Before Monday, read Theorem 2.20.
Before Wednesday, read Example 2.18 (page 114).
Before Friday, read Figure 3.1.
For Week 6 Monday: Page 165-166 Introduction to Section 3.1.
Clearly and unambiguously communicate computational ideas using appropriate formalism. Translate across levels of abstraction.
Describe and use models of computation that don’t involve state machines.
Use context-free grammars and relate them to languages and pushdown automata.
Use precise notation to formally define the state diagram of a Turing machine
Use clear English to describe computations of Turing machines informally.
Design a PDA that recognizes a given language.
Give examples of sets that are context-free (and prove that they are).
State the definition of the class of context-free languages
Explain the limits of the class of context-free languages
Identify some context-free sets and some non-context-free sets
Know, select and apply appropriate computing knowledge and problem-solving techniques. Reason about computation and systems.
Describe and prove closure properties of classes of languages under certain operations.
Apply a general construction to create a new PDA or CFG from an example one.
Formalize a general construction from an informal description of it.
Use general constructions to prove closure properties of the class of context-free langugages.
Use counterexamples to prove non-closure properties of the class of context-free langugages.
Schedule your Test 1 Attempt 1, Test 2 Attempt 1, Test 1 Attempt 2, and Test 2 Attempt 2 times at PrairieTest (http://us.prairietest.com)
Review Quiz 5 on PrairieLearn (http://us.prairielearn.com), complete by Sunday 11/4/2024
Warmup: Design a CFG to generate the language \(\{a^i b^j \mid j \geq i \geq 0\}\)
Sample derivation:
Design a PDA to recognize the language \(\{a^i b^j \mid j \geq i \geq 0\}\)
Theorem 2.20: A language is generated by some context-free grammar if and only if it is recognized by some push-down automaton.
Definition: a language is called context-free if it is the language generated by a context-free grammar. The class of all context-free language over a given alphabet \(\Sigma\) is called CFL.
Consequences:
Quick proof that every regular language is context free
To prove closure of the class of context-free languages under a given operation, we can choose either of two modes of proof (via CFGs or PDAs) depending on which is easier
To fully specify a PDA we could give its \(6\)-tuple formal definition or we could give its input alphabet, stack alphabet, and state diagram. An informal description of a PDA is a step-by-step description of how its computations would process input strings; the reader should be able to reconstruct the state diagram or formal definition precisely from such a descripton. The informal description of a PDA can refer to some common modules or subroutines that are computable by PDAs:
PDAs can “test for emptiness of stack” without providing details. How? We can always push a special end-of-stack symbol, \(\$\), at the start, before processing any input, and then use this symbol as a flag.
PDAs can “test for end of input” without providing details. How? We can transform a PDA to one where accepting states are only those reachable when there are no more input symbols.
Suppose \(L_1\) and \(L_2\) are context-free languages over \(\Sigma\). Goal: \(L_1 \cup L_2\) is also context-free.
Approach 1: with PDAs
Let \(M_1 = ( Q_1, \Sigma, \Gamma_1, \delta_1, q_1, F_1)\) and \(M_2 = ( Q_2, \Sigma, \Gamma_2, \delta_2, q_2, F_2)\) be PDAs with \(L(M_1) = L_1\) and \(L(M_2) = L_2\).
Define \(M =\)
Approach 2: with CFGs
Let \(G_1 = (V_1, \Sigma, R_1, S_1)\) and \(G_2 = (V_2, \Sigma, R_2, S_2)\) be CFGs with \(L(G_1) = L_1\) and \(L(G_2) = L_2\).
Define \(G =\)
Suppose \(L_1\) and \(L_2\) are context-free languages over \(\Sigma\). Goal: \(L_1 \circ L_2\) is also context-free.
Approach 1: with PDAs
Let \(M_1 = ( Q_1, \Sigma, \Gamma_1, \delta_1, q_1, F_1)\) and \(M_2 = ( Q_2, \Sigma, \Gamma_2, \delta_2, q_2, F_2)\) be PDAs with \(L(M_1) = L_1\) and \(L(M_2) = L_2\).
Define \(M =\)
Approach 2: with CFGs
Let \(G_1 = (V_1, \Sigma, R_1, S_1)\) and \(G_2 = (V_2, \Sigma, R_2, S_2)\) be CFGs with \(L(G_1) = L_1\) and \(L(G_2) = L_2\).
Define \(G =\)
Summary
Over a fixed alphabet \(\Sigma\), a language \(L\) is regular
iff it is described by some regular expression
iff it is recognized by some DFA
iff it is recognized by some NFA
Over a fixed alphabet \(\Sigma\), a language \(L\) is context-free
iff it is generated by some CFG
iff it is recognized by some PDA
Fact: Every regular language is a context-free language.
Fact: There are context-free languages that are not nonregular.
Fact: There are countably many regular languages.
Fact: There are countably infinitely many context-free languages.
Consequence: Most languages are not context-free!
Examples of non-context-free languages
\[\begin{aligned} &\{ a^n b^n c^n \mid 0 \leq n , n \in \mathbb{Z}\}\\ &\{ a^i b^j c^k \mid 0 \leq i \leq j \leq k , i \in \mathbb{Z}, j \in \mathbb{Z}, k \in \mathbb{Z}\}\\ &\{ ww \mid w \in \{0,1\}^* \} \end{aligned}\] (Sipser Ex 2.36, Ex 2.37, 2.38)
There is a Pumping Lemma for CFL that can be used to prove a specific language is non-context-free: If \(A\) is a context-free language, there is a number \(p\) where, if \(s\) is any string in \(A\) of length at least \(p\), then \(s\) may be divided into five pieces \(s = uvxyz\) where (1) for each \(i \geq 0\), \(uv^ixy^iz \in A\), (2) \(|uv|>0\), (3) \(|vxy| \leq p\). We will not go into the details of the proof or application of Pumping Lemma for CFLs this quarter.
Recall: A set \(X\) is said to be closed under an operation \(OP\) if, for any elements in \(X\), applying \(OP\) to them gives an element in \(X\).
| True/False | Closure claim |
|---|---|
| True | The set of integers is closed under multiplication. |
| \(\forall x \forall y \left( ~(x \in \mathbb{Z} \wedge y \in \mathbb{Z})\to xy \in \mathbb{Z}~\right)\) | |
| True | For each set \(A\), the power set of \(A\) is closed under intersection. |
| \(\forall A_1 \forall A_2 \left( ~(A_1 \in \mathcal{P}(A) \wedge A_2 \in \mathcal{P}(A) \in \mathbb{Z}) \to A_1 \cap A_2 \in \mathcal{P}(A)~\right)\) | |
| The class of regular languages over \(\Sigma\) is closed under complementation. | |
| The class of regular languages over \(\Sigma\) is closed under union. | |
| The class of regular languages over \(\Sigma\) is closed under intersection. | |
| The class of regular languages over \(\Sigma\) is closed under concatenation. | |
| The class of regular languages over \(\Sigma\) is closed under Kleene star. | |
| The class of context-free languages over \(\Sigma\) is closed under complementation. | |
| The class of context-free languages over \(\Sigma\) is closed under union. | |
| The class of context-free languages over \(\Sigma\) is closed under intersection. | |
| The class of context-free languages over \(\Sigma\) is closed under concatenation. | |
| The class of context-free languages over \(\Sigma\) is closed under Kleene star. | |
We are ready to introduce a formal model that will capture a notion of general purpose computation.
Similar to DFA, NFA, PDA: input will be an arbitrary string over a fixed alphabet.
Different from NFA, PDA: machine is deterministic.
Different from DFA, NFA, PDA: read-write head can move both to the left and to the right, and can extend to the right past the original input.
Similar to DFA, NFA, PDA: transition function drives computation one step at a time by moving within a finite set of states, always starting at designated start state.
Different from DFA, NFA, PDA: the special states for rejecting and accepting take effect immediately.
(See more details: Sipser p. 166)
Formally: a Turing machine is \(M= (Q, \Sigma, \Gamma, \delta, q_0, q_{accept}, q_{reject})\) where \(\delta\) is the transition function \[\delta: Q\times \Gamma \to Q \times \Gamma \times \{L, R\}\] The computation of \(M\) on a string \(w\) over \(\Sigma\) is:
Read/write head starts at leftmost position on tape.
Input string is written on \(|w|\)-many leftmost cells of tape, rest of the tape cells have the blank symbol. Tape alphabet is \(\Gamma\) with \(\textvisiblespace\in \Gamma\) and \(\Sigma \subseteq \Gamma\). The blank symbol \(\textvisiblespace \notin \Sigma\).
Given current state of machine and current symbol being read at the tape head, the machine transitions to next state, writes a symbol to the current position of the tape head (overwriting existing symbol), and moves the tape head L or R (if possible).
Computation ends if and when machine enters either the accept or the reject state. This is called halting. Note: \(q_{accept} \neq q_{reject}\).
The language recognized by the Turing machine \(M\), is \(L(M) = \{ w \in \Sigma^* \mid w \textrm{ is accepted by } M\}\), which is defined as \[\{ w \in \Sigma^* \mid \textrm{computation of $M$ on $w$ halts after entering the accept state}\}\]
2
Formal definition:
Sample computation:
| \(q0\downarrow\) | ||||||
|---|---|---|---|---|---|---|
| \(0\) | \(0\) | \(0\) | \(\textvisiblespace\) | \(\textvisiblespace\) | \(\textvisiblespace\) | \(\textvisiblespace\) |
The language recognized by this machine is …
Describing Turing machines (Sipser p. 185) To define a Turing machine, we could give a
Formal definition: the \(7\)-tuple of parameters including set of states, input alphabet, tape alphabet, transition function, start state, accept state, and reject state; or,
Implementation-level definition: English prose that describes the Turing machine head movements relative to contents of tape, and conditions for accepting / rejecting based on those contents.
High-level description: description of algorithm (precise sequence of instructions), without implementation details of machine. As part of this description, can “call" and run another TM as a subroutine.
Fix \(\Sigma = \{0,1\}\), \(\Gamma = \{ 0, 1, \textvisiblespace\}\) for the Turing machines with the following state diagrams:
Example of string accepted:
Example of string rejected:
Implementation-level description
High-level description
Example of string accepted:
Example of string rejected:
Implementation-level description
High-level description
Example of string accepted:
Example of string rejected:
Implementation-level description
High-level description
Example of string accepted:
Example of string rejected:
Implementation-level description
High-level description