Humans, Computer, and Computational Complexity
... significance. Suppose you are trying to write a computer program. The code quickly becomes too complicated to keep it all in your mind at once. There are too many conditions and too many loops. You want to know whether when you run the program, it will run to completion and produce an output, or whe ...
... significance. Suppose you are trying to write a computer program. The code quickly becomes too complicated to keep it all in your mind at once. There are too many conditions and too many loops. You want to know whether when you run the program, it will run to completion and produce an output, or whe ...
Playing Chess with a Philosopher: Turing and Wittgenstein
... statements about mathematics. Wittgenstein addresses mathematics form the outside, from the structure of mathematical propositions and the use of language. ...
... statements about mathematics. Wittgenstein addresses mathematics form the outside, from the structure of mathematical propositions and the use of language. ...
empty-stack
... The following languages are Turingdecidable: (a) {“M” ”w” : M halts on w after at most 700 steps} (b) {“M” “w”: M halts on w without using more than the first 100 tape squares} For part (b): we only have to simulate M for a finite number of steps and in that time frame it will either halt, hang, us ...
... The following languages are Turingdecidable: (a) {“M” ”w” : M halts on w after at most 700 steps} (b) {“M” “w”: M halts on w without using more than the first 100 tape squares} For part (b): we only have to simulate M for a finite number of steps and in that time frame it will either halt, hang, us ...
Non-deterministic Turing machines Time complexity Time
... • A polynomial reduction of L1 ⊆ Σ∗1 to L2 ⊆ Σ∗2 is a polynomially computable function f : Σ∗1 → Σ∗2 with w ∈ L1 ⇔ f (w) ∈ L2 . • Proposition. If L1 is polynomially reducible to L2 , then 1. L1 ∈ P if L2 ∈ P and L1 ∈ NP if L2 ∈ NP 2. L2 6∈ P if L1 6∈ P and L2 6∈ NP if L1 6∈ NP. ...
... • A polynomial reduction of L1 ⊆ Σ∗1 to L2 ⊆ Σ∗2 is a polynomially computable function f : Σ∗1 → Σ∗2 with w ∈ L1 ⇔ f (w) ∈ L2 . • Proposition. If L1 is polynomially reducible to L2 , then 1. L1 ∈ P if L2 ∈ P and L1 ∈ NP if L2 ∈ NP 2. L2 6∈ P if L1 6∈ P and L2 6∈ NP if L1 6∈ NP. ...
1996TuringIntro
... of what might be supposed to be non-algothmic “divine spark” thinking in mathematics and logic - for example the proof of Gödel’s second theorem. He provides a fascinating insight into what goes on in the mathematician’s mind when attempting to grapple with abstract objects such as infinite sequence ...
... of what might be supposed to be non-algothmic “divine spark” thinking in mathematics and logic - for example the proof of Gödel’s second theorem. He provides a fascinating insight into what goes on in the mathematician’s mind when attempting to grapple with abstract objects such as infinite sequence ...
Extended Analog Computer and Turing machines - Hektor
... time. These are, for example, the Analog Recurrent Neural Network [2] and the machines of Blum-Shub-Smale [3]. The second case are models of computation on real numbers and in continuous time. In this scope the important model is the General Purpose Analog Computer proposed by Shannon in 1941. This ...
... time. These are, for example, the Analog Recurrent Neural Network [2] and the machines of Blum-Shub-Smale [3]. The second case are models of computation on real numbers and in continuous time. In this scope the important model is the General Purpose Analog Computer proposed by Shannon in 1941. This ...
Summary
... Pumping Lemma: Let L be a regular language. There exists an integer p (“pumping length”) for which every w L with |w| p can be written as w = xyz such that ...
... Pumping Lemma: Let L be a regular language. There exists an integer p (“pumping length”) for which every w L with |w| p can be written as w = xyz such that ...
12 Machine-repairmen problem
... pk = k! λ k1 +k2 +···+kr =k Hence the probabilities pk are exactly the same as in the exponential case. This suggests (and it can be proved) that the probabilities pk are insensitive to the distribution of the up times. This is, however, not true for the repair time distribution. ...
... pk = k! λ k1 +k2 +···+kr =k Hence the probabilities pk are exactly the same as in the exponential case. This suggests (and it can be proved) that the probabilities pk are insensitive to the distribution of the up times. This is, however, not true for the repair time distribution. ...
5.8.2 Unsolvable Problems
... Proof To prove that LC is not decidable, we assume that it is decidable by the TM MC and show that this implies the existence of a TM MH that decides LH , which has been shown previously not to exist. Thus, MC cannot exist. We consider two cases, the first in which B∗ is in not C and the second in w ...
... Proof To prove that LC is not decidable, we assume that it is decidable by the TM MC and show that this implies the existence of a TM MH that decides LH , which has been shown previously not to exist. Thus, MC cannot exist. We consider two cases, the first in which B∗ is in not C and the second in w ...
Day33-Reduction - Rose
... for an arbitrary instance I of P1, an instance f(I) of P2, and • f is defined such that for every instance I of P1, I is a yes-instance of P1 if and only if f(I) is a yes-instance of P2. So P1 P2 means "if we have a TM that decides P2, then there is a TM that decides P1. ...
... for an arbitrary instance I of P1, an instance f(I) of P2, and • f is defined such that for every instance I of P1, I is a yes-instance of P1 if and only if f(I) is a yes-instance of P2. So P1 P2 means "if we have a TM that decides P2, then there is a TM that decides P1. ...
slides - Center for Collective Dynamics of Complex Systems (CoCo)
... • It is possible to emulate the behavior of a certain TM using paper and pencil → Emulation of TMs is a naturally computable process! → It must be done by a TM too!! ...
... • It is possible to emulate the behavior of a certain TM using paper and pencil → Emulation of TMs is a naturally computable process! → It must be done by a TM too!! ...
decidable
... • If P is semidecidable then Pyes is called recursively enumerable. • If P is decidable, Pyes and Pno are said to be recursive. • Functions can also be Turing computable • A function f from strings to strings is Turing computable if whenever the Turing machine starts with x on the tape, it ends wit ...
... • If P is semidecidable then Pyes is called recursively enumerable. • If P is decidable, Pyes and Pno are said to be recursive. • Functions can also be Turing computable • A function f from strings to strings is Turing computable if whenever the Turing machine starts with x on the tape, it ends wit ...
Universal language Decision problems Reductions Post`s
... Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. Theorem (Matiyasevich 1970) Hilbert’ ...
... Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. Theorem (Matiyasevich 1970) Hilbert’ ...
ppt
... halt – Otherwise, set to 0 and enter the “carry” state. Then, move backwards trying to find a 0… ...
... halt – Otherwise, set to 0 and enter the “carry” state. Then, move backwards trying to find a 0… ...