Enumerations in computable structure theory
... interesting examples in computable structure theory. Selivanov [17] constructed a family of sets that Goncharov [11] used to produce a structure that is computably categorical but not relatively computably categorical. Manasse [14] used Selivanov’s family of sets to produce a computable structure wi ...
... interesting examples in computable structure theory. Selivanov [17] constructed a family of sets that Goncharov [11] used to produce a structure that is computably categorical but not relatively computably categorical. Manasse [14] used Selivanov’s family of sets to produce a computable structure wi ...
Enumerations in computable structure theory
... σ ⊆ χB iff B = A. The family is effectively discrete if there is a c.e. set E ⊆ 2<ω such that (a) for each A ∈ S, there is σ ∈ E such that σ ⊆ χA , and (b) for all σ ∈ E and all A, B ∈ S, if σ ⊆ χA , χB , then A = B. In [22], Selivanov proved the following. Theorem 2.1 (Selivanov). There exists a fami ...
... σ ⊆ χB iff B = A. The family is effectively discrete if there is a c.e. set E ⊆ 2<ω such that (a) for each A ∈ S, there is σ ∈ E such that σ ⊆ χA , and (b) for all σ ∈ E and all A, B ∈ S, if σ ⊆ χA , χB , then A = B. In [22], Selivanov proved the following. Theorem 2.1 (Selivanov). There exists a fami ...
"The Structure of Constant-Rank State Machines" ()
... Proof. (a) There are three cases of equivalence for invariants a, b: left-, right- and indirect equivalence. In the first two cases of ’direct’ equivalence, rank-lemma 1.1 yields: aLb implies r(a) = r(ab) ≤ r(b) and r(b) = r(ba) ≤ r(a), sothat r(a) = r(b); aRb implies r(a) = r(ba) ≤ r(b) and r(b) = ...
... Proof. (a) There are three cases of equivalence for invariants a, b: left-, right- and indirect equivalence. In the first two cases of ’direct’ equivalence, rank-lemma 1.1 yields: aLb implies r(a) = r(ab) ≤ r(b) and r(b) = r(ba) ≤ r(a), sothat r(a) = r(b); aRb implies r(a) = r(ba) ≤ r(b) and r(b) = ...
CS 208: Automata Theory and Logic
... – Introduced by Alan Turing as a simple model capable of expressing any imaginable computation – Turing machines are widely accepted as a synonyms for algorithmic computability (Church-Turing thesis) – Using these conceptual machines Turing showed that first-order logic validity problem a is non-com ...
... – Introduced by Alan Turing as a simple model capable of expressing any imaginable computation – Turing machines are widely accepted as a synonyms for algorithmic computability (Church-Turing thesis) – Using these conceptual machines Turing showed that first-order logic validity problem a is non-com ...
Foundations of Computation - Department of Mathematics and
... of the propositional variables that they contain. Figure 1.1 is a truth table that compares the value of (p ∧ q) ∧ r to the value of p ∧ (q ∧ r) for all possible values of p, q, and r. There are eight rows in the table because there are exactly eight different ways in which truth values can be assig ...
... of the propositional variables that they contain. Figure 1.1 is a truth table that compares the value of (p ∧ q) ∧ r to the value of p ∧ (q ∧ r) for all possible values of p, q, and r. There are eight rows in the table because there are exactly eight different ways in which truth values can be assig ...
CDM Recursive Functions Klaus Sutner Carnegie Mellon University
... (idealized) human computor, see Turing’s 1936 paper. Alas, they don’t match up well with the way an actual mathematician or computer scientist would define a computable function, or demonstrate that some already known function is in fact computable. Constructing a specific machine for a particular c ...
... (idealized) human computor, see Turing’s 1936 paper. Alas, they don’t match up well with the way an actual mathematician or computer scientist would define a computable function, or demonstrate that some already known function is in fact computable. Constructing a specific machine for a particular c ...
Advanced Logic —
... P ROOF. Suppose IND is true, but for a reductio, suppose that LNP is not. Then there is some subset B ⊆ N such that: (1) B is not empty; and (2) B has no least element. Let A = N\B. Clearly 0 ∈ / B (for then 0 would be the least element), so 0 ∈ A. Moreover, since B has no least element, this just m ...
... P ROOF. Suppose IND is true, but for a reductio, suppose that LNP is not. Then there is some subset B ⊆ N such that: (1) B is not empty; and (2) B has no least element. Let A = N\B. Clearly 0 ∈ / B (for then 0 would be the least element), so 0 ∈ A. Moreover, since B has no least element, this just m ...
The Arithmetical Hierarchy Math 503
... 1. A set is ψ-recursive if and only if it and its complement are ψ-recursively enumerable. 2. A set A is ψ-recursively enumerable if and only if it is the projection of a ψ-recursive relation R, i.e. if and only if A = {x : ∃yR(x, y)}. Note. There is no loss of generality above in considering R a bi ...
... 1. A set is ψ-recursive if and only if it and its complement are ψ-recursively enumerable. 2. A set A is ψ-recursively enumerable if and only if it is the projection of a ψ-recursive relation R, i.e. if and only if A = {x : ∃yR(x, y)}. Note. There is no loss of generality above in considering R a bi ...
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES 1
... K-triviality has become central for the investigation of algorithmic randomness. This property of a set A ∈ 2ω expresses that A is as far from random is possible, in that its initial segments are as compressible as possible: for all n, K(A n ) ≤+ K(n).1 The robustness of this class is expressed by ...
... K-triviality has become central for the investigation of algorithmic randomness. This property of a set A ∈ 2ω expresses that A is as far from random is possible, in that its initial segments are as compressible as possible: for all n, K(A n ) ≤+ K(n).1 The robustness of this class is expressed by ...
Notes on Simply Typed Lambda Calculus
... path(r2 ) and let s1 and s2 be the corresponding sub-terms occurring at σ. Show that s1 and s2 are either both variables, or both applications, or both abstractions. Show that the two parts of 2. in the definition of ≡α are equivalent for r1 and r2 . Exercise 1.4 Give a formal definition of what it ...
... path(r2 ) and let s1 and s2 be the corresponding sub-terms occurring at σ. Show that s1 and s2 are either both variables, or both applications, or both abstractions. Show that the two parts of 2. in the definition of ≡α are equivalent for r1 and r2 . Exercise 1.4 Give a formal definition of what it ...
LOGIC I 1. The Completeness Theorem 1.1. On consequences and
... theory S contains the formula: ψϕ,x := ∃xϕ → ϕ(c/x). That is, every existential formula is witnessed by a constant in the theory S! We will show that every complete consistent theory with the Henkin property has a model. Intuitively, since every existential formula is witnessed by some constant, we ...
... theory S contains the formula: ψϕ,x := ∃xϕ → ϕ(c/x). That is, every existential formula is witnessed by a constant in the theory S! We will show that every complete consistent theory with the Henkin property has a model. Intuitively, since every existential formula is witnessed by some constant, we ...
mass problems associated with effectively closed sets
... simplicial complexes, the diffeomorphism problem for compact manifolds, and the problem of integrability in elementary terms. In an influential 1954 paper [56], Kleene and Post introduced a scheme for classifying unsolvable mathematical problems. Informally, by a real we mean a point in an effective ...
... simplicial complexes, the diffeomorphism problem for compact manifolds, and the problem of integrability in elementary terms. In an influential 1954 paper [56], Kleene and Post introduced a scheme for classifying unsolvable mathematical problems. Informally, by a real we mean a point in an effective ...
Slide 1
... ● There is always a next configuration of M and thus a next row in the tiling iff M does not halt. ● T is in TILES iff there is always a next row. ● So if it were possible to semidecide whether T is in TILES it would be possible to semidecide whether M fails to halt on . But H is not in SD. So ne ...
... ● There is always a next configuration of M and thus a next row in the tiling iff M does not halt. ● T is in TILES iff there is always a next row. ● So if it were possible to semidecide whether T is in TILES it would be possible to semidecide whether M fails to halt on . But H is not in SD. So ne ...
CS383, Algorithms Notes on Asymptotic Time Complexity 1 Growth
... Worst-case complexity provides an upper bound on the cost of processing inputs of a given size. However, the number of steps required by “typical” input instances may be considerably less. An example is the task of searching for a value in an unordered list of length n. If the target value does not ...
... Worst-case complexity provides an upper bound on the cost of processing inputs of a given size. However, the number of steps required by “typical” input instances may be considerably less. An example is the task of searching for a value in an unordered list of length n. If the target value does not ...
Assignment 5 (Exponentiation) Write a function integerPower(base
... integer that evenly divides each of the two numbers. Write function gcd that returns the greatest common divisor of two integers. (Coin Tossing) Write a program that simulates coin tossing. For each toss of the coin the program should print Heads or Tails. Let the program toss the coin 100 times, an ...
... integer that evenly divides each of the two numbers. Write function gcd that returns the greatest common divisor of two integers. (Coin Tossing) Write a program that simulates coin tossing. For each toss of the coin the program should print Heads or Tails. Let the program toss the coin 100 times, an ...
The Complexity of Local Stratification - SUrface
... to be used. Call the finite-register machine programs determined by the above choice instruction together with the standard finite-register machine instructions nondeterministic finite-register machine programs. Operationally, the choice instruction nondeterministically assigns a natural number to X ...
... to be used. Call the finite-register machine programs determined by the above choice instruction together with the standard finite-register machine instructions nondeterministic finite-register machine programs. Operationally, the choice instruction nondeterministically assigns a natural number to X ...
3463: Mathematical Logic
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
The Open World of Super-Recursive Algorithms and
... putability by means of λ-definability. In 1936, Kleene demonstrated that λ-definability is computationally equivalent to general recursive functions. In 1937, Turing showed that λ-definability is computationally equivalent to Turing machines. Church was so impressed by these results that he suggeste ...
... putability by means of λ-definability. In 1936, Kleene demonstrated that λ-definability is computationally equivalent to general recursive functions. In 1937, Turing showed that λ-definability is computationally equivalent to Turing machines. Church was so impressed by these results that he suggeste ...
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 ...
DUBLIN CITY UNIVERSITY
... determines the area of a triangle but first check to see if the three side lengths form a valid triangle. 1(c) ...
... determines the area of a triangle but first check to see if the three side lengths form a valid triangle. 1(c) ...
Slide 1
... • There is no bijection between N and T, but T may be subcountable. • Although both sets are infinite in conclusion there are more infinite sequences of real numbers than that could be mapped with natural numbers. ...
... • There is no bijection between N and T, but T may be subcountable. • Although both sets are infinite in conclusion there are more infinite sequences of real numbers than that could be mapped with natural numbers. ...
Document
... (2.9) Write a monitor to solve the readers/writers problem which works as follows: If readers and writers both are waiting, then it alternates between readers and writers. Otherwise it processes them normally (i.e., readers concurrently and writers serially). Solution: Readers-writers: monitor; Begi ...
... (2.9) Write a monitor to solve the readers/writers problem which works as follows: If readers and writers both are waiting, then it alternates between readers and writers. Otherwise it processes them normally (i.e., readers concurrently and writers serially). Solution: Readers-writers: monitor; Begi ...
lecture05
... least one relation symbol of arity ¸ 2. Then it is undecidable whether a sentence f is finitely satisfiable. ...
... least one relation symbol of arity ¸ 2. Then it is undecidable whether a sentence f is finitely satisfiable. ...
