Towards Formal Verification of Computations and Hypercomputations in Relativistic Physics

It is now more than 15 years since Copeland and Proudfoot introduced the term hypercomputation. Although no hypercomputer has yet been built (and perhaps never will be), it is instructive to consider what properties any such device should possess, and whether these requirements could ever be met. Aside from the potential benefits that would accrue from a positive outcome, the issues raised are sufficiently disruptive that they force us to re-evaluate existing computability theory. From a foundational viewpoint the questions driving hypercomputation theory remain the same as those addressed since the earliest days of computer science, viz. what is computation? and what can be computed? Early theoreticians developed models of computation that are independent of both their implementation and their physical location, but it has become clear in recent decades that these aspects of computation cannot always be neglected. In particular, the computational power of a distributed system can be expected to vary according to the spacetime geometry in which the machines on which it is running are located. The power of a computing system therefore depends on its physical environment and cannot be specified in absolute terms. Even Turing machines are capable of super-Turing behaviour, given the right environment

On the Possibility and Consequences of Negative Mass

We investigate the possibility and consequences of the existence of particles having negative relativistic masses, and show that their existence implies the existence of faster- than-light particles (tachyons). Our proof requires only two postulates concerning such particles: that it is possible for particles of any (positive, negative or zero) relativistic mass to collide inelastically with 'normal' (i.e. positive relativistic mass) particles, and that four-momentum is conserved in such collisions

The halting problem is soluble in Malament-Hogarth spacetimes.

We provide an Isabelle verification that the Halting Problem can be solved in Malament-Hogarth (MH) spacetimes. Our proof is quite general -- rather than assume the full machinery of general relativity, we only assume the existence of a reachability relation defined on an abstract space of locations. An MH spacetime can then be described as a space in which there exists an unboundedly long path together with a location which is reachable from all points on that path. Likewise, we use a very general notion of computation - the current state of a computation is assumed to be representable as a machine configuration containing all the information required to determine how the system changes with the execution of each ensuing instruction. The program is deemed to halt if the system enters a stable configuration. Since this situation is generally detectable by an operating system, we can use its occurrence to trigger events that exploit the nature of MH spacetimes, thereby enabling us to detect whether or not halting will eventually have occurred. Our verification follows existing arguments in the literature, albeit translated into this more general setting

Groups of worldview transformations implied by isotropy of space

Given any Euclidean ordered field, Q, and any 'reasonable' group, G, of (1+3)-dimensional spacetime symmetries, we show how to construct a model MG of kinematics for which the set W of worldview transformations between inertial observers satisfies W=G. This holds in particular for all relevant subgroups of Gal, cPoi, and cEucl (the groups of Galilean, Poincaré and Euclidean transformations, respectively, where c∈Q is a model-specific parameter orresponding to the speed of light in the case of Poincaré transformations). In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set W of worldview transformations satisfies either W⊆Gal, W⊆cPoi, or W⊆cEucl for some c>0. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are Poincaré transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of Q to the field of reals. As part of this work, we also prove the rather surprising result that, for any G containing translations and rotations fixing the time-axis t, the requirement that G be a subgroup of one of the groups Gal, cPoi or cEucl is logically equivalent to the somewhat simpler requirement that, for all g∈G: g[t] is a line, and if g[t]=t then g is a trivial transformation (i.e. g is a linear transformation that preserves Euclidean length and fixes the time-axis setwise)

Comparative Analysis of Statistical Model Checking Tools

Statistical model checking is a powerful and flexible approach for formal verification of computational models like P systems, which can have very large search spaces. Various statistical model checking tools have been developed, but choosing between them and using the most appropriate one requires a significant degree of experience, not only because different tools have different modelling and property specification languages, but also because they may be designed to support only a certain subset of property types. Furthermore, their performance can vary depending on the property types and membrane systems being verified. In this paper we evaluate the performance of various common statistical model checkers against a pool of biological models. Our aim is to help users select the most suitable SMC tools from among the available options, by comparing their modelling and property specification languages, capabilities and performances

Experiencing the Sheffield team software project: a project-based learning approach to teaching agile

Graduates of computer science and software engineering degrees are often expected by employers to possess various technical skills as well as competencies in project management, testing, teamwork, and other soft skills. Extant literature has identified that these competencies are often not addressed by traditional teaching approaches such as lectures and labs. In this paper, we present a project-based learning approach to teaching agile software development where students work in multicultural teams to develop software for clients. This approach to teaching software development addresses some of the competencies required by employers, and the feedback from students, clients, and tutors are discussed and analysed critically

Faster than light motion does not imply time travel

Seeing the many examples in the literature of causality violations based on faster-than-light (FTL) signals one naturally thinks that FTL motion leads inevitably to the possibility of time travel. We show that this logical inference is invalid by demonstrating a model, based on (3+1)-dimensional Minkowski spacetime, in which FTL motion is permitted (in every direction without any limitation on speed) yet which does not admit time travel. Moreover, the Principle of Relativity is true in this model in the sense that all observers are equivalent. In short, FTL motion does not imply time travel after all

Investigations of isotropy and homogeneity of spacetime in first-order logic

We investigate the logical connection between (spatial) isotropy, homogeneity of space, and homogeneity of time within a general axiomatic framework. We show that isotropy not only entails homogeneity of space, but also, in Image 1, homogeneity of time. In turn, homogeneity of time implies homogeneity of space in general, and the converse also holds true in Image 2 An important innovation in our approach is that formulations of physical properties are simultaneously empirical and axiomatic (in the sense of first-order mathematical logic). In this case, for example, rather than presuppose the existence of spacetime metrics – together with all the continuity and smoothness apparatus that would entail – the basic logical formulas underpinning our work refer instead to the sets of (idealised) experiments that support the properties in question, e.g., isotropy is axiomatized by considering a set of experiments whose outcomes remain unchanged under spatial rotation. Higher-order constructs are not needed

Groups of worldview transformations implied by Einstein’s special principle of relativity over arbitrary ordered fields

In 1978, Yu. F. Borisov presented an axiom system using a few basic assumptions and four explicit axioms, the fourth being a formulation of the relativity principle; and he demonstrated that this axiom system had (up to choice of units) only two models: a relativistic one in which worldview transformations are Poincare transformations and a classical one in which they are Galilean. In this paper, we reformulate Borisov’s original four axioms within an intuitively simple, but strictly formal, first-order logic framework, and convert his basic background assumptions into explicit axioms. Instead of assuming that the structure of physical quantities is the field of real numbers, we assume only that they form an ordered field. This allows us to investigate how Borisov’s theorem depends on the structure of quantities. We demonstrate (as our main contribution) how to construct Euclidean, Galilean, and Poincare models of Borisov’s axiom system over every non-Archimedean field. We also demonstrate the existence of an infinite descending chain of models and transformation groups in each of these three cases, something that is not possible over Archimedean fields. As an application, we note that there is a model of Borisov’s axioms that satisfies the relativity principle, and in which the worldview transformations are Euclidean isometries. Over the field of reals it is easy to eliminate this model using natural axioms concerning time’s arrow and the absence of instantaneous motion. In the case of non-Archimedean fields, however, the Euclidean isometries appear intrinsically as worldview transformations in models of Borisov’s axioms and neither the assumption of time’s arrow, nor the rejection of instantaneous motion, can eliminate them