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The Halting Problem centers around creating a machine, H, that can determine if any given machine, M, will halt or loop indefinitely.
The Halting Problem centers around creating a machine, H, that can determine if any given machine, M, will halt or loop indefinitely.
True (A)
If machine H predicts that machine M will halt, then M must, in reality, halt; otherwise, H has failed to solve the halting problem.
If machine H predicts that machine M will halt, then M must, in reality, halt; otherwise, H has failed to solve the halting problem.
True (A)
A machine R configured to enter an infinite ‘loop’ state if H predicts M will halt, and halt if H predicts M will loop, is an example of an implementation of solving the halting problem.
A machine R configured to enter an infinite ‘loop’ state if H predicts M will halt, and halt if H predicts M will loop, is an example of an implementation of solving the halting problem.
False (B)
Constructing a universal halting detector, exemplified by machine H, is possible with current computational algorithms.
Constructing a universal halting detector, exemplified by machine H, is possible with current computational algorithms.
The machine, D, takes another machine, M, as input and if M halts when given its own source code as input, D loops indefinitely, and halts if M loops when given its own source code.
The machine, D, takes another machine, M, as input and if M halts when given its own source code as input, D loops indefinitely, and halts if M loops when given its own source code.
A Turing Machine, as conceptualized in 1927, serves as a theoretical model of computation.
A Turing Machine, as conceptualized in 1927, serves as a theoretical model of computation.
In the context of decidability, the symbol 'K' universally denotes an input alphabet consisting of all real integers.
In the context of decidability, the symbol 'K' universally denotes an input alphabet consisting of all real integers.
The Halting Problem revolves around predicting whether a Machine 'M' will enter an infinite loop state for a given input 'D'.
The Halting Problem revolves around predicting whether a Machine 'M' will enter an infinite loop state for a given input 'D'.
In the context of the Halting Problem, if a machine M with input D is determined to 'halt', it implies the machine will execute without errors and provide a valid result.
In the context of the Halting Problem, if a machine M with input D is determined to 'halt', it implies the machine will execute without errors and provide a valid result.
If a universal resolver, 'R', is created that always correctly predicts whether another machine, H, will halt, it inherently solves the Halting Problem.
If a universal resolver, 'R', is created that always correctly predicts whether another machine, H, will halt, it inherently solves the Halting Problem.
The Halting Problem is decidable for all possible Turing machines.
The Halting Problem is decidable for all possible Turing machines.
The 'wff' symbol represents 'well-formed formula', representing the structure of logical sentences that could be part of algorithms being assessed.
The 'wff' symbol represents 'well-formed formula', representing the structure of logical sentences that could be part of algorithms being assessed.
A Turing Machine's primary purpose is to physically construct powerful computing hardware using vacuum tubes.
A Turing Machine's primary purpose is to physically construct powerful computing hardware using vacuum tubes.
The class P encompasses problems solvable by algorithms with polynomial time complexity, denoted as $O(P(n))$ where n is the input size.
The class P encompasses problems solvable by algorithms with polynomial time complexity, denoted as $O(P(n))$ where n is the input size.
NP-complete problems are a subset of NP problems, and if a polynomial-time algorithm is found for any NP-complete problem, it would imply that $P = NP$.
NP-complete problems are a subset of NP problems, and if a polynomial-time algorithm is found for any NP-complete problem, it would imply that $P = NP$.
If problem K is proven to be NP-complete, and another problem J can be polynomially reduced from K, then J is also NP-complete.
If problem K is proven to be NP-complete, and another problem J can be polynomially reduced from K, then J is also NP-complete.
Exhaustive search, often visualized as a tree search, guarantees finding a solution if one exists but is efficient for large problem spaces.
Exhaustive search, often visualized as a tree search, guarantees finding a solution if one exists but is efficient for large problem spaces.
In propositional logic, a well-formed formula (wff) is considered falsifiable if there exists at least one interpretation (assignment of truth values to its variables) that makes the formula false.
In propositional logic, a well-formed formula (wff) is considered falsifiable if there exists at least one interpretation (assignment of truth values to its variables) that makes the formula false.
The expression $¬(B ∨ D ∨ ¬(A ∧ C))$ is equivalent to $(¬B ∧ ¬D ∧ (A ∧ C))$.
The expression $¬(B ∨ D ∨ ¬(A ∧ C))$ is equivalent to $(¬B ∧ ¬D ∧ (A ∧ C))$.
Determining the satisfiability of a propositional logic formula with n variables using exhaustive search has a time complexity of $O(n^2)$.
Determining the satisfiability of a propositional logic formula with n variables using exhaustive search has a time complexity of $O(n^2)$.
If an algorithm runs in $O(n^3)$ time, it belongs to the class NP.
If an algorithm runs in $O(n^3)$ time, it belongs to the class NP.
A problem is NP-complete if it is in NP and every problem in NP is Turing-reducible to it.
A problem is NP-complete if it is in NP and every problem in NP is Turing-reducible to it.
If $P = NP$, then all problems in NP can be solved in exponential time.
If $P = NP$, then all problems in NP can be solved in exponential time.
The time complexity $O(2^n)$ indicates that the algorithm's runtime grows quadratically with the size of the input n.
The time complexity $O(2^n)$ indicates that the algorithm's runtime grows quadratically with the size of the input n.
A propositional formula is unsatisfiable if and only if there is no assignment of truth values to its variables that makes the formula true.
A propositional formula is unsatisfiable if and only if there is no assignment of truth values to its variables that makes the formula true.
If a problem is decidable, it means that there exists an algorithm that will always halt and provide a correct 'yes' or 'no' answer for any input.
If a problem is decidable, it means that there exists an algorithm that will always halt and provide a correct 'yes' or 'no' answer for any input.
The tree search method for satisfiability is an efficient way to solve NP-complete problems in polynomial time.
The tree search method for satisfiability is an efficient way to solve NP-complete problems in polynomial time.
If a problem is in NP, then any proposed solution to the problem can be verified in polynomial time.
If a problem is in NP, then any proposed solution to the problem can be verified in polynomial time.
Entailment can be checked by assessing the satisfiability of the original knowledge base alone.
Entailment can be checked by assessing the satisfiability of the original knowledge base alone.
If $Γ ⊣ φ$, then $w(Γ ∪ {∼φ})$ must be an empty set.
If $Γ ⊣ φ$, then $w(Γ ∪ {∼φ})$ must be an empty set.
In the context of entailment, $w(Γ)$ represents the set of all possible worlds.
In the context of entailment, $w(Γ)$ represents the set of all possible worlds.
The transformation of entailment into satisfiability involves verifying if $w(Γ ∪ {φ})$ is equivalent to the universal set of possible worlds.
The transformation of entailment into satisfiability involves verifying if $w(Γ ∪ {φ})$ is equivalent to the universal set of possible worlds.
If $w(Γ ∪ {∼φ}) = ∅$, it indicates that $Γ$ does not entail $φ$.
If $w(Γ ∪ {∼φ}) = ∅$, it indicates that $Γ$ does not entail $φ$.
The expression $w(Γ ∪ \{∼φ\}) = w(Γ) ∪ w(\{∼φ\})$ is a valid representation of how models combine in entailment checking.
The expression $w(Γ ∪ \{∼φ\}) = w(Γ) ∪ w(\{∼φ\})$ is a valid representation of how models combine in entailment checking.
Satisfiability checking is irrelevant in determining whether one statement entails another.
Satisfiability checking is irrelevant in determining whether one statement entails another.
A Turing Machine is a theoretical model of computation that cannot be used to simulate any computer algorithm.
A Turing Machine is a theoretical model of computation that cannot be used to simulate any computer algorithm.
If adding the negation of a sentence to a knowledge base results in a contradiction, then the original knowledge base entails the sentence.
If adding the negation of a sentence to a knowledge base results in a contradiction, then the original knowledge base entails the sentence.
In the context of entailment as satisfiability, the goal is to find a model that satisfies both the knowledge base $Γ$ and the conclusion $φ$.
In the context of entailment as satisfiability, the goal is to find a model that satisfies both the knowledge base $Γ$ and the conclusion $φ$.
The computational complexity theory is completely unrelated to the concept of algorithms.
The computational complexity theory is completely unrelated to the concept of algorithms.
Entailment ($Γ ⊣ φ$) means $φ$ is always false whenever $Γ$ is true.
Entailment ($Γ ⊣ φ$) means $φ$ is always false whenever $Γ$ is true.
If $w(Γ) = W$ (the set of all possible worlds), then $Γ$ is a contradiction.
If $w(Γ) = W$ (the set of all possible worlds), then $Γ$ is a contradiction.
If entails , then entails .
If entails , then entails .
In checking entailment via satisfiability, we transform the problem into checking the validity of {}.
In checking entailment via satisfiability, we transform the problem into checking the validity of {}.
Flashcards
Turing Machine
Turing Machine
A theoretical model of computation that defines an abstract machine.
Decidability
Decidability
Determining whether a problem can be solved by an algorithm.
Halting Problem
Halting Problem
The problem of determining whether a Turing machine will halt or run forever on a given input.
Machine Input
Machine Input
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Machine M
Machine M
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Halting Machine H
Halting Machine H
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Machine H
Machine H
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Machine R
Machine R
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Halting Problem Machine (H)
Halting Problem Machine (H)
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"Halt"
"Halt"
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"Loop"
"Loop"
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What is Entailment?
What is Entailment?
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Entailment as Satisfiability
Entailment as Satisfiability
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What does w(Γ) represent?
What does w(Γ) represent?
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Γ ⊨ φ implies what about w(Γ) and w(¬φ)?
Γ ⊨ φ implies what about w(Γ) and w(¬φ)?
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Problem Transformation
Problem Transformation
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What is an Algorithm?
What is an Algorithm?
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What is Conjunction (∧)?
What is Conjunction (∧)?
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What is Satisfiability?
What is Satisfiability?
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What is Unsatisfiability?
What is Unsatisfiability?
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What is a Turing Machine?
What is a Turing Machine?
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Computational Complexity Theory
Computational Complexity Theory
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What does ¬φ mean?
What does ¬φ mean?
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What is a well-formed formula (wff)?
What is a well-formed formula (wff)?
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What are Models (in logic)?
What are Models (in logic)?
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What is Γ ∪ {¬φ}?
What is Γ ∪ {¬φ}?
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What is a WFF?
What is a WFF?
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What is the Complexity Class P?
What is the Complexity Class P?
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What is the Complexity Class NP?
What is the Complexity Class NP?
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What is NP-complete?
What is NP-complete?
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NP-complete Reduction
NP-complete Reduction
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Implication of P = NP
Implication of P = NP
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What does P != NP mean?
What does P != NP mean?
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What is Exhaustive Search?
What is Exhaustive Search?
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Satisfiability in Logic
Satisfiability in Logic
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Unsatisfiability in Logic
Unsatisfiability in Logic
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Truth Table method
Truth Table method
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Complexity of Truth Table
Complexity of Truth Table
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What is a Decidability task?
What is a Decidability task?
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Truth Assignment Example
Truth Assignment Example
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What is Exponential time?
What is Exponential time?
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Study Notes
- Artificial Intelligence is a course about foundations.
- This lecture is about entailment and algorithms.
Entailment as Satisfiability
- The decision problem “Γ ⊨ φ ?” can be transformed into a satisfiability problem.
- Γ ⊨ φ if and only if Γ ∪ {¬φ} is not satisfiable.
- This is the first step in transforming the initial decision problem.
- The problem is "is Γ ∪ {¬φ} satisfiable?" which can be transformed into a wff satisfiability problem.
- Then Γ∪ {¬φ} can be transformed into a single formula: ∧(Γ∪{¬φ}).
- ∧(Γ∪{¬φ}) combines all wffs in Γ∪ {¬φ} with ∧ and it is called the conjunctive closure of the set Γ∪ {¬φ}.
Algorithm
- Algorithm is related to computational complexity theory
Turing Machine (A. Turing, 1937)
- It consists of a non-empty and finite set of states S.
- The machine transitions between states s ∈ S at each instant.
- It has a non-empty and finite alphabet of symbols Q, including a blank, default symbol b.
- Each cell in the tape contains a symbol q ∈ Q.
- It uses a partial transition function τ: S × Q → S × Q × {Left, None, Right}.
- The function takes the current state and input symbol to produce the next state, output symbol, and head move direction.
- It is partial because it needs not be defined on any input tuple.
- A subset of terminal states T ⊆ S and an initial state s0 ∈ S are defined for the machine.
- Busy beaver is an example of a Turing Machine with 3 states.
- S = {A, B, C, HALT}
- s0 = A
- T = {HALT}
- Q = {0, 1}
- b= 0
- Assuming the tape is infinite and has plenty of blank symbols.
- The transition table τ = {< A, 0 > → < B, 1, Right >, < A, 1 > → < C, 1, Left >, < B,0 > → < A, 1, Left >, < B,1 > → < B, 1, Right >, < C, 0 > → < B, 1, Left >, < C, 1 > → < HALT, 1, Right >}
Decisions and Decidability
- A problem is a relation between inputs and outputs (= solutions), such that K ⊆ I × S.
- A search problem may associate one input to many solutions.
- Optimization problems have a search problem plus an objective or cost function c: S → ℝ, where ℝ is the set of real numbers.
- The task is to find the solution(s) with maximal or minimal cost.
- A decision problem has a binary solution space S = {0, 1}.
- K associates each input to a unique solution: K: I → {0, 1}.
- An example is Γ ⊨ φ ?, where the input space I contains all possible combinations of set Γ of wffs with individual wffs φ so the solution is uniquely defined for any instance of such problems in I.
- For a decidable problem K there exists an algorithm, that is, a Turing machine.
- Alternative definitions of an algorithm include an effective procedure.
- It always terminates and produces the right answer in finite time.
- An undecidable problem is the Halting Problem.
- Given a formal description of a Turing machine and a specific input, is it possible to tell if it will either halt eventually or run forever?
- The answer is no, such Turing Machine cannot exist.
- The intuitive proof behind the undecidability of this problem assumes there exists a Turing machine H that, given Turing machine M with input D, always terminates producing a "halt" or "loop" output depending on whether M with input D will terminate or not.
Computational Complexity
- These notions only apply decidable problems.
- The benchmark is a Turing Machine that computes the correct answer in worst-case scenarios.
Time complexity
- The number of steps the Turing machine requires for computing the answer.
- It is a function of some numerical dimension of the input i.e the number of atoms in a wff.
Memory complexity
- The number of tape cells that the Turing machine requires for computing the answer.
- It is a function of some numerical dimension of the input
Big-O Notation
- f(x) = O(g(x)) means that ∃M > 0, ∃x0 > 0 such that |f(x)| ≤ M|g(x)|, ∀x > x0.
Classes P, NP and NP-complete
Class P
- The class of problems for which there is a Turing machine that requires O(P(n)) time.
- P is a polynomial of finite degree, n is the dimension of the worst-case input.
Class NP
- The class of all problems with a method for enumerating all possible answers recursive enumerability.
- It requires an algorithm in class P that verifies if a possible answer is also a solution.
- It includes all problems in class P, so P ⊆ NP..
Class NP- complete
- It is a subclass of NP.
- A problem K is NP-complete if every problem in class NP is reducible to K.
Reducibility
- Consider a problem K for which a decision algorithm M(K) is known
- The problem J is reducible to K if there exist a decision algorithm M(J) so algorithm M(K) is called just once, as a subroutine at the end of M(J) and apart from M(K), M(J) has polynomial complexity.
The problem SAT
- It is NP-complete.
- If we had a polynomial decision algorithm for SAT, we would also have that P = NP.
- It is believed that P ≠ NP.
Satisfiability and Decidability
- Determining if the wff φ is satisfiable, is transformable into a search problem.
- In this case, that involves finding a possible world within the set of all possible worlds that satisfies φ.
- This problem is referred to as SAT in scientific literature.
- To solve this problem, one can try every possible value assignment for the atoms in φ
- It is an NP-complete problem.
- Exhaustive tree search can be used to find solutions but this method has 𝑂(2𝑛) time complexity, where n is the number of propositional symbols
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Description
Explore the Halting Problem, which questions the possibility of creating a machine that can determine if any given machine will halt or loop indefinitely. This exploration covers Turing Machines, decidability, and the implications for computational algorithms. Understanding the Halting Problem is crucial in computer science.
