Pushdown Automata and Turing Machine

Pushdown Automata and Turing Machine

Contents

Recall Pushdown Automata 1

Describe Acceptance of CFL by PDA: Final state and Empty store 5

Construct PDA for CFLs accepted by Final state 6

Construct PDA for CFLs accepted by Empty store 9

Convert a CFG into PDA and vice-a-versa 10

Define Two-Stack PDA 10

Describe DPDA and DCFL 10

Describe Turning Machine with Model 10

Recall representation of Turing Machine 10

Describe String acceptance by a TM 10

Recall TM Head Pointer movements 10

Recall the rules for designing the Turing Machine 10

Design TM for the Strings and Languages 10

Design TM for the Functions 10

Recall the following: i. Universal Turing Machine ii. Multi-Tape Turing Machine iii. Multi-dimensional Turing Machine iv. Multi-head Turing Machine 10

Describe Church’s Thesis 10

Recall Recursive and Recursively Enumerable Languages 10

List properties of Recursive and Recursively Enumerable Languages 10

Describe Counter Machines and their similarities with TM 10

Recall Undecidable problem of TMs 10

Describe PCP and MPCP Problems 10

Describe Recursive Functions Theory 10

Recall Initial, Partial, and Primitive Recursive Functions 10

Recall Pushdown Automata

Pushdown Automata (PDA) is a type of automaton that extends the capabilities of a finite automaton (FA) by including a stack. A PDA can be formally defined as a 6-tuple (Q, Σ, Γ, δ, q0, F),

where:

  • Q is a finite set of states
  • Σ is a finite set of input symbols
  • Γ is a finite set of stack symbols
  • δ: Q × (Σ ∪ {ε}) × Γ → P(Q × Γ*) is the transition function
  • q0 ∈ Q is the initial state
  • F ⊆ Q is the set of final (or accepting) states.

At any given time, a PDA is in a state, has a finite input string to be processed, and has a stack that can be used to store symbols. The PDA reads input symbols from the input string and uses the stack to keep track of information about the input string that it has already seen. The transition function, δ, determines how the PDA can change its state and/or modify the stack based on the current input symbol and the top of the stack.

Alternately, A pushdown automaton (PDA) is a type of automaton that extends the capabilities of a finite automaton with an additional stack. It is used to recognize context-free languages (CFLs) by utilizing a stack to keep track of the context or memory of the language.

A pushdown automaton consists of the following components:

  1. Input alphabet: A finite set of symbols that represent the input to the automaton.
  2. Stack alphabet: A finite set of symbols that represent the stack symbols.
  3. Transition function: A set of rules that define the transitions of the PDA based on the current state, input symbol, and top symbol of the stack. The transition function determines how the PDA reads the input and manipulates the stack.
  4. Start state: The initial state of the PDA.
  5. Accept states: The set of states in which the PDA accepts the input.

The operation of a PDA can be summarized as follows:

  1. Initially, the PDA is in the start state with an empty stack.
  2. The PDA reads an input symbol from the input tape and checks the top symbol of the stack.
  3. Based on the current state, input symbol, and top stack symbol, the PDA uses the transition function to determine the next state and how to manipulate the stack.
  4. The PDA repeats steps 2 and 3 until it reaches an accept state or there are no more input symbols to read.
  5. If the PDA reaches an accept state and the stack is empty, it accepts the input; otherwise, it rejects the input.

The stack in a PDA allows it to remember previously seen symbols and perform stack operations such as push (adding a symbol to the stack) and pop (removing the top symbol from the stack). This additional memory enables PDAs to recognize context-free languages, which have a higher level of complexity than regular languages.

PDAs are a fundamental concept in the theory of formal languages and automata. They provide a formal framework for understanding and analyzing context-free grammars and languages.

Applications of PDA:

Pushdown automata (PDAs) have various applications in computer science and related fields.

Here are some common applications of PDAs:

  1. Language Recognition: PDAs are used to recognize and validate context-free languages (CFLs). They can be employed in compilers, interpreters, and language processors to analyze and process programming languages, formal grammars, and syntax structures.
  2. Parsing: PDAs are utilized in parsing algorithms to analyze the structure of a given input string based on a given grammar. They can perform top-down or bottom-up parsing, constructing parse trees or abstract syntax trees.
  3. Compiler Design: PDAs play a crucial role in various stages of compiler design, including lexical analysis, syntactic analysis, and semantic analysis. PDAs can help tokenize input strings, perform syntax checking, and enforce language-specific rules.
  4. Natural Language Processing (NLP): PDAs are employed in NLP applications to process and analyze natural language structures, such as sentence parsing, semantic analysis, and machine translation. They assist in understanding the grammatical structure and meaning of natural language sentences.
  5. XML Processing: PDAs are used in XML processing applications, such as XML parsing, validation, and transformation. PDAs can ensure that XML documents adhere to specific XML schemas or DTDs and perform operations like document transformation or extraction.
  6. Code Generation: PDAs can be used in code generation tasks to generate executable code or intermediate representations from high-level language specifications. They can enforce language-specific rules, perform type checking, and generate optimized code.
  7. Automated Theorem Proving: PDAs can be employed in automated theorem proving systems to verify the correctness of mathematical proofs and logical statements. They assist in checking the syntax and structure of formal proofs.
  8. Model Checking: PDAs are used in model checking algorithms to verify the correctness of finite-state systems, such as hardware designs or software protocols. They help analyze the state transitions and properties of the system.

These are just a few examples of the applications of PDAs. Their ability to handle context-free languages and stack-based operations makes them valuable in various areas of computer science, including language processing, formal methods, and artificial intelligence.

Types of PDA and its representation techniques:

PDAs (Pushdown Automata) can be classified based on different criteria, such as the type of stack, the type of acceptance, and the determinism of their transitions. Here are some common classifications of PDAs:

  1. Deterministic PDA (DPDA):

A DPDA is a PDA where for each combination of current state, input symbol, and stack symbol, there is at most one possible transition. The transitions are determined deterministically, without requiring any choices. DPDAs recognize the class of deterministic context-free languages (DCFLs).

  1. Non-deterministic PDA (NPDA):

An NPDA is a PDA where for a given combination of current state, input symbol, and stack symbol, there may be multiple possible transitions. The transitions can have non-deterministic choices, allowing the NPDA to explore multiple paths during its computation. NPDAs recognize the class of context-free languages.

  1. ε-NPDA (Epsilon-NPDA):

An ε-NPDA is a variation of NPDA where ε-transitions, also known as epsilon transitions, are allowed. ε-transitions can be taken without reading any input symbol, and they can also involve stack operations. The inclusion of ε-transitions increases the expressive power of the PDA.

  1. Two-way PDA:

A two-way PDA is a PDA that can read the input symbols in both directions, either left to right or right to left. The stack is still accessed from one end, but the input tape can be traversed bidirectionally. Two-way PDAs recognize the same class of languages as ordinary PDAs.

Representation Techniques:

PDAs can be represented using various formalisms and notations, including:

  • Transition Diagrams: Transitions between states are represented using directed arrows labeled with input symbols, stack symbols, and stack operations.
  • Transition Tables: A tabular representation of transitions, listing the current state, input symbol, stack symbol, and the next state along with stack operations.
  • Transition Functions: A mathematical notation that defines the transitions in terms of a function that takes the current state, input symbol, and stack symbol as arguments and returns the next state and stack operations.
  • Formal Grammars: PDAs can be described using formal grammars such as context-free grammars (CFGs) or augmented grammars.

These representation techniques help in visualizing and understanding the behavior of PDAs, and they serve as a basis for designing and implementing PDA-based algorithms and systems.

Describe Acceptance of CFL by PDA: Final state and Empty store

Acceptance of context-free languages (CFLs) by a pushdown automaton (PDA) can be determined using two different criteria: final state acceptance and empty store acceptance.

  1. Final State Acceptance:

In this criterion, a PDA accepts a string if it reaches a final state after processing the entire input while the stack may still contain symbols. The PDA does not have to empty the stack completely.

The acceptance process can be summarized as follows:

  • Initially, the PDA is in the start state with an empty stack.
  • It reads the input symbols one by one and performs state transitions based on the current state, input symbol, and top symbol of the stack.
  • After processing the entire input, if the PDA reaches a final state, it accepts the string regardless of the remaining symbols on the stack.
  1. Empty Store Acceptance:

In this criterion, a PDA accepts a string if it empties the stack completely after processing the entire input. The final state of the PDA is not considered for acceptance.

The acceptance process can be summarized as follows:

  • Initially, the PDA is in the start state with an empty stack.
  • It reads the input symbols one by one and performs state transitions based on the current state, input symbol, and top symbol of the stack.
  • After processing the entire input, if the PDA reaches an accepting state and the stack becomes empty, it accepts the string.

It’s important to note that the choice of acceptance criterion can affect the class of languages recognized by the PDA. For example, PDAs that use final state acceptance recognize the class of all context-free languages, while PDAs that use empty store acceptance recognize a subclass of context-free languages known as the deterministic context-free languages (DCFLs).

The acceptance criterion of a PDA is determined by its design and the requirements of the specific application or language being recognized.

Construct PDA for CFLs accepted by Final state

To construct a PDA for a CFL accepted by final state, we can follow these steps:

  1. Read the input string and push a special symbol (say $) onto the stack to mark the bottom of the stack.
  2. Transition from the start state to a new state by reading the first input symbol and pushing a non-terminal onto the stack.
  3. Use the stack to keep track of which non-terminal is being expanded.
  4. For each non-terminal on the stack, use a transition to expand it to one of its production rules.
  5. If the stack is empty and the input string is also empty, accept the input string by transitioning to a final state.

Example 1: Design a PDA for the language {anb2n | n>=1} accepted by final state.

Explanation: In this language, n number of a’s should be followed by 2n number of b’s. Hence, we will apply a very simple logic, and that is if we read single ‘a’, we will push two a’s onto the stack. As soon as we read ‘b’ then for every single ‘b’ only one ‘a’ should get popped from the stack.

The ID can be constructed as follows:

  1. δ(q0, a, Z) = (q0, aaZ)
  2. δ(q0, a, a) = (q0, aaa)

Now when we read b, we will change the state from q0 to q1 and start popping corresponding ‘a’. Hence,

3. δ(q0, b, a) = (q1, ε)

Thus this process of popping ‘b’ will be repeated unless all the symbols are read. Note that popping action occurs in state q1 only.

4. δ(q1, b, a) = (q2, ε)

After reading all b’s, all the corresponding a’s should get popped. Hence when we read ε as input symbol then there should be nothing in the stack. Hence the move will be:

5. δ(q2, b, a) = (q1, ε)

Where

PDA = ({q0, q1, q2}, {a, b}, {a, Z}, δ, q0, Z, {q2})

We can summarize the ID as:

  1. δ(q0, a, Z) = (q0, aaZ)
  2. δ(q0, a, a) = (q0, aaa)
  3. δ(q0, b, a) = (q1, ε)
  4. δ(q1, b, a) = (q1, ε)
  5. δ(q1, ε, Z) = (q2, ε)

Now we will simulate this PDA for the input string “aaabbbbbb”.

δ(q0, aaabbbbbb, Z) ⊢ δ(q0, aabbbbbb, aZ)

⊢ δ(q0, abbbbbb, aaZ)

⊢ δ(q0, bbbbbb, aaaZ)

⊢ δ(q1, bbbbb, aaaZ)

⊢ δ(q2, bbbb, aaZ)

⊢ δ(q1, bbb, aaZ)

⊢ δ(q2, bb, aZ)

⊢ δ(q1, b, aZ)

⊢ δ(q2, ε, Z)

Accepted

Example 2: Design a PDA for the language {0n1m0n | m, n>=1} accepted by final state.

Explanation: In this PDA, n number of 0’s are followed by any number of 1’s followed n number of 0’s. Hence the logic for design of such PDA will be as follows:

Push all 0’s onto the stack on encountering first 0’s. Then if we read 1, just do nothing. Then read 0, and on each read of 0, pop one 0 from the stack.

This scenario can be written in the ID form as:

  1. δ(q0, 0, Z) = δ(q0, 0Z)
  2. δ(q0, 0, 0) = δ(q0, 00)
  3. δ(q0, 1, 0) = δ(q1, 0)
  4. δ(q1, 0, 0) = δ(q2, ε)
  5. δ(q2, 0, 0) = δ(q2, ε)

(ACCEPT state)

Now we will simulate this PDA for the input string “0011100”.

δ(q0, 0011100, Z) ⊢ δ(q0, 011100, 0Z)

⊢ δ(q0, 11100, 00Z)

⊢ δ(q0, 1100, 00Z)

⊢ δ(q1, 100, 00Z)

⊢ δ(q1, 00, 00Z)

⊢ δ(q2, 0, 0Z)

⊢ δ(q2, ε, Z)

Accepted

Construct PDA for CFLs accepted by Empty store

To construct a PDA for a CFL accepted by empty store, we can follow these steps:

  1. Read the input string and push a special symbol (say Z0) onto the stack to mark the bottom of the stack.
  2. Transition from the start state to a new state by reading the first input symbol and pushing a non-terminal onto the stack.
  3. Use the stack to keep track of which non-terminal is being expanded.
  4. For each non-terminal on the stack, use a transition to expand it to one of its production rules.
  5. If the stack is empty and the input string is also empty, accept the input string by transitioning to a final state and then pop the bottom marker symbol Z0.

Example 1: Design a PDA for the language {anb2n | n>=1} accepted by Empty store.

Explanation: In this language, n number of a’s should be followed by 2n number of b’s. Hence, we will apply a very simple logic, and that is if we read single ‘a’, we will push two a’s onto the stack. As soon as we read ‘b’ then for every single ‘b’ only one ‘a’ should get popped from the stack.

The ID can be constructed as follows:

  1. δ(q0, a, Z) = (q0, aaZ)
  2. δ(q0, a, a) = (q0, aaa)

Now when we read b, we will change the state from q0 to q1 and start popping corresponding ‘a’. Hence,

3. δ(q0, b, a) = (q1, ε)

Thus this process of popping ‘b’ will be repeated unless all the symbols are read. Note that popping action occurs in state q1 only.

4. δ(q1, b, a) = (q1, ε)

After reading all b’s, all the corresponding a’s should get popped. Hence when we read ε as input symbol then there should be nothing in the stack. Hence the move will be:

5. δ(q1, ε, Z) = (q2, ε)

Where,

PDA = ({q0, q1, q2}, {a, b}, {a, Z}, δ, q0, Z, {q2})

We can summarize the ID as:

  1. δ(q0, a, Z) = (q0, aaZ)
  2. δ(q0, a, a) = (q0, aaa)
  3. δ(q0, b, a) = (q1, ε)
  4. δ(q1, b, a) = (q1, ε)
  5. δ(q1, ε, Z) = (q2, ε)

Now we will simulate this PDA for the input string “aaabbbbbb”.

  1. δ(q0, aaabbbbbb, Z) ⊢ δ(q0, aabbbbbb, aaZ)
  2. ⊢ δ(q0, abbbbbb, aaaaZ)
  3. ⊢ δ(q0, bbbbbb, aaaaaaZ)
  4. ⊢ δ(q1, bbbbb, aaaaaZ)
  5. ⊢ δ(q1, bbbb, aaaaZ)
  6. ⊢ δ(q1, bbb, aaaZ)
  7. ⊢ δ(q1, bb, aaZ)
  8. ⊢ δ(q1, b, aZ)
  9. ⊢ δ(q1, ε, Z)
  10. ⊢ δ(q2, ε)
  11. ACCEPT

Example 2: Design a PDA for the language {0n1m0n | m, n>=1}.

Explanation: In this PDA, n number of 0’s are followed by any number of 1’s followed n number of 0’s. Hence the logic for design of such PDA will be as follows:

Push all 0’s onto the stack on encountering first 0’s. Then if we read 1, just do nothing. Then read 0, and on each read of 0, pop one 0 from the stack.

This scenario can be written in the ID form as:

  1. δ(q0, 0, Z) = δ(q0, 0Z)
  2. δ(q0, 0, 0) = δ(q0, 00)
  3. δ(q0, 1, 0) = δ(q1, 0)
  4. δ(q0, 1, 0) = δ(q1, 0)
  5. δ(q1, 0, 0) = δ(q1, ε)
  6. δ(q0, ε, Z) = δ(q2, Z) (ACCEPT state)

Now we will simulate this PDA for the input string “0011100”.

  1. δ(q0, 0011100, Z) ⊢ δ(q0, 011100, 0Z)
  2. ⊢ δ(q0, 11100, 00Z)
  3. ⊢ δ(q0, 1100, 00Z)
  4. ⊢ δ(q1, 100, 00Z)
  5. ⊢ δ(q1, 00, 00Z)
  6. ⊢ δ(q1, 0, 0Z)
  7. ⊢ δ(q1, ε, Z)
  8. ⊢ δ(q2, Z)
  9. ACCEPT

Convert a CFG into PDA and vice-a-versa

A context-free grammar (CFG) can be converted into a pushdown automaton (PDA) and vice versa. The conversion process involves following steps:

Conversion of CFG to PDA:

  1. For every production rule of the CFG, create a corresponding transition in the PDA.
  2. For every non-terminal symbol in the CFG, push a unique symbol on the stack and transition to the corresponding state in the PDA.
  3. For every terminal symbol in the CFG, consume the input symbol and transition to the next state in the PDA.
  4. For every epsilon rule in the CFG, create a transition that pops the stack symbol without consuming any input symbol.

Conversion of PDA to CFG:

  1. For every transition in the PDA, create a corresponding production rule in the CFG.
  2. For every state in the PDA, create a corresponding non-terminal symbol in the CFG.
  3. For every symbol pushed on the stack in the PDA, create a corresponding terminal symbol in the CFG.
  4. For every transition that pops a symbol from the stack in the PDA, create a production rule that generates the corresponding string of terminal symbols.

The PDA generated from a CFG accepts the same language as the CFG, and vice versa.

Alternately,

To convert a Context-Free Grammar (CFG) into a Pushdown Automaton (PDA), and vice versa, we can follow the following procedures:

Conversion from CFG to PDA:

  1. Create states: Each non-terminal symbol in the CFG will correspond to a state in the PDA.
  2. Set the initial state: Designate one state as the initial state of the PDA.
  3. Set the final state(s): Determine which state(s) will be the final state(s) of the PDA.
  4. Transitions: For each production rule in the CFG, create transitions in the PDA based on the following rules:

a. If the production rule has the form A -> a, where A is a non-terminal and a is a terminal symbol, create a transition that reads the terminal symbol a from the input and pops the top symbol from the stack.
b. If the production rule has the form A -> BC, where A, B, and C are non-terminals, create a transition that reads an input symbol (if necessary) and pops the top symbol from the stack. Then, push B onto the stack and transition to a new state corresponding to B. Finally, transition to another state corresponding to C.
c. If the production rule has the form A -> ε, where A is a non-terminal, create a transition that does not read any input symbol and pops the top symbol from the stack.

Conversion from PDA to CFG:

  1. Variables: Each state of the PDA corresponds to a variable in the CFG.
  2. Terminals: Each input symbol in the PDA corresponds to a terminal symbol in the CFG.
  3. Start symbol: Set the start symbol of the CFG to be the initial state of the PDA.
  4. Productions: Create productions for each transition in the PDA based on the following rules:

a. For transitions that read a terminal symbol and pop a symbol from the stack, create a production that matches the terminal symbol and replaces the variable with the symbols on top of the stack.
b. For transitions that push a symbol onto the stack and transition to a new state, create a production that replaces the variable with the new state variable and the symbols on top of the stack.
c. For transitions that pop a symbol from the stack without reading any input symbol, create a production that replaces the variable with the symbols on top of the stack.
d. For transitions that do not read any input symbol and do not pop any symbol from the stack, create a production that replaces the variable with the symbols on top of the stack.

Note: The conversion from CFG to PDA and from PDA to CFG may not always be one-to-one, meaning there can be multiple PDAs corresponding to the same CFG and vice versa. The procedures provided above are general guidelines and may need to be adapted based on the specific requirements and constraints of the given CFG and PDA.

Alternately,

Algorithm to find CFG corresponding to a given PDA

Input: A CFG, G = (V, T, P, S)

Output: Equivalent PDA, P = (Q, ∑, S, δ, q0, I, F) such that the non- terminals of the grammar G will be {Xwx | w,x ∈ Q} and the start state will be Aq0,F.

Step 1: For every w, x, y, z ∈ Q, m ∈ S and a, b ∈ ∑, if δ (w, a, ε) contains (y, m) and (z, b, m) contains (x, ε), add the production rule Xwx → a Xyzb in grammar G.

Step 2: For every w, x, y, z ∈ Q, add the production rule Xwx → XwyXyx in grammar G.

Step 3: For w ∈ Q, add the production rule Xww → ε in grammar G.

Example 1: Convert the following grammar to a PDA that accepts the same language.

  1. S → 0S1 | A
  2. A → 1A0 | S | ε

Explanation:

The CFG can be first simplified by eliminating unit productions:

  1. S → 0S1 | 1S0 | ε

Now we will convert this CFG to GNF:

  1. S → 0SX | 1SY | ε
  2. X → 1
  3. Y → 0

The PDA can be:

R1: δ(q, ε, S) = {(q, 0SX) | (q, 1SY) | (q, ε)}

R2: δ(q, ε, X) = {(q, 1)}

R3: δ(q, ε, Y) = {(q, 0)}

R4: δ(q, 0, 0) = {(q, ε)}

R5: δ(q, 1, 1) = {(q, ε)}

Example 2: Construct PDA for the given CFG, and test whether 0104 is acceptable by this PDA.

  1. S → 0BB
  2. B → 0S | 1S | 0

Explanation:

The PDA can be given as:

  1. A = {(q), (0, 1), (S, B, 0, 1), δ, q, S, ?}

The production rule δ can be:

R1: δ(q, ε, S) = {(q, 0BB)}

R2: δ(q, ε, B) = {(q, 0S) | (q, 1S) | (q, 0)}

R3: δ(q, 0, 0) = {(q, ε)}

R4: δ(q, 1, 1) = {(q, ε)}

Testing 0104 i.e. 010000 against PDA:

  1. δ(q, 010000, S) ⊢ δ(q, 010000, 0BB)
  2. ⊢ δ(q, 10000, BB) R1
  3. ⊢ δ(q, 10000,1SB) R3
  4. ⊢ δ(q, 0000, SB) R2
  5. ⊢ δ(q, 0000, 0BBB) R1
  6. ⊢ δ(q, 000, BBB) R3
  7. ⊢ δ(q, 000, 0BB) R2
  8. ⊢ δ(q, 00, BB) R3
  9. ⊢ δ(q, 00, 0B) R2
  10. ⊢ δ(q, 0, B) R3
  11. ⊢ δ(q, 0, 0) R2
  12. ⊢ δ(q, ε) R3
  13. ACCEPT

Thus 0104 is accepted by the PDA.

Example 3: Design a PDA for the CFG given below:

  1. S → aSb
  2. S → a | b | ε

Solution:

The PDA can be given as:

P = {(q), (a, b), (S, a, b, z0), δ, q, z0, q}

The mapping function δ will be:

R1: δ(q, ε, S) = {(q, aSb)}

R2: δ(q, ε, S) = {(q, a) | (q, b) | (q, ε)}

R3: δ(q, a, a) = {(q, ε)}

R4: δ(q, b, b) = {(q, ε)}

R5: δ(q, ε, z0) = {(q, ε)}

Simulation: Consider the string aaabb

  1. δ(q, εaaabb, S) ⊢ δ(q, aaabb, aSb) R3
  2. ⊢ δ(q, εaabb, Sb) R1
  3. ⊢ δ(q, aabb, aSbb) R3
  4. ⊢ δ(q, εabb, Sbb) R2
  5. ⊢ δ(q, abb, abb) R3
  6. ⊢ δ(q, bb, bb) R4
  7. ⊢ δ(q, b, b) R4
  8. ⊢ δ(q, ε, z0) R5
  9. ⊢ δ(q, ε)
  10. ACCEPT

Example of Converting a PDA to a CFG

The PDA:

M = ({q0,q1}, {0,1}, {X, Zo}, D, q0, Zo, {})

with D (delta):

d(q0,0,Z0) = (q0,XZo) (1a)

d(q0,0,X) = (q0,XX) (1b)

d(q0,1,X) = (q1,e) (2a)

d(q1,1,X) = (q1,e) (2b)

d(q1,e,X) = (q1,e) (2c)

d(q1,e,Zo) = (q1,e) (2d)

The CFG

Obviously, T = {0,1} (same as sigma from the PDA)

We will call the Start Symbol “S”.

V = (S, [q0,X,q0], [q0,X,q1], [q1,X,q0], [q1,X,q1],

[q0,Zo,q0], [q0,Zo,q1], [q1,Zo,q0], [q1,Zo,q1])

The Productions

First, generate the productions from S:

S -> [q0,Zo,q0]

S -> [q0,Zo,q1]

(the first two components of the RHS are fixed, the last piece is every possible state from the machine).

We will derive forward, only building productions for variables that can be derived.

So, for the q0,Zo,q0 variable, generate the productions based on the 1a rule above:

[q0,Zo,q0] -> 0[q0,X,q0][q0,Zo,q0] (1a’)

-> 0[q0,X,q1][q1,Zo,q0] (1a”)

next come the matching productions from the second variable:

[q0,Zo,q1] -> 0[q0,X,q0][q0,Zo,q1] (1a”’)

-> 0[q0,X,q1][q1,Zo,q1] (1a””)

Then, working from the two variables that mention X, and the matching rule 1b, we get the following collection of productions:

[q0,X,q0] -> 0[q0,X,q0][q0,X,q0] (1b’)

-> 0[q0,X,q1][q1,X,q0] (1b”)

[q0,X,q1] -> 0[q0,X,q0][q0,X,q1] (1b”’)

-> 0[q0,X,q1][q1,X,q1] (1b””)

Then for the other four delta moves, we get the following productions:

[q0,X,q1] -> 1 (2a’)

[q1,Zo,q1] -> e (2b’)

[q1,X,q1] -> e (2c’)

[q1,X,q1] -> 1 (2d’)

Now, note that we can tidy up a bit:

The variables [q1,X,q0] and [q1,Zo,q0] never appear as a LHS, so we can drop these productions: 1a”, 1b”.

Similarly, [q0,X,q0] and [q0,Zo,q0] always appear on the RHS of any of their own productions (or just on the RHS), so we can drop these productions: 1a’, 1a”’, 1b’, 1b”’, as well as the first production from the start symbol.

Which leaves us with

S -> [q0,Zo,q1]

[q0,Zo,q1] -> 0[q0,X,q1][q1,Zo,q1]

[q0,X,q1] -> 0[q0,X,q1][q1,X,q1]

and the four 2a-d’ rules.

And all that can be renamed to:

S -> A

A -> 0BC

B -> 0BD | 1

C -> e

D -> 1 | e

Define Two-Stack PDA

A Two-Stack Pushdown Automaton (PDA) is a variation of a PDA that uses two stacks instead of one. It is a computational model used in formal language theory and automata theory. In a Two-Stack PDA, the additional stack allows for increased computational power and can recognize certain languages that cannot be recognized by a standard PDA.

A Two-Stack PDA consists of the following components:

  1. Input Alphabet: The set of symbols that represent the input to the machine.
  2. Stack Alphabets: Two separate sets of symbols that represent the contents of the two stacks.
  3. States: A finite set of states that the machine can be in at any given time.
  4. Initial State: The initial state of the machine where the computation begins.
  5. Accepting States: A set of states that indicate the machine has accepted the input.
  6. Transitions: A set of rules that define how the machine can transition from one state to another based on the input symbol and the top symbols of the two stacks.

The transition rules in a Two-Stack PDA are defined as follows:

δ(q, a, X, Y) -> (p, α)

δ(q, a, X, Y) -> (p, α, β)

δ(q, ε, X, Y) -> (p, α)

δ(q, ε, X, Y) -> (p, α, β)

Here, q and p represent states, a represents an input symbol, X and Y represent the top symbols of the two stacks, and α and β represent sequences of symbols to be pushed or popped onto the stacks.

The operation of a Two-Stack PDA involves reading input symbols, manipulating the contents of the two stacks, and transitioning between states based on the input and stack symbols. The machine accepts an input string if it reaches an accepting state after processing the entire input.

Two-Stack PDAs are more powerful than standard PDAs and can recognize certain context-sensitive languages that cannot be recognized by a standard PDA. However, they are still less powerful than Turing machines, which can recognize all recursively enumerable languages.

The use of two stacks in a Two-Stack PDA provides additional computational capabilities and allows for more complex language recognition and parsing tasks.

Example: construct a PDA using 2 stacks for accepting the language L={a^n b^n c^n d^n | n>=1}.