Notions of Computability at Higher Type

... §1. Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium. In elementary recursion theory, one begins with the question: what do ...

... §1. Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium. In elementary recursion theory, one begins with the question: what do ...

MARTIN`S CONJECTURE, ARITHMETIC EQUIVALENCE, AND

... restricted to the class of uniformly Turing invariant functions. Theorem 1.2 (Slaman and Steel [25]). Part I of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem 1.3 (Steel [26]). Part II of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem ...

... restricted to the class of uniformly Turing invariant functions. Theorem 1.2 (Slaman and Steel [25]). Part I of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem 1.3 (Steel [26]). Part II of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem ...

Computability theoretic classifications for classes of structures

... On the one end are the classes which have some global property restricting the behavior of their structures. On the other end are the classes which are complete in the sense that they allow all possible behaviors to happen. Let us say a bit more about these two extremes. Tame classes. In Section 2, ...

... On the one end are the classes which have some global property restricting the behavior of their structures. On the other end are the classes which are complete in the sense that they allow all possible behaviors to happen. Let us say a bit more about these two extremes. Tame classes. In Section 2, ...

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 ...

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

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 ...

... 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 ...

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 ...

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 ...

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 (

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. ...

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 ...

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 ...

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) . ...

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 ...

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 ...

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 ...

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 ...

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.) ...

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 ...

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 ...

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: ...

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 ...