3: S→ SS a G4: S SS aa G5: S→ Sala G6: SaSa aala G7: S SAS | E →> Describe the language of each of these grammars. Then, identify from the list below a pair of grammars that define the same language. a) G5, G4 b) G6, G2 c) G1, G2 d) G1, G3 4. Here is a context-free grammar G: S →> AB A → 0A1 | 2 B1B3A Which of the following strings is in L(G)? a) 00213021 b) 0021113002111 c) 21113021 d) 021131021 5. Consider the languge L={a}. Which grammar defines L? a) G₁:SACIC, A → b b) G₁:S ACla, A → b →> c) G₁:Sacla, A → clblɛ d) G₁:S →dla, A → clblɛ Submit Homework

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 14E: Assume that is a binary operation on a non empty set A, and suppose that is both commutative and...

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