Quiz 3 303

Background on tutoring machines Parts of a Turing Machine (TM):

: Turing machine designed by Alan Turing in 1936. It is not a real machine, but a model of a general purpose computer. You can think of it as a finite automata with unbounded and unrestricted memory. It forms the theoretical limit of computation. 1. Memory 2. Read/write head 3. CPU/brain

What Does "Countable" Mean?

A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers 𝑁 = { 1 , 2 , 3 , ... } N={1,2,3,...}. In other words, you can "list" the elements of the set in a sequence, even if the set is infinite. Countable sets are classified into two types: Finite Countable: A set that has a finite number of elements. Infinite Countable: A set that has an infinite number of elements but can still be listed in a sequence (e.g., integers). Natural Numbers ( 𝑁 N): { 1 , 2 , 3 , ... } {1,2,3,...} Clearly countable because they are the definition of a counting sequence. Integers ( 𝑍 Z): { ... , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , ... } {...,−3,−2,−1,0,1,2,3,...} You can list them like this: 0 , 1 , − 1 , 2 , − 2 , 3 , − 3 , ... 0,1,−1,2,−2,3,−3,.... Rational Numbers ( 𝑄 Q): Numbers of the form 𝑝 𝑞 q p ​ , where 𝑝 p and 𝑞 q are integers, and 𝑞 ≠ 0 q =0. Even though 𝑄 Q seems "denser" than integers, you can systematically list them (e.g., using a diagonalization argument).

Church-Turing Thesis: Strong Church-Turing Thesis:

Church-Turing Thesis: Every physically realizable computation device can be simulated by a Turing Machine. (This has not been proven, but it is a general belief.) Strong Church-Turing Thesis: A TM can simulate all other computational models efficiently (with polynomial overhead). Essentially, the set of "computable" problems is fully captured by Turing Machines

Enumerators

Concept: A TM variant with a printer that outputs strings as it operates. Operation: Starts with a blank tape. Outputs strings in any order, with possible repetitions. The set of strings it outputs is the language of the enumerator. Equivalence: Enumerators have the same computational power as Turing Machines; they can generate any Turing-recognizable language.

Multi-Tape Turing Machine

Concept: A TM with multiple tapes and separate heads, which can make computation more efficient. Simulation on a Single Tape: A single-tape TM can simulate a multi-tape TM by: Arranging all tapes side by side on a single tape. Using special markers to separate each virtual tape. If a tape needs to grow, the single-tape TM can shift contents to make space. Equivalence: The multi-tape TM is no more powerful than a single-tape TM; they can recognize and decide the same languages.

Stay-Put Turing Machine

Concept: A Turing Machine variant where the tape head can stay in place (denoted by "S") instead of always moving left ("L") or right ("R"). Transition Function: Original: δ: Q × Γ → Q × Γ × {L, R} Modified: δ: Q × Γ → Q × Γ × {L, R, S} Proof of Equivalence: A Stay-Put Turing Machine can be simulated by a standard TM: To "stay put," the original TM transitions right, does nothing, and then transitions left. This simulates staying in the same spot using two moves.

Non-Deterministic Turing Machine (NTM)

Concept: NTMs have a transition function that allows multiple possible actions at each step, effectively exploring multiple paths simultaneously. Transition Function: Deterministic TM: δ: Q × Γ → Q × Γ × {L, R} Non-Deterministic TM: δ: Q × Γ → P(Q × Γ × {L, R}) (power set notation) Acceptance Condition: An NTM accepts an input string if any branch of computation leads to an accepting state. Theorem 3.16: Every NTM has an equivalent deterministic TM. Proof Outline: Construct a deterministic TM with three tapes: The first tape holds the input string. The second tape simulates the NTM's behavior. The third tape keeps track of states and branches. The deterministic TM systematically explores all possible computation branches of the NTM. If an accepting branch is found, it accepts; otherwise, it rejects or loops. Key Observations: If the NTM accepts a string, the deterministic TM will eventually find that branch and accept. If the NTM does not accept, the deterministic TM will reject or run indefinitely.

Decidable Problems: Turing-decidable languages Recognizable Problems:

Decidable Problems: A problem is decidable if we can construct an algorithm to always correctly answer Yes or No in finite time. A TM will halt on every input with accept/reject. (Turing Decidable) Turing-decidable languages Undecidable Problems: For these problems, no TM correctly decides (halts and says "accept" or "reject") for every input in the language. Recognizable Problems: A problem is recognizable if there is an algorithm to always correctly answer Yes in finite time. A TM will accept every input in the language. (Turing Recognizable)

What Is Diagonalization? How Does Diagonalization Work?

Diagonalization is a mathematical technique, famously introduced by Georg Cantor, to prove that certain sets, like the real numbers, are uncountable. It shows that no matter how you attempt to list all elements of these sets, there will always be elements left out. This method is often used in proofs related to the sizes of infinite sets. How Does Diagonalization Work? Diagonalization constructs an element that is guaranteed not to be in the list, by carefully altering entries along the "diagonal" of the list.

Let's look at the Factoring Problem Q:What are the possible implications of solving the factoring problems efficiently?

Input: A product of pq of primes p and q Output: p and q Ex: Input: 15; Output: 3,5 It is believed that the factoring problem has no polynomial-time algorithm. (It is so difficult to crack that many encryption schemes today are "secured" assuming this is true, e.g., RSA) So, believed there's no TM to solve this problem. But in 1994 Peter Shor showed that a quantum computer could factor efficiently. 1. The Church-Turing Thesis is false 2. Quantum computers are physically realizable 3. There exists an efficient, classical algorithm for the factoring problem Only time will tell the answer..

One to One ONTO bijection

One-to-one: Never maps two different elements in A to same element in B, so a 6 = b implies f (a) 6 = f (b). Onto: For every b ∈ B there is a ∈ A with f (a) = b. If you have both one-to-one and onto, then you have a one-to-one correspondence also called a bijection.

Uncountable Sets

Some sets cannot be listed or matched one-to-one with 𝑁 N. These sets are called uncountable. For example: Real Numbers ( 𝑅 R): The set of all real numbers is uncountable, proven by Cantor's diagonalization argument. The Power Set of Natural Numbers ( 𝑃 ( 𝑁 ) P(N)): The set of all subsets of 𝑁 N is uncountable

Universal Turing Machine Idea:

There exists a Turing Machine that can simulate other Turing Machines as "software". We can encode TM as strings: Write down the entire transition function table, then pick an encoding that satisfies.

Robustness of Turing Machines

Turing Machines are robust because their computational power is unchanged by small changes in the definition of the model.

Turing-decideable closed under:

UNION,INTERSECTION,CONCATENATION, STAR, COMPLENET

Turing-recognizable closed under:

UNION,INTERSECTION,CONCATENATION, STAR, Note: Turing-recognizable languages are not closed under complementation. That is, if a language L is Turing-recognizable, its complement ‾ may not be Turing-decideable.

Defn: A language L is Turing-decidable

if some Turing Machine "decides" it. Here decides means: 1. If input x ∈ L, the Turing Machine halts and accepts 2. If input x /∈ L, the Turing Machine halts and rejects (looping forever is forbidden)

Defn: A language is Turing-Recognizable

if some Turing Machine "recognizes" it. Here recognizes means: 1. If input x ∈ L, the Turing Machine halts and accepts 2. If input x /∈ L, the Turing Machine either: a. Halts and rejects or b. Loops forever