tag

#computability

17 pages tagged

Maths › 03. Logic and Foundations ›

14.5 Extensions

14.5 Extensions and Refinements Gödel’s original theorems admit stronger forms and alternative formulations. These refinements clarify exactly which assumptions are needed and how incompleteness behaves across different systems. Rosser’s Improvement Gödel’s first theorem used a strengthened consistency assumption. Rosser showed that this can be weakened. Instead of the original Gödel sentence, Rosser constructs a sentence $R$ that balances proofs of a statement and its negation. Informally, $R$ says: “if there...
logic godel proof theory computability foundations May 2, 2026
Maths › 03. Logic and Foundations ›

12.5 Structure of Degrees

12.5 Structure of Degrees Turing degrees organize problems by computational power. After Post’s problem, it becomes clear that this structure is highly complex. It is not a simple linear hierarchy but a rich partially ordered system. Partial Order Turing degrees are ordered by reducibility: $$ \mathbf{a} \leq_T \mathbf{b} $$ means every set in degree $\mathbf{a}$ is computable using an oracle for any set in degree $\mathbf{b}$. This order has three...
computability degree structure turing degrees logic May 2, 2026
Maths › 03. Logic and Foundations ›

12.2 Turing Degrees

12.2 Turing Degrees Turing reducibility compares sets by computational power. If $$ A \leq_T B $$ then an oracle for $B$ is strong enough to decide $A$. This comparison naturally groups sets into equivalence classes. Two sets $A$ and $B$ have the same Turing degree if each can compute the other: $$ A \equiv_T B $$ where: $$ A \leq_T B \quad \text{and} \quad B \leq_T A $$ A Turing...
computability turing degrees oracle computation degree theory May 2, 2026
Maths › 03. Logic and Foundations ›

12.3 Recursively Enumerable Sets

12.3 Recursively Enumerable Sets A set is recursively enumerable if there is an algorithm that can list its elements. The older name is recursively enumerable , often abbreviated as r.e. The modern name is computably enumerable , often abbreviated as c.e. Both terms refer to the same idea. A set $A \subseteq \mathbb{N}$ is recursively enumerable if there is a Turing machine that prints exactly the elements of $A$: $$...
computability recursively enumerable semi-decidable halting problem May 2, 2026
Maths › 03. Logic and Foundations ›

12.4 Post’s Problem

12.4 Post’s Problem (Overview) The halting problem shows that some sets are not computable. Turing degrees refine this by measuring how hard such sets are. A natural question arises: is there any degree strictly between computable sets and the halting problem? Let: $$ \mathbf{0} $$ be the degree of computable sets, and: $$ \mathbf{0'} $$ be the degree of the halting problem. Post’s problem asks whether there exists a recursively...
computability post problem priority method turing degrees May 2, 2026
Maths › 03. Logic and Foundations ›

12.1 Reducibility

12.1 Reducibility Reducibility is a way to compare problems by asking whether one problem can be solved using another. Suppose we have two decision problems, $A$ and $B$. Each problem asks whether an input belongs to a certain set: $$ x \in A $$ or $$ x \in B $$ We say that $A$ reduces to $B$ if a method for solving $B$ would also give us a method for...
computability reducibility decision problems turing machines May 2, 2026
Maths › 03. Logic and Foundations ›

12. Degrees of Unsolvability

Degrees of unsolvability study how undecidable problems can be compared according to their relative computational difficulty, rather than treating all undecidable problems as having the same level of complexity. The chapter begins with reducibility, where one problem is transformed into another in a computable way, so that a solution to the second problem would also give a solution to the first. Turing degrees are then introduced as equivalence classes of...
logic computability turing degrees reducibility recursively enumerable sets May 2, 2026
Maths › 03. Logic and Foundations ›

11.5 Undecidability Results

11.5 Undecidability Results The halting problem is the first major example of an undecidable problem. Once we know that one problem is undecidable, we can use it to prove that many other problems are undecidable too. A decision problem asks for a yes-or-no answer. In computability theory, we usually represent a decision problem as a language: $$ L \subseteq \Sigma^* $$ A machine decides $L$ if it always halts and...
undecidability reductions rice theorem computability logic May 2, 2026
Maths › 03. Logic and Foundations ›

11.4 Halting Problem

11.4 Halting Problem The halting problem asks whether a Turing machine eventually stops. Given a machine $M$ and an input word $w$, we ask: $$ \text{Does } M \text{ halt on input } w? $$ Here “halt” means that the computation eventually reaches a stopping state, such as: $$ q_{\mathrm{accept}} $$ or: $$ q_{\mathrm{reject}} $$ If $M$ reaches one of these states after finitely many steps, then $M$ halts on...
halting problem undecidability computability logic theory May 2, 2026
Maths › 03. Logic and Foundations ›

11. Turing Machines

Turing machines provide a simple formal model of computation, where an algorithm is represented by a finite set of instructions acting on a tape, a head position, and an internal state. The chapter begins with machine definitions, including the tape alphabet, states, transition function, starting configuration, accepting and rejecting states, and the precise meaning of one step of computation. Computation traces are then introduced to describe the full sequence of...
logic computability turing machines undecidability May 2, 2026
Maths › 03. Logic and Foundations ›

10.5 Equivalence

The previous section introduced several formal models of computation and stated that they compute the same class of functions. In this section, we give a more systematic account of this equivalence by presenting explicit constructions that translate computations from one model into another. The central result is that all standard models define exactly the same class of partial functions: $$ f : \mathbb{N}^n \rightharpoonup \mathbb{N}. $$ This class is called...
logic computability equivalence turing machines recursive functions May 2, 2026
Maths › 03. Logic and Foundations ›

10.4 Formal Models

The Church Turing thesis says that the informal idea of effective computation is captured by several equivalent formal models, and this section describes the main models that are used in computability theory. The purpose is not to develop every model in full technical detail, but to explain what each model treats as a computation, how it represents data, and why it defines the same class of computable functions. A formal...
logic computability formal models turing machines lambda calculus May 2, 2026
Maths › 03. Logic and Foundations ›

10.3 Church Turing

The purpose of computability theory is to make precise the idea of an effective procedure. Informally, an effective procedure is a method that can be followed step by step, without guesswork, and which produces an answer after finitely many steps whenever it is meant to halt. The central question is whether this informal idea can be captured by a formal mathematical definition. Recursive functions give one such definition. Turing machines...
logic computability church turing thesis turing machines recursive functions May 2, 2026
Maths › 03. Logic and Foundations ›

10. Computable Functions

Computability theory studies which functions can be calculated by a precise mechanical procedure, and it gives a mathematical meaning to the informal idea of an effective algorithm. The chapter begins with recursive functions, where computation is described by basic initial functions together with operations that generate more complex functions, such as composition, primitive recursion, and minimization. Partial and total functions are then distinguished, since some computations produce an output for...
logic computability recursive functions computation May 2, 2026
Maths › 03. Logic and Foundations ›

10.1 Recursive Functions

Recursive function theory gives a mathematical definition of computable numerical functions by specifying basic functions and closure operations that generate increasingly complex functions from simple ones, and this framework allows one to reason about computation purely within arithmetic. All functions considered in this section have inputs and outputs in the natural numbers: $$ \mathbb{N} = {0,1,2,\dots}. $$ An $n$ ary function is written: $$ f : \mathbb{N}^n \to \mathbb{N}, $$...
logic computability recursive functions primitive recursion May 2, 2026
Maths › 03. Logic and Foundations ›

10.2 Partial vs Total

A computable process may fail to produce an output for some inputs, not because the input is malformed, but because the process never halts. This distinction leads to two kinds of functions: total functions, which are defined on every input, and partial functions, which may be undefined on some inputs. In computability theory, partial functions are essential because algorithms are allowed to run forever. A computation that never halts produces...
logic computability partial functions total functions May 2, 2026
Maths ›

03. Logic and Foundations

This volume develops formal logic, set theory, computability, and the foundations of mathematics in a unified way, where syntax, semantics, proof, and computation are treated as precise mathematical objects. Part I. Propositional and First-Order Logic Chapter 1. Propositional Logic This chapter introduces the basic language of propositional logic, its semantics, equivalence laws, normal forms, and the relation between proof and truth. 1.1 Syntax: Variables, Connectives 1.2 Semantics: Truth Tables 1.3...
logic foundations set-theory computability proof-theory May 2, 2026