ppt

... we have enough information to find it. However, actually getting the answer from the inputs is not feasible due to the complexity of the problem. This is not like asking “will the human race exist in 5,000 years” – we do not have enough information to answer that ...

... we have enough information to find it. However, actually getting the answer from the inputs is not feasible due to the complexity of the problem. This is not like asking “will the human race exist in 5,000 years” – we do not have enough information to answer that ...

Predicates

... and structures we might want to avoid presenting all the dirty details for the actual creation of the Turing Machine. • Imagine for example that you want to examine whether a polynomial has integer roots!!! • A high-level description of a Turing Machine is a list of instructions which we are confide ...

... and structures we might want to avoid presenting all the dirty details for the actual creation of the Turing Machine. • Imagine for example that you want to examine whether a polynomial has integer roots!!! • A high-level description of a Turing Machine is a list of instructions which we are confide ...

Document

... do we care? • Are any of these functions, ones that we would actually want to compute? – The argument does not even give any example of something that can’t be done, it just says that such an example exists ...

... do we care? • Are any of these functions, ones that we would actually want to compute? – The argument does not even give any example of something that can’t be done, it just says that such an example exists ...

1996TuringIntro

... the mathematician’s mind when attempting to grapple with abstract objects such as infinite sequences, but dismisses the idea that the “spark”, when it comes, is really “divine” or in any other way essentially resistant to mechanical modelling. It may indeed not be strictly algorithmic - but that is ...

... the mathematician’s mind when attempting to grapple with abstract objects such as infinite sequences, but dismisses the idea that the “spark”, when it comes, is really “divine” or in any other way essentially resistant to mechanical modelling. It may indeed not be strictly algorithmic - but that is ...

5 COMPUTABLE FUNCTIONS Computable functions are deﬁned on

... EXAMPLE 4 The function f (n) = n + 3 is computable. The input is W = 1n+1 . Thus we need only add two 1’s to the input. A Turing machine M which computes f follows: M = {q1 , q2 , q3 } = {s0 1s0 L, s0 B 1s1 L, s1 B 1sH L} Observe that: (1) q1 moves the machine M to the left. (2) q2 writes 1 in the b ...

... EXAMPLE 4 The function f (n) = n + 3 is computable. The input is W = 1n+1 . Thus we need only add two 1’s to the input. A Turing machine M which computes f follows: M = {q1 , q2 , q3 } = {s0 1s0 L, s0 B 1s1 L, s1 B 1sH L} Observe that: (1) q1 moves the machine M to the left. (2) q2 writes 1 in the b ...

Day33-Reduction - Rose

... To Use Reduction for Undecidability 1. Choose a language L1: ● that is already known not to be in D, and ● show that L1 can be reduced to L2. 2. Define the reduction R. 3. Describe the composition C of R with Oracle. 4. Show that C does correctly decide L1 iff Oracle exists. We do this by showing: ...

... To Use Reduction for Undecidability 1. Choose a language L1: ● that is already known not to be in D, and ● show that L1 can be reduced to L2. 2. Define the reduction R. 3. Describe the composition C of R with Oracle. 4. Show that C does correctly decide L1 iff Oracle exists. We do this by showing: ...

0,1 - Duke University

... Proof Each function {0,1}* {0,1} can be represented as a binary string: f(0) f(1) f(00) f(01) f(10) f(11) … Suppose this set is countable. Then we are able to create a list of all problems 1. a11a12 a13 ...

... Proof Each function {0,1}* {0,1} can be represented as a binary string: f(0) f(1) f(00) f(01) f(10) f(11) … Suppose this set is countable. Then we are able to create a list of all problems 1. a11a12 a13 ...

decidable

... – The blank tape halting problem is semidecidable if there is a Turing machine M that, given an encoding of TM T, halts and says “yes” if T halts on blank tape, but M fails to halt if T fails to halt on blank tape – The passing problem is semidecidable if there is a Turing machine M that, given an ...

... – The blank tape halting problem is semidecidable if there is a Turing machine M that, given an encoding of TM T, halts and says “yes” if T halts on blank tape, but M fails to halt if T fails to halt on blank tape – The passing problem is semidecidable if there is a Turing machine M that, given an ...

Playing Chess with a Philosopher: Turing and Wittgenstein

... than propositions about physical objects – or about sense impressions, need an analysis. What mathematical propositions do stand in need of is a clarification of their grammar, just as those other propositions”. (Wittgenstein, R.F.M., VII-16 : 378.) ...

... than propositions about physical objects – or about sense impressions, need an analysis. What mathematical propositions do stand in need of is a clarification of their grammar, just as those other propositions”. (Wittgenstein, R.F.M., VII-16 : 378.) ...

Extended Analog Computer and Turing machines - Hektor

... In the theory of analog computation two different families of models of analog computation that come from different behaviors of computation processes in time have been considered. The first family contains models of computation on real numbers but in discrete time. These are, for example, the Analo ...

... In the theory of analog computation two different families of models of analog computation that come from different behaviors of computation processes in time have been considered. The first family contains models of computation on real numbers but in discrete time. These are, for example, the Analo ...

Slides

... point where things don't make sense You should always mark such proofs clearly Start your proof with the words Proof by contradiction Write Negation of conclusion as the justification for the negated conclusion Clearly mark the line when you have both p and ~p as a contradiction Finally, s ...

... point where things don't make sense You should always mark such proofs clearly Start your proof with the words Proof by contradiction Write Negation of conclusion as the justification for the negated conclusion Clearly mark the line when you have both p and ~p as a contradiction Finally, s ...

The Open World of Super-Recursive Algorithms and

... that were stronger than Turing machines were fruitless. Equivalence with Turing machines has been proved for many models of algorithms. That is why the majority of mathematicians and computer scientists have believed that the Church-Turing Thesis was true. Many logicians assume that the Thesis is an ...

... that were stronger than Turing machines were fruitless. Equivalence with Turing machines has been proved for many models of algorithms. That is why the majority of mathematicians and computer scientists have believed that the Church-Turing Thesis was true. Many logicians assume that the Thesis is an ...

Step back and look at the Science

... Won a Prize in 1936 for work on probability theory Became interested in Hilbert’s Entscheidungsproblem (decision problem) of 1928 1936, Turing came up with proof of impossibility …but Alonzo Church published independent paper also showing that it is impossible 1937 Turing’s "On computable ...

... Won a Prize in 1936 for work on probability theory Became interested in Hilbert’s Entscheidungsproblem (decision problem) of 1928 1936, Turing came up with proof of impossibility …but Alonzo Church published independent paper also showing that it is impossible 1937 Turing’s "On computable ...

Thesis statements

... planner, in which you discuss the prompt/thesis you selected. Be sure to support your thesis statement with details and examples (3 reasons) . ...

... planner, in which you discuss the prompt/thesis you selected. Be sure to support your thesis statement with details and examples (3 reasons) . ...

mathematical logic: constructive and non

... 'computation procedures' in which the computer is to perform steps depending on some unpredictable future state of his mind, or in which the 'procedure' is somehow to vary with the argument of the function. But for the thesis, ' computation ' is intended to mean of a predetermined function independe ...

... 'computation procedures' in which the computer is to perform steps depending on some unpredictable future state of his mind, or in which the 'procedure' is somehow to vary with the argument of the function. But for the thesis, ' computation ' is intended to mean of a predetermined function independe ...

Recursive Enumerable

... A language is decidable if and only if both it and its complement are r.e. Theorem A language L is r.e. if and only if there is a decidable two-argument predicate P such that x is in L there exists y such that P(x,y). This P is a verifier: if you are given y, then you can use P to verify that x is ...

... A language is decidable if and only if both it and its complement are r.e. Theorem A language L is r.e. if and only if there is a decidable two-argument predicate P such that x is in L there exists y such that P(x,y). This P is a verifier: if you are given y, then you can use P to verify that x is ...

Advanced Topics in Theoretical Computer Science

... M has finitely many transitions and the alphabet is finite, this conjunction is finite as well, and thus a formula of first order logic. ...

... M has finitely many transitions and the alphabet is finite, this conjunction is finite as well, and thus a formula of first order logic. ...

Computing functions with Turing machines

... we don’t know for sure if it is going to accept the input or not. • It turns out that this problem is not solvable! • In other words we can prove that the predicate Halt is not computable (there is no Turing Machine that takes as input the pair (, x) and decides if M is
going to accept on x.
...

... we don’t know for sure if it is going to accept the input or not. • It turns out that this problem is not solvable! • In other words we can prove that the predicate Halt is not computable (there is no Turing Machine that takes as input the pair (

Annals of Pure and Applied Logic Ordinal machines and admissible

... second author recast the proof of the Sacks–Simpson theorem using the computational paradigm instead of constructibility theory. The crucial point involved was how the informally presented recursions in the argument of Sacks and Simpson [11, 12] (and a recursion method presented by Shore [13]) can b ...

... second author recast the proof of the Sacks–Simpson theorem using the computational paradigm instead of constructibility theory. The crucial point involved was how the informally presented recursions in the argument of Sacks and Simpson [11, 12] (and a recursion method presented by Shore [13]) can b ...

God, the Devil, and Gödel

... remain adherents of p and adherents of q (and alas, sometime also adherents of ( p q ) as well). However, in such a case what is usually alleged to have been disproved by Gödel is either that p or that q. It therefore requires not only the meta-mathematical result, but also considerable philosoph ...

... remain adherents of p and adherents of q (and alas, sometime also adherents of ( p q ) as well). However, in such a case what is usually alleged to have been disproved by Gödel is either that p or that q. It therefore requires not only the meta-mathematical result, but also considerable philosoph ...

Sets, Logic, Computation

... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...

... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...

Computability and Incompleteness

... procedures. Showing that something is computable is easier: you just describe an algorithm, and assume it will be recognized as such. Showing that something is not computable needs more conceptual groundwork. Surprisingly, formal models of computation did not arise until the 1930’s, and then, all of ...

... procedures. Showing that something is computable is easier: you just describe an algorithm, and assume it will be recognized as such. Showing that something is not computable needs more conceptual groundwork. Surprisingly, formal models of computation did not arise until the 1930’s, and then, all of ...

HONEST ELEMENTARY DEGREES AND DEGREES OF RELATIVE

... only if there is a g ∈ b that eventually dominates every f ∈ a. We refer the reader to [16, 17] for more information concerning the E relation, including its original definition in terms of universal functions. We remark that although

... only if there is a g ∈ b that eventually dominates every f ∈ a. We refer the reader to [16, 17] for more information concerning the E relation, including its original definition in terms of universal functions. We remark that although

... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...