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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 ›

11.2 Computation Traces

11.2 Computation Traces A Turing machine does not compute in one large jump. It computes step by step. A computation trace records these steps. Each step changes the machine from one configuration to the next. A configuration contains all information needed to continue the computation: Component Meaning current state the machine’s internal state tape contents the symbols written on the tape head position the cell currently being scanned Once these...
turing machines computation traces configurations algorithms May 2, 2026
Maths › 03. Logic and Foundations ›

11.3 Universal Machines

11.3 Universal Machines A universal Turing machine is a Turing machine that can simulate other Turing machines. This is one of the most important ideas in computability theory. It means that computation can be represented as data. A machine can be written down, encoded as a string, and given as input to another machine. In ordinary programming terms, a universal machine is like an interpreter. It receives two things: $$...
turing machines universality simulation programs computation May 2, 2026
Maths › 03. Logic and Foundations ›

11.1 Machine Definitions

11.1 Machine Definitions A Turing machine is a formal model of an algorithm. It describes computation using a finite set of rules and an unbounded tape. The machine has three main parts: Part Role Tape stores symbols Head reads and writes one cell at a time Finite control stores the current state The tape is divided into cells: $$ \cdots,\ c_{-2},\ c_{-1},\ c_0,\ c_1,\ c_2,\ \cdots $$ Each cell contains...
turing machines formal models computation automata 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