A version of the Church-Turing Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, non-well-founded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard Church-Turing Thesis.

this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) "hypothetic" physical devices can compute NP-hard problems faster than in exponential time

Nowadays, we are using mainly computer of fourth generation, and we are designing fifth-generation computers. It is reasonable to ask: what is the perspective? What will the computers of generation omega look like? ffl As the speed of data processing increases, we face a natural limitation of causality, according to which the speed of all processes is limited by the speed of light. ffl Lately, a new area of acausal (causality violating) processes has entered mainstream physics. This area has important astrophysical applications. In this paper, we show: ffl how non-equilibrium thermodynamics makes these processes consistent, ffl how these processes can be used in computations, and ffl how the very possibility of these processes lead to the granularity of the physical world. KEYWORDS: computer generations, granularity, non-equilibrium thermodynamics, quantum computing, acausal processes 1. GENERATIONS OF COMPUTERS: WE NEED FASTER AND FASTER COMPUTERS No matter how fast modern co...

In Newtonian physics, if we know the state of the world at some moment of time, then we can precisely predict the state of the world at all future times. In this sense, Newtonian physics is deterministic. In modern physics (starting with quantum mechanics), theories are usually non-deterministic in the sense that even if we know exactly the initial state of the world, we cannot uniquely predict the future state of the world. In quantum mechanics (and in most modern quantum-based physical theories), the best we can get is probabilities of different future states.

Accepted (Day Month Year) Communicated by (xxxxxxxxxx) In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation.

In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation.