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 {0ⁿ1^m0ⁿ | 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 {0ⁿ1^m0ⁿ | 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}.

Describe DPDA and DCFL

DPDA (Deterministic Pushdown Automaton):

A DPDA, or Deterministic Pushdown Automaton, is a variation of a Pushdown Automaton (PDA) that operates deterministically. It is a computational model used in formal language theory and automata theory. A DPDA extends the capabilities of a PDA by incorporating determinism, meaning that for each input symbol and top stack symbol, there is at most one valid transition.

Components of a DPDA:

1. Input Alphabet: The set of symbols that represent the input to the machine.
2. Stack Alphabet: The set of symbols that can be pushed onto the stack.
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 stack symbol.

In a DPDA, the transition function is deterministic, which means that for each combination of current state, input symbol, and top stack symbol, there is at most one valid transition. This determinism ensures that the machine follows a unique computation path for any given input.

DCFL (Deterministic Context-Free Language):

A DCFL, or Deterministic Context-Free Language, is a subset of the context-free languages (CFLs) that can be recognized by a DPDA. A language is considered deterministic if there exists a DPDA that recognizes it. In other words, a language is deterministic if it can be generated by a deterministic grammar or accepted by a deterministic automaton.

DCFLs have certain properties that make them more predictable and easier to work with than general CFLs. Determinism allows for unambiguous parsing of the input strings, as there is at most one valid computation path for any given input. This property simplifies the analysis and processing of DCFLs compared to non-deterministic CFLs.

DCFLs can be represented using deterministic grammars or recognized by DPDAs. Deterministic grammars are a subset of context-free grammars (CFGs) that produce deterministic languages. These grammars have deterministic productions, which means that for each non-terminal and input symbol combination, there is at most one production rule.

The deterministic nature of DPDAs and DCFLs provides advantages in terms of efficiency and ease of analysis. However, not all CFLs are deterministic, and some languages require non-deterministic PDA models for their recognition.

Describe Turning Machine with Model

A Turing Machine (TM) is a theoretical computational model invented by Alan Turing in the 1930s. It is a mathematical abstraction used to study the concept of computation and the limits of what can be computed algorithmically. The Turing Machine serves as the foundation of modern computer science and the theory of computation.

A Turing Machine is defined by a set of tuples that describe its behavior. These tuples specify the transition rules and properties of the Turing Machine.

The standard tuples used to define a Turing Machine are as follows:

1. State Set (Q): It is a finite set of states that the Turing Machine can be in. Each state represents a particular configuration of the machine.
2. Alphabet (Σ): It is a finite set of symbols that can appear on the tape. The alphabet includes both input symbols and tape symbols.
3. Tape Alphabet (Γ): It is a finite set of symbols that can be written on the tape. The tape alphabet is a superset of the input alphabet.
4. Initial State (q0): It is the initial state of the Turing Machine when the computation starts.
5. Accept State (F): It is a set of states in which the Turing Machine halts and accepts the input.
6. Reject State (R): It is a set of states in which the Turing Machine halts and rejects the input.
7. Transition Function (δ): It defines the behavior of the Turing Machine. The transition function is defined as δ: Q × Γ → Q × Γ × {L, R}, where δ(q, a) = (p, b, D) means that if the Turing Machine is in state q and reads symbol a on the tape, it transitions to state p, writes symbol b on the tape, and moves the head in direction D (L for left, R for right).

The tuples provide a formal and concise representation of the Turing Machine’s characteristics and behavior. They define the set of possible states, the symbols that can appear on the tape, the initial and accepting/rejecting states, and the rules for transitioning between states based on the current state and the symbol read.

By defining these tuples, we can precisely describe the behavior of a Turing Machine and analyze its computational capabilities. Turing Machines can be used to solve various computational problems and simulate the behavior of other computational models.

Model of a Turing Machine:

A Turing Machine consists of the following components:

1. Tape: The tape is an infinite, one-dimensional strip divided into cells. Each cell can hold a symbol from a finite alphabet. The tape serves as the primary storage for input, output, and intermediate results during computation.
2. Head: The head is a device that reads and writes symbols on the tape. It can move left or right along the tape to access different cells.
3. State Register: The state register holds the current state of the Turing Machine. It determines the behavior of the machine based on the current symbol being read and the current state.
4. Transition Function: The transition function defines the behavior of the Turing Machine. It specifies the next state to transition to, the symbol to write on the tape, and the direction in which the head should move (left or right) based on the current state and the symbol being read.

The operation of a Turing Machine involves the following steps:

1. Initialization: The Turing Machine starts in an initial state with an input string written on the tape.
2. Read and Write: The head reads the symbol from the current cell on the tape. Based on the current state and the symbol being read, the machine determines the next state and the symbol to write on the tape. The head writes the new symbol in the current cell.
3. Move: The head moves either left or right along the tape based on the transition specified by the current state.
4. State Transition: The state of the Turing Machine changes according to the transition function. The machine transitions to a new state based on the current state and the symbol being read.
5. Repeat: Steps 2-4 are repeated until the machine reaches a halting state, indicating the completion of the computation.

The Turing Machine is a powerful computational model that can simulate any algorithm or computation that can be described in an algorithmic manner. It can solve a wide range of computational problems, from simple calculations to complex simulations. The Turing Machine concept provides a theoretical foundation for the design and analysis of modern computers and algorithms.

Applications of Turing Machines:

Turing Machines (TMs) have various applications in theoretical computer science and beyond. Here are some notable applications:

1. Computability and Complexity Theory: Turing Machines are fundamental tools for studying computability and complexity. They help define and analyze the limits of what can be computed and how efficiently it can be done. TMs are used to classify problems into different complexity classes, such as P, NP, and undecidable problems.
2. Algorithm Design and Analysis: Turing Machines provide a theoretical framework for designing and analyzing algorithms. By simulating a problem on a TM, its computational steps can be modeled and studied, leading to insights into algorithmic properties, time complexity, space complexity, and overall efficiency.
3. Compiler and Programming Language Design: Turing Machines serve as a theoretical basis for designing programming languages and building compilers. The design of programming languages involves considering the syntax, semantics, and execution of programs, which can be formalized using TMs. Compilers convert high-level code into machine-readable instructions, and TMs help in understanding and optimizing this translation process.
4. Artificial Intelligence and Machine Learning: Turing Machines form the theoretical foundation of artificial intelligence and machine learning. They are used to study and develop algorithms for various AI tasks, such as problem-solving, pattern recognition, natural language processing, and decision-making. TMs are also used in the analysis of learning algorithms and the study of computational models for intelligence.
5. Formal Language Theory: Turing Machines are used to define and study formal languages, including regular languages, context-free languages, and recursively enumerable languages. TMs can determine the membership of strings in languages, help in language recognition, and provide insights into language generation and manipulation.
6. Cryptography: TMs have applications in cryptography, particularly in the analysis and design of cryptographic protocols and algorithms. They help in studying the security properties of cryptographic systems, analyzing their resistance to attacks, and developing new encryption and decryption methods.
7. Computational Biology: TMs are used in computational biology to model biological processes, analyze DNA and protein sequences, simulate genetic algorithms, and study molecular interactions. TMs provide a framework for understanding the computational aspects of biological systems.

These are just a few examples of the wide-ranging applications of Turing Machines. Their versatility and theoretical underpinnings make them a fundamental tool for studying computation and its various aspects.

Recall representation of Turing Machine

A Turing Machine can be represented using various notations and diagrams to visualize its components and behavior. Here are some common representations of Turing Machines:

1. Transition Table: A Turing Machine can be represented using a transition table. The table lists all the possible combinations of the current state and the symbol being read, along with the next state, symbol to write, and direction to move the tape head. Each row in the table represents a transition rule.
2. State Diagram: A state diagram represents the states of the Turing Machine as nodes and the transitions between states as directed edges. Each edge is labeled with the input symbol being read, the symbol to write, and the direction to move the tape head. The initial state is usually marked with an arrow entering it, and the accepting or rejecting states are marked differently.
3. Turing Machine Description (TM Description): A Turing Machine can also be described using a textual representation that specifies its components and behavior. It typically includes the set of states, the input alphabet, the tape alphabet, the initial state, the accepting and rejecting states, and the transition rules.
4. Turing Machine Diagram: A Turing Machine can be visually represented using a diagram that shows the tape, tape head, and the current state. The tape is represented as an infinite strip divided into cells, and the tape head position is indicated by an arrow. The current state is usually displayed nearby.

These representations help in understanding the structure and functioning of a Turing Machine. They provide a visual or textual representation of its components, transitions, and behavior. Depending on the context and the purpose of the representation, different notations may be used to effectively communicate the details of the Turing Machine.

Describe String acceptance by a TM

A Turing Machine (TM) accepts a string if it reaches an accepting state after processing the input string on its tape. The acceptance of a string by a TM depends on the final state the machine enters after executing its transition rules.

Here is the process of string acceptance by a Turing Machine:

1. Initialization: The Turing Machine is initialized with an input string written on its tape. The tape head is positioned at the leftmost cell of the input.
2. Execution: The Turing Machine starts executing its transition rules based on the current state and the symbol being read from the tape. It follows the rules specified in the transition function to update the tape contents, move the tape head, and transition to the next state.
3. Tape Read and Write: The Turing Machine reads the symbol from the current tape cell under the tape head. It then writes a new symbol on the tape according to the transition rule. The tape head may move left or right on the tape based on the transition.
4. State Transition: The Turing Machine changes its current state based on the transition rule. It updates the state to the next state specified in the transition.
5. Repeat: Steps 2-4 are repeated until the Turing Machine reaches either an accepting state or a rejecting state.
6. Acceptance: If the Turing Machine enters an accepting state after processing the entire input string, it accepts the string. This indicates that the string is recognized by the Turing Machine as belonging to the language defined by its transition rules.
7. Rejection: If the Turing Machine enters a rejecting state or halts without reaching an accepting state, it rejects the string. This indicates that the string does not belong to the language defined by the Turing Machine.

The acceptance or rejection of a string by a Turing Machine is determined by the final state it reaches after processing the entire input. If the machine enters an accepting state, the string is accepted; otherwise, it is rejected. The Turing Machine acts as a recognizer for the language it is designed to accept, and it can be used to determine whether a given string belongs to that language or not.

Recall TM Head Pointer movements

The Turing Machine (TM) head pointer can move in three different directions: left (L), right (R), and stay in place (S). The head pointer determines the current position on the tape and is responsible for reading and writing symbols during the execution of the TM.

Here are the possible movements of the TM head pointer:

1. Left (L): When the head pointer moves left, it shifts one cell to the left on the tape. This is denoted by the symbol “L” in the TM transition rules. The head reads the symbol on the current cell, moves left, and positions itself on the previous cell.
2. Right (R): When the head pointer moves right, it shifts one cell to the right on the tape. This is denoted by the symbol “R” in the TM transition rules. The head reads the symbol on the current cell, moves right, and positions itself on the next cell.
3. Stay (S): When the head pointer stays in place, it does not move horizontally on the tape. This is denoted by the symbol “S” in the TM transition rules. The head reads the symbol on the current cell and remains in the same position without any horizontal movement.

The movement of the TM head pointer is specified in the transition rules of the Turing Machine. Each transition rule defines the new state, the symbol to write on the tape, and the direction for the head pointer to move (L, R, or S) based on the current state and the symbol being read.

The TM head pointer movements allow the machine to access different cells on the tape, read symbols, and write new symbols during the computation. By controlling the movement of the head pointer, the Turing Machine can manipulate the tape contents and perform various computations on input strings.

Let’s consider an example of a Turing Machine (TM) with its head pointer movements:

Suppose we have a TM that recognizes the language L = {0^n 1^n | n ≥ 0}, which consists of strings of 0s followed by an equal number of 1s.

The TM has the following transition rules:

1. If the current state is q0 and the symbol under the head is ‘0’, write ‘X’, move the head right, and transition to state q1.
2. If the current state is q1 and the symbol under the head is ‘0’, move the head right and remain in state q1.
3. If the current state is q1 and the symbol under the head is ‘1’, write ‘Y’, move the head left, and transition to state q2.
4. If the current state is q2 and the symbol under the head is ‘1’, move the head left and remain in state q2.
5. If the current state is q2 and the symbol under the head is ‘X’, move the head right and transition to state q3.
6. If the current state is q3 and the symbol under the head is ‘Y’, move the head right and remain in state q3.
7. If the current state is q3 and the symbol under the head is blank (‘_’), move the head left and transition to state q4.

Let’s assume our input string is “000111”.

Initial Configuration:

State: q0

1. Tape: 0 0 0 1 1 1 (Blank symbols are represented by ‘_’)

Head: ^

Step 1:

State: q1

1. Tape: X 0 0 1 1 1

Head: ^

Step 2:

State: q1

1. Tape: X X 0 1 1 1

Head: ^

Step 3:

State: q1

1. Tape: X X X 1 1 1

Head: ^

Step 4:

State: q2

1. Tape: X X X 1 Y 1

Head: ^

Step 5:

State: q2

1. Tape: X X X Y Y 1

Head: ^

Step 6:

State: q3

1. Tape: X X X Y Y 1

Head: ^

Step 7:

State: q3

1. Tape: X X X Y Y 1 _

Head: ^

Step 8:

State: q4

1. Tape: X X X Y Y 1 _
2. Head: ^

The TM halts in state q4, indicating that the input string “000111” is accepted. In this example, the head pointer moves left, right, and stays in place based on the transition rules and the symbols read from the tape. This movement allows the TM to perform the necessary computations to recognize the language.

Recall the rules for designing the Turing Machine

When designing a Turing Machine (TM), several rules or considerations should be kept in mind. Here are the key rules for designing a TM:

1. States: Determine the set of states that the TM will have. This includes at least one initial state and one or more accepting (final) states. Each state represents a particular configuration of the machine.
2. Alphabet: Define the alphabet or set of symbols that the TM can read from the tape. This includes input symbols, tape symbols, and special symbols like blanks.
3. Transition Function: Define the transition function that specifies how the TM moves from one state to another based on the current state and the symbol read from the tape. Each transition consists of the following components:
   • Current State: The state the TM is currently in.
   • Current Symbol: The symbol read from the tape at the current position.
   • Next State: The state to transition to after performing the specified actions.
   • Write Symbol: The symbol to write on the tape at the current position.
   • Move Direction: The direction for the TM’s head pointer to move (left, right, or stay).
4. Initial Configuration: Define the initial configuration of the TM, which includes the initial state, the input string on the tape, and the position of the head pointer.
5. Halting Condition: Specify the halting condition or criteria for the TM to stop its computation. This can be reaching an accepting (final) state or entering a specific state that indicates the rejection of the input.
6. Tape Usage: Decide how the TM will utilize the tape for reading input, writing output, and performing internal computations. This includes determining the tape alphabet, the initial input on the tape, and the tape’s unbounded nature.

These rules guide the design of a Turing Machine, allowing it to perform computations on input strings based on the specified transition rules and configurations. The design should ensure that the TM can effectively process inputs and produce the desired output or behavior.

Design TM for the Strings and Languages

To design a Turing Machine (TM) for a specific language or set of strings, we need to define the TM’s states, alphabet, transition function, initial configuration, and halting condition.

Example 1: Construct a TM for the language L = {0ⁿ1ⁿ2ⁿ | where n≥1}

Explanation:

L = {0ⁿ1ⁿ2ⁿ | n≥1} represents language where we use only 3 character, i.e., 0, 1 and 2. In this, some number of 0’s followed by an equal number of 1’s and then followed by an equal number of 2’s. Any type of string which falls in this category will be accepted by this language.

The TM can be represented by Transition Diagram:

Example 2: Construct a TM machine for checking the palindrome of the string of even length.

Explanation:

Firstly we read the first symbol from the left and then we compare it with the first symbol from right to check whether it is the same.

Again we compare the second symbol from left with the second symbol from right. We repeat this process for all the symbols. If we found any symbol not matching, we cannot lead the machine to HALT state.

Example 3: Construct a TM machine for checking the palindrome of the string of odd length.

Explanation:

Firstly we read the first symbol from left and then we compare it with the first symbol from right to check whether it is the same.

Again we compare the second symbol from left with the second symbol from right. We repeat this process for all the symbols. If we found any symbol not matching, we lead the machine to HALT state.

The same TM can be represented by Transition Diagram:

Design TM for the Functions

Designing a Turing Machine (TM) for a specific function involves defining the TM’s states, alphabet, transition function, initial configuration, and halting condition. Here’s a general outline of the design process:

1. Define the Function:
   • Specify the function that the TM should compute. For example, let’s consider the function f(x) = 2x, where x is a binary number.
2. Determine the Alphabet:
   • Identify the alphabet or set of symbols that the TM will use. In this case, the alphabet includes {0, 1}, the input symbols representing binary digits.
3. Define the States:
   • Determine the set of states that the TM will have. This includes at least one initial state and one or more accepting (final) states. For simplicity, let’s use three states: q0, q1, and q2.
4. Specify the Transition Function:
   • Define the transition function that specifies how the TM moves from one state to another based on the current state and the symbol read from the tape. We need to consider the rules and steps required to compute the given function.
   • For the function f(x) = 2x, the TM needs to multiply the input binary number by 2.
     • In state q0, scan the tape from left to right until the first blank symbol is encountered, indicating the end of the input. Transition to state q1.
     • In state q1, scan the tape from right to left, doubling each binary digit encountered. If a carry is generated, propagate it to the leftmost digit. Transition to state q2.
     • In state q2, scan the tape from left to right to check if all digits have been processed. If any non-blank symbol is found, repeat the process from state q1. If only blank symbols are present, accept the input.
5. Determine the Initial Configuration:
   • Define the initial configuration of the TM, which includes the initial state, the input string on the tape, and the position of the head pointer. In this case, set the initial state to q0, place the binary number on the tape, and position the head at the first symbol.
6. Specify the Halting Condition:
   • Define the halting condition or criteria for the TM to stop its computation. In this case, the TM halts in state q2. If the TM reaches q2, it accepts the input.

This is a general outline of designing a TM for a specific function. The actual implementation and details may vary depending on the specific requirements and constraints of the function.

Example 1: Construct TM for the addition function for the unary number system.

Explanation:

The unary number is made up of only one character, i.e. The number 5 can be written in unary number system as 11111. In this TM, we are going to perform the addition of two unary numbers.

For example

2 + 3
i.e. 11 + 111 = 11111

If you observe this process of addition, you will find the resemblance with string concatenation function.

In this case, we simply replace + by 1 and move ahead right for searching end of the string we will convert last 1 to Δ.

Example 2: Construct a TM for subtraction of two unary numbers f(a-b) = c where a is always greater than b.

Solution: Here we have certain assumptions as the first number is greater than the second one. Let us assume that a = 3, b = 2, so the input tape will be:

We will move right to – symbol as perform reduction of a number of 1’s from the first number.

The Turing machine will look like this:

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

i. Universal Turing Machine:

A Universal Turing Machine (UTM) is a theoretical construct that can simulate any other Turing Machine. It is a Turing Machine designed to take as input the description of another Turing Machine and the input for that Turing Machine, and simulate its behavior. The UTM has a special tape called the “input tape” where the description of the Turing Machine and its input is provided. It can read the description, simulate the behavior of the specified Turing Machine, and produce the same output as the simulated machine.

Example:

Let’s say we have a Turing Machine M1 that computes the function f(x) = x + 1. We can construct a Universal Turing Machine UTM that takes as input the description of M1 and an input value, and simulates the behavior of M1. The UTM can execute the steps of M1 and compute the output of f(x) for any given input x.

ii. Multi-Tape Turing Machine:

A Multi-Tape Turing Machine is a variant of the Turing Machine that has multiple tapes instead of a single tape. Each tape can independently move left or right, read or write symbols. The transition function of a Multi-Tape Turing Machine takes into account the states, symbols read from each tape, and the movement and writing actions for each tape.

Example:

Consider a Multi-Tape Turing Machine with two tapes. One tape represents the input string, and the other tape is used as a working tape for computations. The machine can perform operations like copying, rearranging, or manipulating the contents of the tapes simultaneously.

iii. Multi-dimensional Turing Machine:

A Multi-dimensional Turing Machine extends the concept of a Turing Machine to operate on multiple dimensions, where each cell can have multiple states. It allows movements in multiple directions (up, down, left, right) and provides the ability to access neighboring cells in multiple dimensions.

Example:

Imagine a Multi-dimensional Turing Machine used for image processing. The machine can analyze and manipulate pixels in a two-dimensional grid. It can perform operations like blurring, edge detection, or color transformations by examining the neighboring pixels in different directions.

iv. Multi-head Turing Machine:

A Multi-head Turing Machine is a variant of the Turing Machine that has multiple read/write heads instead of a single head. Each head can independently move left or right, read or write symbols on the tape. The transition function of a Multi-head Turing Machine takes into account the states, symbols read from each head, and the movement and writing actions for each head.

Example:

Consider a Multi-head Turing Machine with two heads. One head is responsible for reading the input string, and the other head is used for auxiliary computations. The machine can perform parallel computations, compare different parts of the input, or perform complex operations that require multiple points of reference on the tape.

Describe Church’s Thesis

Church’s Thesis, also known as Church’s Hypothesis or Church’s Thesis, is a foundational principle in the field of computer science and mathematical logic. It was proposed by the mathematician and logician Alonzo Church in the 1930s.

Church’s Thesis states that any effectively calculable function can be computed by a Turing Machine. In other words, any function that can be computed by an algorithm or by a step-by-step computational process can be simulated by a Turing Machine. The thesis asserts that Turing Machines capture the essence of computability and can solve any problem that can be solved algorithmically.

This thesis is important because it provides a theoretical foundation for the study of computability and the limits of what can be computed. It implies that any computational device or programming language that can compute effectively calculable functions is equivalent in power to a Turing Machine.

While Church’s Thesis is widely accepted and has formed the basis for much of theoretical computer science, it is still a hypothesis that has not been proven or disproven. It is based on the intuitive notion of effective computability and is supported by the fact that Turing Machines can solve a wide range of computational problems. However, the thesis remains an open question and a subject of ongoing research in computability theory and the philosophy of computation.

Example:

While Church’s Thesis is a foundational principle in computer science, it does not lend itself to direct examples in the same way that algorithms do. Church’s Thesis is a hypothesis about the nature of computation and states that any effectively calculable function can be computed by a Turing Machine. It is a broad and fundamental idea that underlies much of theoretical computer science.

However, we can provide an example that illustrates the concept of effective calculability, which is central to Church’s Thesis:

Consider the problem of determining whether a given number is prime. The question of primality is a computational problem that can be approached with various algorithms. One such algorithm is the Sieve of Eratosthenes, which systematically eliminates composite numbers to identify primes.

While the Sieve of Eratosthenes is a practical algorithm for finding prime numbers, Church’s Thesis suggests that any effective method for determining primality could be simulated by a Turing Machine. In other words, Church’s Thesis implies that there exists a Turing Machine that can perform the same calculations and make the same decisions as any algorithm designed to solve the prime testing problem.

Church’s Thesis provides a theoretical foundation for the study of computability and helps guide the development of algorithms and computational models. While it does not offer specific examples in the same way as algorithms do, it shapes our understanding of what is computable and what is not.

Church’s Thesis vs Algorithms:

Church’s Thesis and algorithms are related concepts, but they address different aspects of computation.

Church’s Thesis is a foundational principle in computer science and mathematical logic that states that any effectively calculable function can be computed by a Turing Machine. It provides a theoretical framework for understanding the limits of computation and asserts that Turing Machines capture the essence of computability. Church’s Thesis is a hypothesis that has not been proven or disproven, but it is widely accepted as a guiding principle in the field.

On the other hand, an algorithm is a specific, concrete procedure or set of rules used to solve a particular problem or perform a task. It is a practical, step-by-step description of how to accomplish a computation. Algorithms can be implemented on various computational systems, including Turing Machines, but they are not limited to them. Algorithms can be implemented in programming languages, hardware circuits, or even carried out manually by humans.

While Church’s Thesis provides a theoretical foundation for computability, algorithms are the practical tools used to implement computations. Algorithms are designed to solve specific problems efficiently and effectively, taking into account the available resources and constraints. They are the practical manifestation of computational procedures.

In summary, Church’s Thesis addresses the theoretical notion of computability and the limits of computation, while algorithms are practical tools used to perform specific computations and solve problems. Church’s Thesis provides a theoretical framework for understanding what can be computed, while algorithms provide the practical means for carrying out computations.

Recall Recursive and Recursively Enumerable Languages

Recursive Languages:

Recursive languages are a class of languages that can be recognized by a Turing machine that halts on all inputs. In other words, a language is recursive if there exists an algorithm or Turing machine that can determine whether a given input string belongs to the language or not. The Turing machine for a recursive language will always halt and provide a definite answer.

Examples:

• The language of all even-length strings over the alphabet {0, 1}.
• The language of all palindromes over the alphabet {a, b}.

Recursively Enumerable Languages:

Recursively enumerable languages, also known as recursively enumerable or computably enumerable languages, are a class of languages that can be recognized by a Turing machine that may halt or loop indefinitely on certain inputs. In other words, a language is recursively enumerable if there exists a Turing machine that can recognize or accept all the strings in the language, but it may not halt or provide a definite answer for strings not in the language.

Examples:

• The language of all strings over the alphabet {0, 1} that encode a valid Turing machine and input, where the Turing machine halts and accepts on that input.
• The language of all strings that represent a well-formed arithmetic expression.

In summary, recursive languages can be recognized by Turing machines that always halt and provide a definite answer, while recursively enumerable languages can be recognized by Turing machines that may halt or loop indefinitely and accept all strings in the language but not necessarily reject strings outside the language.

List properties of Recursive and Recursively Enumerable Languages

Closure Properties of Recursive Languages and Recursive Enumerable Languages:

1. Union: If L1 and L2 are two recursive languages, their union L1∪L2 will also be recursive. This is because if a Turing machine halts on L1 and halts on L2, it will also halt on L1∪L2.
2. Concatenation: If L1 and L2 are two recursive languages, their concatenation L1.L2 will also be recursive. For example, if L1 is the language of strings with n number of “a”s followed by n number of “b”s followed by n number of “c”s, and L2 is the language of strings with m number of “d”s followed by m number of “e”s followed by m number of “f”s, then their concatenation L1.L2 will be the language of strings with n number of “a”s followed by n number of “b”s followed by n number of “c”s followed by m number of “d”s followed by m number of “e”s followed by m number of “f”s. This can be decided by a Turing machine, hence it is recursive.
3. Kleene Closure: If L1 is a recursive language, its Kleene closure L1* will also be recursive. For example, if L1 is the language of strings with n number of “a”s followed by n number of “b”s followed by n number of “c”s, then L1* will be the language of strings that can be formed by concatenating any number of strings from L1. This can be decided by a Turing machine, making it recursive.
4. Intersection and Complement: If L1 and L2 are two recursive languages, their intersection L1 ∩ L2 will also be recursive. For example, if L1 is the language of strings with n number of “a”s followed by n number of “b”s followed by n number of “c”s followed by m number of “d”s, and L2 is the language of strings with n number of “a”s followed by n number of “b”s followed by n number of “c”s followed by n number of “d”s, then their intersection L1 ∩ L2 will be the language of strings with n number of “a”s followed by n number of “b”s followed by n number of “c”s followed by n number of “d”s. This can be decided by a Turing machine, hence it is recursive. Similarly, the complement of a recursive language L1, which is ∑*-L1, will also be recursive.

Note: Unlike recursive languages, recursively enumerable (RE) languages are not closed under complementation. This means that the complement of an RE language may not necessarily be an RE language.

Examples of Recursive Languages:

1. L1 = {a^nb^nc^n | n ≥ 0}: This language consists of strings with equal numbers of ‘a’s, ‘b’s, and ‘c’s in the form of anbn cn. It is a recursive language because it can be recognized by a Turing machine that can count the number of ‘a’s, ‘b’s, and ‘c’s and verify their equality.
2. L2 = {ww | w ∈ {a, b}*}: This language contains strings that are repeated twice, such as “aa”, “bb”, “abab”, etc. It is recursive because a Turing machine can compare the first half of the string with the second half to determine if they are identical.

Examples of Recursively Enumerable Languages:

1. L3 = {w#w^R | w ∈ {a, b}*}: This language consists of strings in the form of wwR, where w is a string and wR is its reverse. The ‘#’ symbol is used as a separator. It is recursively enumerable because a Turing machine can generate all possible pairs of strings and check if their reverse matches the original string.
2. L4 = {w | w is a valid program in a Turing-complete programming language}: This language includes all valid programs written in a Turing-complete programming language. It is recursively enumerable because a Turing machine can simulate the execution of the program and halt if it produces the correct output or diverge if it does not terminate.

Describe Counter Machines and their similarities with TM

Counter machines are a type of abstract computing device that extends the capabilities of Turing machines (TMs) by incorporating a finite set of counters. A counter machine consists of a set of counters, an instruction set, and a control unit.

Similarities between Counter Machines and Turing Machines:

1. Computational Power: Both counter machines and Turing machines are computationally equivalent. This means that any computation that can be performed by a Turing machine can also be performed by a counter machine, and vice versa. Both models are capable of simulating each other.
2. Memory: Both counter machines and Turing machines have memory. In a Turing machine, the memory is represented by an infinite tape divided into cells, while in a counter machine, the memory is represented by a finite set of counters. Both models can read from and write to their respective memory structures.
3. Instruction Execution: Both counter machines and Turing machines operate based on a set of instructions. These instructions define the behavior of the machine, such as moving the head, reading from or writing to memory, and changing the state.
4. Computation Steps: Both counter machines and Turing machines perform computation by executing a sequence of steps. In each step, the machine reads the current symbol, performs an action based on the instruction and the current state, and transitions to the next state.
5. Computational Universality: Both counter machines and Turing machines are capable of expressing and computing any computable function. They are both universal computing models, meaning that any algorithm or computation that can be expressed algorithmically can be carried out by either model given enough time and resources.

Despite these similarities, there are also differences between counter machines and Turing machines. Counter machines have a more limited memory structure with a finite number of counters, whereas Turing machines have an infinite memory tape. This difference affects the types of problems each model is most suitable for and the complexity of certain computations.

Recall Undecidable problem of TMs

One well-known undecidable problem of Turing machines is the Halting Problem. The Halting Problem is the problem of determining, given a description of a Turing machine and an input, whether that Turing machine will eventually halt (i.e., stop) or run indefinitely. In other words, it asks whether the Turing machine will reach a halting state or continue to run forever on a particular input.

The Halting Problem was proven to be undecidable by Alan Turing in 1936. The proof involves constructing a hypothetical Turing machine that, when given another Turing machine and an input, determines whether the given Turing machine halts on that input. Through a clever argument, Turing showed that no such Turing machine can exist.

The implication of the undecidability of the Halting Problem is that there is no general algorithm or procedure that can determine whether an arbitrary Turing machine will halt on a given input. This result has important implications in computer science and theoretical computer science, as it sets limits on what can be computationally solved. It also demonstrates the existence of inherent limitations in formal systems and the concept of computability.

Here are a few examples of undecidable problems related to Turing machines:

1. Halting Problem: Given a description of a Turing machine M and an input string w, determine whether M halts (stops) on input w. This problem is undecidable, meaning there is no algorithm that can solve it for all possible Turing machines and inputs.
2. Post’s Correspondence Problem (PCP): Given a set of dominoes with strings on the top and bottom, determine whether there exists a sequence of dominoes that can be stacked such that the concatenation of the top strings matches the concatenation of the bottom strings. The PCP is undecidable, meaning there is no algorithm that can determine a solution for all possible sets of dominoes.
3. Rice’s Theorem: Let P be any non-trivial property of Turing machines’ languages. The problem of determining whether a given Turing machine’s language has property P is undecidable. This means that for any interesting and non-trivial property of languages, there is no algorithm that can determine whether an arbitrary Turing machine’s language has that property.
4. Turing Machine Equivalence: Given two Turing machines M1 and M2, determine whether their languages are the same, i.e., they accept exactly the same set of strings. This problem is undecidable, meaning there is no algorithm that can determine whether two arbitrary Turing machines accept the same language.

These are just a few examples of undecidable problems related to Turing machines. There are many more undecidable problems in the field of computability theory and theoretical computer science.

Describe PCP and MPCP Problems

Post’s Correspondence Problem (PCP):

The Post’s Correspondence Problem (PCP) is a famous undecidable problem in computer science, named after Emil Post. It involves a set of dominoes, where each domino has a string written on the top and another string written on the bottom. The problem is to determine whether there exists a sequence of dominoes that can be stacked in a particular order such that the concatenation of the top strings matches the concatenation of the bottom strings.

Formally, given a set of dominoes {(t1, b1), (t2, b2), …, (tn, bn)}, where ti and bi are strings for each domino i, the PCP asks if there exists a sequence of dominoes i1, i2, …, ik (1 <= i1, i2, …, ik <= n) such that the concatenation of the top strings ti1, ti2, …, tik is equal to the concatenation of the bottom strings bi1, bi2, …, bik.

The PCP is known to be undecidable, which means that there is no algorithm that can always determine whether a given set of dominoes has a solution or not.

Modified Post’s Correspondence Problem (MPCP):

The Modified Post’s Correspondence Problem (MPCP) is an extension of the PCP where we are allowed to use multiple copies of each domino in the sequence. In other words, we can repeat the same domino multiple times to form the sequence.

The MPCP asks if there exists a sequence i1, i2, …, ik (1 <= i1, i2, …, ik <= n) such that the concatenation of the top strings ti1, ti2, …, tik is equal to the concatenation of the bottom strings bi1, bi2, …, bik, allowing repetition of the same domino in the sequence.

Similar to the PCP, the MPCP is also undecidable, meaning there is no algorithm that can solve it for all possible sets of dominoes.

The PCP and MPCP are significant problems in theoretical computer science as they showcase the existence of undecidable problems and the limitations of algorithmic solutions. They have been extensively studied and used as building blocks for proving undecidability results in various areas of computer science and mathematics.

Example 1: Consider the correspondence system as given below

A = (b, bab³, ba) and B = (b³, ba, a). The input set is ∑ = {0, 1}. Find the solution.

Solution:

A solution is 2, 1, 1, 3. That means w₂w₁w₁w₃ = x₂x₁x₁x₃

The constructed string from both lists is bab³b³a.

Example 2:

Does PCP with two lists x = (b, a, aba, bb) and y = (ba, ba, ab, b) have a solution?

Solution: Now we have to find out such a sequence that strings formed by x and y are identical. Such a sequence is 1, 2, 1, 3, 3, 4. Hence from x and y list

Example 3:

Obtain the solution for the following system of posts correspondence problem. A = {100, 0, 1}, B = {1, 100, 00}

Solution: Consider the sequence 1, 3, 2. The string obtained from A = babababb. The string obtained from B = bababbbb. These two strings are not equal. Thus if we try various combination from both the sets to find the unique sequence, we could not get such a sequence. Hence there is no solution for this system.

Example 4:

Obtain the solution for the following system of posts correspondence problem, X = {100, 0, 1}, Y = {1, 100, 00}.

Solution: The solution is 1, 3, 1, 1, 3, 2, 2. The string is

X1X3X1X1X3X2X2 = 100 + 1 + 100 + 100 + 1 + 0 + 0 = 1001100100100
Y1Y3Y1Y1Y3Y2Y2 = 1 + 00 + 1 + 1 + 00 + 100 + 100 = 1001100100100

Describe Recursive Functions Theory

Recursive Functions Theory, also known as Recursion Theory or Computability Theory, is a branch of theoretical computer science that deals with the study of computability and the limitations of computation. It explores the notion of what can and cannot be computed by formal models of computation, such as Turing machines and recursive functions.

The central concept in Recursive Functions Theory is the notion of a recursive function. Recursive functions are a class of functions defined using a set of basic functions and a set of recursive operations. These functions can be thought of as computable functions, i.e., functions that can be effectively computed by an algorithm or a computational model.

Recursive Functions Theory aims to understand the fundamental properties of recursive functions and their relationship to other computational models. It investigates questions such as:

1. Computability: Which functions are computable? Can they be computed by Turing machines or other formal models of computation?
2. Undecidability: Are there problems that cannot be solved by any algorithm or computational model?
3. Complexity: How difficult is it to compute certain functions? Can we classify problems based on their computational complexity?
4. Hierarchies: Can we classify functions or sets based on their level of computational complexity or computability?
5. Reducibility: Can one problem be reduced to another? Is there a way to transform one problem into another while preserving computability?
6. Oracles and Degrees: Can we extend the computational power of existing models by introducing new elements or oracles? Can we compare the relative computational power of different models?

Recursive Functions Theory has deep connections to other areas of computer science, mathematics, and philosophy. It has been used to prove the undecidability of various problems, establish the limits of computation, and provide insights into the nature of computation itself.

Key figures in the development of Recursive Functions Theory include Alonzo Church, Alan Turing, Kurt Gödel, and Emil Post, among others. Their pioneering work laid the foundations for the study of computability and influenced the development of theoretical computer science as a discipline.

Recall Initial, Partial, and Primitive Recursive Functions

Initial, Partial, and Primitive Recursive Functions are classes of computable functions defined within the framework of Recursive Functions Theory. These classes represent increasingly powerful classes of computable functions, with the primitive recursive functions being the most expressive.

1. Initial Functions:

Initial functions are the simplest class of computable functions. They consist of basic functions that are predefined and serve as the building blocks for more complex functions. These basic functions typically include constant functions, successor function, and projection functions. Initial functions are also closed under composition and primitive recursion, which means that by combining basic functions using these operations, we can construct new computable functions.

1. Partial Recursive Functions:

Partial recursive functions are an extension of initial functions that allows for more complex computations. In addition to the basic functions and operations allowed in initial functions, partial recursive functions also allow for the use of minimization or unbounded search. This means that a partial recursive function can include a loop that searches for the smallest input value that satisfies a certain condition. However, the loop may not terminate for some inputs, resulting in a partial function that is undefined for those inputs.

1. Primitive Recursive Functions:

Primitive recursive functions are a further extension of partial recursive functions that adds additional computational capabilities. In addition to the basic functions and operations allowed in initial and partial recursive functions, primitive recursive functions allow for primitive recursion over multiple variables. Primitive recursion allows a function to refer to its own previous values during computation. This additional capability allows for more complex computations and the construction of more expressive functions.

Primitive recursive functions are a subset of partial recursive functions and form a strict hierarchy within the class of computable functions. While all initial functions are partial recursive, and all partial recursive functions are computable, not all computable functions are primitive recursive. There exist computable functions that cannot be defined within the framework of primitive recursion.

The distinction between initial, partial, and primitive recursive functions helps to understand the computational power and limitations of different classes of computable functions within the theory of recursive functions.

Here are examples of initial, partial recursive, and primitive recursive functions:

1. Initial Functions:
   • Constant function: f(x) = 3 (returns a constant value of 3 for any input).
   • Successor function: f(x) = x + 1 (increments the input by 1).
   • Projection functions: f(x, y) = x (returns the first component of a pair).
2. Partial Recursive Functions:
   • Addition function: f(x, y) = x + y (computes the sum of two numbers).
   • Multiplication function: f(x, y) = x * y (computes the product of two numbers).
   • Factorial function: f(x) = x! (computes the factorial of a non-negative integer).
3. Primitive Recursive Functions:
   • Fibonacci sequence: f(0) = 0, f(1) = 1, f(n) = f(n-1) + f(n-2) (computes the nth Fibonacci number).
   • Exponential function: f(x, y) = x^y (computes the power of x raised to the exponent y).
   • Ackermann function: A(m, n) (a highly recursive function that grows rapidly, used as a benchmark in computability theory).

It’s important to note that these examples are just a small subset of the functions in each class and are intended to provide a glimpse into the types of functions that fall into each category. The primitive recursive functions are more expressive and can compute more complex functions than the initial and partial recursive functions.