CFG: Why is this grammar ambiguous?

Question

The grammar is as follows.

S -> SS' | a | b
S' -> a | b

The way I understand it, derivations from this grammar will be like SS'S'S'S'... (0 or more S') , where each S or S' will generate a or b .

Can someone provide an example that shows this grammar is ambiguous? (The solution says it is.)

Answer 1

It isn't ambiguous. Your analysis is correct.

Here's a mechanical check of your grammar (reshaped for our tool):

S = S Sprime ;
S = a ;
S = b ;
Sprime = a ;
Sprime = b ;

Execution of tool:

C:\DMS\Domains\DMSStringGrammar\Tools\ParserGenerator>run ParserGenerator.P0B -interactive C:\
DMS GLR Parser Generator 2.4.1
Copyright (C) 1997-2018 Semantic Designs, Inc.
Opening C:\temp\Example.bnf
*** EOF seen
<<<Rule Collection Completed>>>
NTokens = 5 NRules = 5
LR(1) Parser Generator -- Find Follow and SLR Lookahead sets
Computing MemberSets for Nonterminal Tokens...

What next? ambiguities 100
Print results where (<CR> defaults to console)?
Default paper width: 80
How wide should the printout be (<CR> selects default)?
*** Search for ambiguities to depth 100

Nonterminal < Sprime > is not ambiguous
*** Search for ambiguities to depth 1; trying 2 rule pairs...
*** Search for ambiguities to depth 2; trying 2 rule pairs...
*** Search for ambiguities to depth 3; trying 2 rule pairs...
*** Search for ambiguities to depth 4; trying 2 rule pairs...
Nonterminal < S > is not ambiguous [modulo rule derivation loops]

*** 0 ambiguities found ***
*** All ambiguities in grammar detected ***

This tool is rather overkill for grammar with two nonterminals. But when somebody gives a set of 200 nonterminals it is much harder to do by hand.

(For theorists: this tool obviously can't decide this for all grammars. It uses a recursive iterative deepening search in the space of nonterminal expansions to look of duplicate/ambiguous expansions. That's works pretty well in pratice).