Construct a grammar over {a, b} whose language contains precisely the strings with the same numb

Construct a grammar over {a, b} whose language contains...

Key Concepts

Context-Free Grammar

A context?free grammar (CFG) is a formal system used to define languages by specifying a set of production rules that describe how strings in the language can be generated from a start symbol. CFGs are particularly useful for languages that have recursive structures, as they can express nested or balanced patterns that are not possible with regular grammars.

Production Rules

Production rules are the building blocks of a grammar that dictate how symbols can be replaced or combined to form strings in a language. These rules typically involve nonterminal symbols being expanded into sequences of terminal and/or nonterminal symbols, allowing for the recursive construction of strings with certain properties.

Balanced Strings

Balanced strings refer to strings that exhibit a specific type of symmetry or equivalence, such as having an equal number of two different symbols. In the context of a grammar over the alphabet {a, b}, the goal is to generate strings that contain the same number of a's and b's, which often requires carefully designed production rules that ensure any introduction of one symbol is balanced by the introduction of the other.

Transcript

00:01 We are given languages and we are asked to find phrase structure grammars for each of these languages.

00:11 In part a, we're given the set consisting of the bit strings 0, 1, and 1.

00:20 So we know that our grammar g is going to be a 4 -touple with components v, t, s, and p, where v is the vocabulary.

00:46 T is a set of terminal symbols, s is the start symbol, and p is the set of productions.

00:59 So i'm actually going to identify the set of productions first, and then i'll use these to find v and t.

01:11 So to obtain the bit string 0, i want to include the production s produces 0, to obtain the bitstream 1, we want to also include the production s produces 1.

01:28 And to obtain the bit string 1 -1, we want to include the production s produces 1 -1.

01:37 So we have 3 productions, and we see that our vocabulary v is going to include 0, 1, and the start symbol s.

01:51 And the set of terminals is simply going to be 0 and 1.

02:02 Now, in part b, we are given the set of bit strings containing only ones.

02:19 Once again, our grammar g is a four tuple with components v, t, s, and p, where v is the vocabulary, t is the set of terminal symbols, s is the start symbol, and p is the set of productions.

02:38 So to determine our vocabulary first, i'll determine the set of production.

02:43 We know that our set of terminal symbols is going to be simply one, since our strings are only going to contain ones.

03:00 Now to obtain the set of bit strings containing only ones, i want to include, well, we could have an empty string, so i want to include the production s produces lambda, and we also want to include the production s produces 1s.

03:21 So this enables us with this, along with lambda, to produce any string of arbitrary length, which is all ones.

03:32 And so these are all the productions we need.

03:37 And we see that our vocabulary is simply going to be the set 1s...

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