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Translating the Hypergame Paradox - UvA-DARE
Translating the Hypergame Paradox - UvA-DARE

... the identity map as a productive function. Actually, in recursion theory it is not hard to prove more: every set A containing all indices of the empty set and no index of A4r is productive*. For instance, call a number z $&e if there exists an n that bounds the length of any descending &chain starti ...
Chapter 2: Sets
Chapter 2: Sets

... Definition 3. Two sets A and B are equal (written A = B) if they have exactly the same members or elements. Two finite sets A and B are said to be equivalent (written A ∼ = B) if they have the same number of elements: that is, n(A) = n(B) a) {a, b, c} and {c, b, a} are equal. b) the set of all names ...
ordinal logics and the characterization of informal concepts of proof
ordinal logics and the characterization of informal concepts of proof

... is also a proof predicate, but Gonx8 is provable in (S) itself. To give a precise treatment of this idea of recognizing a proof predicate as such we shall consider formal systems whose constants are not only numerical terms and function symbols, but also proof predicates. This is independently justi ...
Inclusion-Exclusion Principle
Inclusion-Exclusion Principle

... Let the n presents be {1, 2, 3, …, n}, where the present i is owned by person i. Now a random ordering of the presents means a permutation of {1, 2, 3, …, n}. e.g. (3,2,1) means the person 1 picks present 3, person 2 picks present 2, etc. And the question whether someone picks his/her own present be ...
Recursive Functions
Recursive Functions

... Introducing Recursive Functions • We have seen so far that a function, such as main, can call another function to perform some computation. • In C, a function can also call itself. Such types of functions are called recursive functions. A function, f, is also said to be recursive if it calls anothe ...
int main(void)
int main(void)

... Introducing Recursive Functions • We have seen so far that a function, such as main, can call another function to perform some computation. • In C, a function can also call itself. Such types of functions are called recursive functions. A function, f, is also said to be recursive if it calls anothe ...
A,B
A,B

... by P(S), sometimes written 2S. • If a (finite) set has n elements then its power set has 2n elements. – You can determine this by the fundamental principle of counting. • Think about that. Can you see why |P(S)| = 2|S| ? – NB the number of rows in truth table that represents all possible truth value ...
An Introduction to Elementary Set Theory
An Introduction to Elementary Set Theory

... In this project we will learn elementary set theory from the original historical sources by two key figures in the development of set theory, Georg Cantor (1845–1918) and Richard Dedekind (1831– 1916). We will learn the basic properties of sets, how to define the size of a set, and how to compare di ...
Arithmetic as a theory modulo
Arithmetic as a theory modulo

... In natural deduction and in sequent calculus, the cut elimination theorem and the analysis of the structure of cut free proofs is the key to many results about predicate logic with no axioms: analyticity and non-provability results, completeness results for proof search algorithms, decidability resu ...
Set Theory
Set Theory

... Definition 28 Let U = {x1 , x2 , . . . , xn } be the universe and A be some set in this universe. Then we can represent A by a sequence of bits (0s and 1s) of length n in which the i-th bit is 1 if xi ∈ A and 0 otherwise. We call this string the characteristic vector of A. EXAMPLE ! If U = {x1 , x2 ...
Automata and formal languages ∗
Automata and formal languages ∗

... The second section deals with the second stage Chomsky’s hierarchy, namely context-free languages. The link between context-free grammars and pushdown automata is underlined, together with recent results on deterministic pushdown automata. As in the case of finite automata, the link with group theor ...
Chapter 2: Sets
Chapter 2: Sets

... Definition 3. Two sets A and B are equal (written A = B) if they have exactly the same members or elements. Two finite sets A and B are said to be equivalent (written A ∼ = B) if they have the same number of elements: that is, n(A) = n(B) a) {a, b, c} and {c, b, a} are equal. b) {4, 5, 6} and {4, 4, ...
Chapter 6 Recursion
Chapter 6 Recursion

... Computes 4 * 3! = 24 and returns to main Starts up the 2-Fact machine Waits for return value (2!) from 2-Fact. Computes 3 * 2! = 6 and returns to 4-Fact Starts up the 1-Fact machine Waits for return value (1!) from 1-Fact. Computes 2 * 1! = 2 and returns to 3-Fact Starts up the 0-Fact machine Waits ...
Infinite Sets and Infinite Sizes
Infinite Sets and Infinite Sizes

... As with the negative integers, the fractions are not natural numbers; for example, “one-third” is not an admissible answer to a how-many question. On the other hand, it can serve as an answer to an howmany-of question, as in “how many of the students got an A?” Similarly it can serve as an answer to ...
8
8

... For each number n ≥ 1, the prime factorization of n is a finite product of form Πi≤k p(i)ki . We know from the fundamental theorem of arithmetic that the prime factorization exists and is unique, for each number n ≥ 1. Think of the prime factorization of n as encoding a finite sequence as follows: 1 ...
Contents
Contents

degrees of recursively saturated models
degrees of recursively saturated models

... Lemma 1.3(i) and (ii), M must realize T(v) by some «. This « codes/,. Similar arguments show that S is closed under joins and has the tree property. Lemma 1.3 also implies M is S-saturated. Suppose that M is also S'-saturated. Then S' G S. Let s' G S'. Coding arguments similar to those above show th ...
Sequences, Sums, Cardinality
Sequences, Sums, Cardinality

... Let A and B be finite sets, i.e. |A| and |B| are finite. • What is the size of A × B? |A × B| = |A| · |B| • How many binary relations R ⊆ A × B from A to B are there? The number of relations from A to B is the number of subsets of A × B.Thus the answer is 2|A|·|B| • How many total functions f : A → ...
Computability theoretic classifications for classes of structures
Computability theoretic classifications for classes of structures

... low property, the categoricity property and the Turing ordinal. As more evidence towards its naturalness, we will see in Theorems 3.14 and 3.15 how Σ-smallness induces a strong dichotomy on classes. 1 In [Mon10b] we used the phrase “K has a computable 1-back-and-forth structure” for what we now say ...
1 Chapter III Set Theory as a Theory of First Order Predicate Logic
1 Chapter III Set Theory as a Theory of First Order Predicate Logic

... Set Theory was invented in large part to analyse the concept of infinity, and to develop systematic means of studying and describing its different manifestations in different contexts. Because of this it is in the curious situation that what it has to say about infinity is constitutive of the very s ...
Computational foundations of basic recursive function theory
Computational foundations of basic recursive function theory

... become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and theorems in recursion theory may be classified according to which axioms of computation are needed to prove them. The theory is about typed functions over the ...
Comparing Infinite Sets - University of Arizona Math
Comparing Infinite Sets - University of Arizona Math

... keep going forever on the number of elements in the finite set. In an infinite set, you still have the first element, second element, and so on. However there is no last element because the infinite set will contain a kth element and there will always be a (k+1)th element because there is no limit t ...
equivalents of the compactness theorem for locally finite sets of
equivalents of the compactness theorem for locally finite sets of

... As it is known (see [2]) Ff in is equivalent to some statement about propositional calculus. We consider the language {¬, ∧, ∨} and accept standard definitions of propositional formulae. A set X of propositional formulas is said to be locally satisfiable iff every finite subset X0 of X is satisfiabl ...
How Many Recursive Calls Does a Recursive Function
How Many Recursive Calls Does a Recursive Function

... convince a student that the growth increases exponentially (i.e. something like O(2n)), is it possible to calculate the exact n u m b e r o f calls for a given n? This is what m y student asked me. The answer turns out to be yes, and the result is only slightly more complicated than calculating the ...
A Basis for a Mathematical Theory of Computation
A Basis for a Mathematical Theory of Computation

... irrelevant to the statement being proved. 5. A way of defining new data spaces in terms of given base spaces and of defining functions on the new spaces in terms of functions on the base spaces. Lack of such a formalism is one of the main weaknesses of ALGOL but the business data processing language ...
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Computability theory

Computability theory, also called recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.The basic questions addressed by recursion theory are ""What does it mean for a function on the natural numbers to be computable?"" and ""How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?"". The answers to these questions have led to a rich theory that is still being actively researched. The field has since grown to include the study of generalized computability and definability. Invention of the central combinatorial object of recursion theory, namely the Universal Turing Machine, predates and predetermines the invention of modern computers. Historically, the study of algorithmically undecidable sets and functions was motivated by various problems in mathematics that turned to be undecidable; for example, word problem for groups and the like. There are several applications of the theory to other branches of mathematics that do not necessarily concentrate on undecidability. The early applications include the celebrated Higman's embedding theorem that provides a link between recursion theory and group theory, results of Michael O. Rabin and Anatoly Maltsev on algorithmic presentations of algebras, and the negative solution to Hilbert's Tenth Problem. The more recent applications include algorithmic randomness, results of Soare et al. who applied recursion-theoretic methods to solve a problem in algebraic geometry, and the very recent work of Slaman et al. on normal numbers that solves a problem in analytic number theory.Recursion theory overlaps with proof theory, effective descriptive set theory, model theory, and abstract algebra.Arguably, computational complexity theory is a child of recursion theory; both theories share the same technical tool, namely the Turing Machine. Recursion theorists in mathematical logic often study the theory of relative computability, reducibility notions and degree structures described in this article. This contrasts with the theory of subrecursive hierarchies, formal methods and formal languages that is common in the study of computability theory in computer science. There is a considerable overlap in knowledge and methods between these two research communities; however, no firm line can be drawn between them. For instance, parametrized complexity was invented by a complexity theorist Michael Fellows and a recursion theorist Rod Downey.
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