Characterising Computational Devices with Logical Systems

In this thesis we shall present and develop the concept of a theory machine. Theory machines describe computation via logical systems, providing an overarching formalism for characterising computational systems such as Turing machines, type-2 machines, quantum computers, infinite time Turing machines, and various physical computation devices.

Notably we prove that the class of finite problems that are computable by a finite theory machine acting in first-order logic is equal to the class Turing machine computable problems. Whereas the class infinite problems that are computable by a finite first-order theory machine is equal to the class type-2 machine computable problems.

A key property of a theory machine computation is that it does not have to occur in a causally ordered manner. A consequence of this fact is that the class of problems that are computable by finite first-order theory machine in polynomial resources is equal to $NP \cap co-NP$. Since there are problems which appear to lie in $NP \cap co-NP \setminus P$ that are efficiently solvable by a quantum computer (such as the factorisation problem), this gives weight to the argument that there is an atemporal/non-causal component to the apparent speed-up offered by quantum computers.