Question

Show that the following languages are not context-free: a) \(\left\\{a^{n} b^{m} c^{k} \mid n>m>k\right\\}\) b) \(\left\\{w \in\\{a, b, c\\}^{*} \mid n_{a}(w)>n_{b}(w)>n_{c}(w)\right\\}\) c) \(\left\\{w w w \mid w \in\\{a, b\\}^{*}\right\\}\) d) \(\left\\{a^{n} b^{m} c^{k} \mid n, m \in \mathbb{N}\right.\) and \(\left.k=m * n\right\\}\) e) \(\left\\{a^{n} b^{m} \mid m=n^{2}\right\\}\)

Short answer

All the given languages are not context-free. Each one violates certain conditions of the pumping lemma for context-free languages.

Step-by-step solution

Proving language a is not context-free

Attempting to use Pumping Lemma for \( a^{n} b^{m} c^{k} \mid n > m > k \). Assume for contradiction this language is context-free. Then there exists a pumping length \( p \) for it. Consider the word \( w = a^{p} b^{p} c^{p/2} \), which clearly is in the language (since \( p > p > p/2 \)). The lemma tells us that this word can be decomposed into 5 parts \( w = uvxyz \), with |vxy| <= \( p \), |vy| >= 1, and \( u(v^{i}x)(y^{i}z) \) is in the language for every natural number \( i \). However, no matter how we decompose the word, with \( |vxy| <= p \), the pumped word \( u(v^{2}x)(y^{2}z) \) cannot be in the language, since pumping will either change the amount of \( a \)'s, \( b \)'s, or \( c \)'s without affecting the others, violating one of the strict inequality conditions. Therefore our initial assumption that the language is context-free must be wrong.

Proving language \( b \) is not context-free

For \( w \in \{a, b, c\}^{*} \mid n_{a}(w)>n_{b}(w)>n_{c}(w) \), give a similar reasoning as in Step 1. In this case, take a word \( w = a^{p} b^{p/2} c^{p/4} \). No matter how we decompose this word under the pumping conditions, the result of the pumping operation will violate the inequality conditions for the numbers of \( a \), \( b \), and \( c \). Thus, this language is likewise not context-free.

Proving language \( c \) is not context-free

Language \( \{www \mid w \in \{a, b\}^{*}\} \) is not context-free. This can be proven by applying the pumping lemma, using a word \( w = (ab)^{p}(ab)^{p}(ab)^{p} \). Because a context-free grammar can not keep track of three distinct counts on variables \( w \), the pumping operator will violate the repeating requirement of the language.

Proving language \( d \) is not context-free

For \( a^{n} b^{m} c^{k} | n, m \in \mathbb{N} \) and \( k=m*n \), again we apply the pumping lemma. Take a word w=p^{p}q^{1}r^{p}, which is within the language because p=p*1. As in previous steps, the pumping operation will violate the multiplication requirement.

Proving language \( e \) is not context-free

Finally, for \( a^{n} b^{m} | m=n^{2} \), we refer to the pumping lemma for proof. Using the word \( w=a^{p}b^{p^{2}} \), and acknowledging that p^2 = p*p shows that \( w \) is within the language. However, any decomposition and subsequent pumping operation will violate the square requirement.

Key concepts

Pumping Lemma

The pumping lemma is a crucial tool used in the field of formal languages to demonstrate that a particular language is not context-free. It relies on the principle that for any context-free language, there exists a length, say *p*, beyond which we can repeatedly "pump" (or repeat) parts of the word and the new word will still belong to the language. Specifically, for any string *w* of this language, if it is sufficiently long (at least *p* characters or more), it can be divided into five sections: \( uvxyz \). The conditions held are:

• |vxy| \( \leq p \)
• |vy| \( \geq 1 \)

This means that no matter how *w* is split under these rules, when we apply the pumping power and form a new string \( u(v^{i}x)(y^{i}z) \) for any integer *i*, this new string must also belong to the language. By showing that any decomposed string can violate given inequalities or logical constraints in some repeating sequences, we prove that the language is not context-free. This idea is applied in each part of the original exercise to show that they cannot meet the conditions laid out by the pumping lemma.

Formal Languages

Formal languages are mathematical representations of sets of sequences or strings. They are crucial in computer science, linguistics, artificial intelligence, and defines a language in terms of specific rules and symbols. These languages can represent anything from simple search patterns used in computing to complex programming languages.

• A formal language is usually defined over an alphabet, which is a finite set of symbols.
• It includes a set of rules or grammar that determines the combination of these symbols into valid strings.

These rules ensure the sequences produced are well-formed according to particular conditions. Moreover, formal languages are categorized based on their complexity and the kind of rules applied, such as regular languages, context-free languages, etc. In the context of this exercise, we are predominantly dealing with languages that satisfy specific numeric conditions, complex enough to often be outside the realm of context-free languages.

Context-Free Grammars

Context-free grammars (CFGs) are a type of formal grammar that consists of a set of recursive rewriting rules used to generate patterns of strings. CFGs are predominantly used to describe the syntax of programming languages and other structured forms.

• They provide a framework for defining hierarchical structures seen in many naturally occurring and computer languages.
• A CFG consists of variables (or non-terminals), terminals (symbols that appear in the strings of the language), a start variable, and production rules.
• The grammar derives languages by starting with the initial variable and using the production rules repeatedly.

The incapability of CFGs to express certain constraints like overlapping patterns, specific arithmetic progression of counts, or deeply nested structures frequently means that certain languages are not context-free, as showcased in the given exercise examples. For instance, a language requiring equal multiple counts of its symbols cannot typically be represented by a CFG.

Inequalities in Formal Languages

Inequalities in formal languages refer to constraints placed on the quantities or occurrences of symbols within strings. In particular, these constraints often involve comparisons such as "greater than" or "less than," either directly on symbol counts or through mathematical relationships.

• Formal languages can be designed to require such inequalities within their structure, as seen in the original exercise.
• Implementing such constraints often demands intricate grammar rules that fall outside context-free capabilities.
• Languages where symbols need to maintain specific order, or adhere to calculated relationships, like exponential growth, often provide examples of these inequalities.

In the context of the problem, fairly complex inequalities like \( n > m > k \) or \( m = n^2 \) demonstrate the limits of context-free grammars. Such relationships between symbols involve arithmetic operations or comparisons that CFGs generally cannot enforce, hence revealing them as non-context-free through techniques like the pumping lemma.