10.5 Equivalence of Models
Detailed equivalence proofs between Turing machines, recursive functions, and lambda calculus, with explicit constructions and simulations.
17 notes
Detailed equivalence proofs between Turing machines, recursive functions, and lambda calculus, with explicit constructions and simulations.
Distinction between partial and total computable functions, undefined values, domains of definition, and the role of nontermination.
Primitive recursive functions, general recursive functions, minimization, and the formal construction of computable numerical functions.
Stronger forms of incompleteness, Rosser’s improvement, Löb’s theorem, and connections to computability.
Global properties of the Turing degrees including incomparability, density, and jump structure.
Existence of intermediate degrees between computable sets and the halting problem.
Sets that can be enumerated by algorithms and their role in semi-decidability and computability theory.
Equivalence classes of sets under Turing reducibility and the ordering of computational power.
Formal methods for comparing decision problems using many-one and Turing reducibility.
Reducibility, Turing degrees, recursively enumerable sets, Post's problem, and the structure of degrees.
Extensions of undecidability using reductions and general results such as Rice’s theorem.
The undecidable problem of determining whether a Turing machine halts on a given input.
Turing machine definitions, computation traces, universal machines, the halting problem, and undecidability results.
Turing machines, register machines, lambda calculus, recursive functions, and the precise mathematical models used to define computation.
The Church Turing thesis, formal models of computation, equivalence of models, and the distinction between mathematical theorem and foundational principle.
Recursive functions, partial and total functions, the Church-Turing thesis, formal models of computation, and equivalence of models.
Formal logic, set theory, computability, and the foundations of mathematics treated as a formal system.