# Semantic Spacetime - What is it?

A semantic spacetime is a discrete model of space and time, which may be applied at arbitrary scale, to describe systems functional and non-functional systems[1]. The idea of semantic spacetime was coined (by me) as a way of discussing the properties of position, time, and their functional properties, within network model of space. The idea draws on a synthesis of pre-existing ideas about space and time, and reformulates them in a traditional graph theoretical language. Although unconventional as a concept, it is conservative in the forms and claims it makes, in an effort to remain close to the traditions of physics.

Semantic spacetime is a discrete model of spacetime, but it is not intended as a theory of quantum gravity, in spite of some affinity with quantum systems. Nor does it make assumptions about spacetime, such as the existence of a manifold structure, or the existence of particular invariances. Rather, its structure rooted in simple information concepts. In particular, it incorporates Milner's bigraph model of computational interactions in a form that is more familiar outside of category theory[5].

The idea of semantic spacetimes is outlined in a series of papers, discussing spacetime ideas, from scaling to symmetry, and topology[2,3,4]. It adopts a discrete spacetime model, with an extreme form of locality, and relates observer semantics to the issues of knowledge from observables. Interesting aspects of well-known physics emerge from basic requirements for information propagation.

## Motivation

The formulation on semantic spacetime aims to address the issues that arise in describing configuration, dynamics, and purpose, in both fundamental science, and in technology. These aspects are normally not reconcilable within a common language. Thus, it may span viewpoints from elementary particle physics to cities[5], indeed information systems at any scale. Although not a complete theory, it lays out guidance on the formulation of the basic issues of information propagation, with some proofs left to the reader.

The motivation for semantic spacetime came from information technology, where the inevitability enjoyed in descriptions classical physics cannot be assumed. An object may not continue in a state of uniform motion unless the locations or points of spacetime have homogeneous behaviours. In information technology, each node is independent and information flows only if there is cooperation at every location; nothing corresponds to a the assumption of conservation of momentum through spacetime, for instance.

Physics has many tools for describing observed behaviours, but traditionally it suppresses all but the simplest semantics, in its effort to be impartial and universal. It does not commonly try to generalise the concept of propagation to include inhomogeneities in spacetime semantics (e.g. spatial variations of physical law) or intentionality, except in material science, where they are assumed to arise only on the macroscopic scale. For example, it does not deal with interpretations of relative functional purpose and meaning, which are essential to understanding engineering, biology, and technology.

The separation of spacetime, as an inert background, is a key part of the Enlightenment tradition of science, as an impartial system for describing the world. However, semantics do arise in science, in the form of complementary names, types and labels, which signify what we know about the behaviours of the things within it. For instance, the semantic labels of elementary forces are electric charge, nuclear charge, mass, etc. These form virtual networks too, which generalise the simple notion of points being next to one another, which are important in biology. Semantic spacetime posits that, by ignoring such semantics, we may be doing a disservice to a complete scientific understanding, and that there exists an impartial formulation, based on the formalisation of `promise theory’, in which subjective aspects of measurement can play a formal role, leading to new insights. The introduction of semantics requires little formalism, and mimics standard formulations of spacetime, suggesting that formalisation of semantics may be of wider importance than conventionally assumed.

## Semantics of space and time

The semantics of ordinary space and time are diverse in interpretation. For space, we think of distance, trajectory, adjacency (topology), neighbourhood, continuity, direction, etc. For time, we have clock time, duration, time of day, partial ordering, etc.

Promise theory is used as a representation for semantics. Directed adjacency is a logical primitive, as in graph theory, but with the caveat that each node must both emit and absorb adjacency relations, cooperatively, making `probable spacetime adjacency’ similar in structure to quantum probabilities and transitions. Thus space is made up of cooperating nodes and edges.

In promise theory, which extends graphs with semantic labels, each adjacency therefore requires multiple promises from each node, so the formulation could be argued more fundamental than an ordinary graph theory. Although not normally considered relevant in physics, these stronger requirements for spacetime continuity may still play a role in quantum descriptions of physical spacetime.

Local and global time are measured from within the interior of a semantic spacetime, and are interpreted through transitions from one state to another. This can only happen at a fixed rate, because every system essentially measures its own clock. The ability to observe remote events is a non-trivial problem in this model. A consequence of this is that there is no concept of variable velocity, nor momentum, and thus a discrete spacetime with finite number of states is not obviously a canonical system. This does not exclude the existence of canonical subsystems, however. There is therefore an implicit challenge to view continuum formulations of physics as some limit of discrete transition systems. The connection with canonical systems remains unknown.

In physics, semantics are usually limited to force labels and particle types, and space is classically neutral (this may have to change for a quantum theory of spacetime), but at a larger scale we distinguish, for instance, between parking spaces and fields. These different semantics play a role in the interactions between spatial regions, and thus lead to new physics and new scales.

Two notable things to come out of the work on semantic spacetimes are:

- Many familiar ideas learned over decades of thinking about artificial intelligence emerge from a discussion of spacetime semantics, without any implementation details needing to be presupposed. So, although one could say that `nothing really new' comes out of it, many standard things emerge very easily from very few assumptions.
- The semantic spacetime (or promise theory) view is that logic emerges from reasoning, rather than vice versa (which was the starting point of most computer science).

## Semantic space and materials

There are some conflicts of emphasis between a view of spacetime based on information and one based on dynamics. Historically, spacetime has been a backdrop (like a Cartesian theatre) in which the play of physics took place. In modern times, quantum theory has thrown doubt on this view. A model of space, time, and observation that attempts to take into account the semantics of locations and events, was explored as a way of developing a physics of more macro-level systems, like machinery, organisations, and even cities.

Robin Milner's description of space as functional machinery is revealing as an entirely separate way of describing space, through containment rather than by invariance and long range order[6]. A simple question asked by semantic spacetime is whether it makes sense to distinguish between matter and space, e.g. does matter fill space, replace space, or simply relabel space? From an information perspective, labelling is the only independent concept. Thus, why would one need to introduce space and matter as separate substances, rather than different states of the same? There is an analogy here, at fundamental scales, with the discussion of the quantum vacuum and its relationship to the old concept of the aether. There cannot directly be acceleration in a closed discrete spacetime, since all transitions happen at the same speed, so one cannot define inertial mass, except as an effect of spacetime geometry. In elementary physics, this would be controversial. In technology, there is little controversy about digital transition systems, and the formulation is immediately helpful to understanding software behaviours.

## Points as functional agents

The idea of a spacetime made up of `autonomous agents' as reducible or irreducible points, embraces locality and thus a the key principle of relativity. In such a formulation, the nature of space and time is intrinsically linked to the problem of measurement, because every point acts like an independent observer. Even the propagation of a message cannot be assumed as a necessity. If applied to physical spacetime, this challenges the idea of universal physical law.

Agents are autonomous and inert, except for the promises they make. Elementary agents may have no capabilities beyond representing fields of force, but coarse-grained regions may lead to new semantics, such as molecules, materials, and even biological processes; a large (human) scale space may represent a shopping district with diverse functional capabilities, with points representing different shops. The contention of Semantic Spaces is that it makes sense to carry concepts from the physics of the small to the large, by scaling across this full spectrum of scales, not just across a few orders of magnitude in which the semantics are inert.

## Measurement and the semantics of spacetime

The different capabilities of agents or points at different scales leads to the possibility of non-trivial semantics. In measurement, observational data may be gathered and averaged for stability, in one of two ways, which reflect spacelike and timelike ensembles:

- Repeated trials with constant state and semantics, in which time plays no role.
- Continuously adapting accumulation of state, whose semantics define change in real time.

The first of these corresponds to the normal experimental trial approach in science. The second corresponds to cognitive realtime input. The two approaches are inequivalent. They correspond roughly to the Frequentist and Bayesian interpretations of statistical probability (or limited memory, sliding window Markov processes). A more intuitive way of referring to them is as spacelike ensemble measurement (objective) and timelike `cognitive' measurement (subjective), respectively. The quantitative and qualitative values of the approaches may differ by an indeterminate amount, yet both are valid estimators of some notion of reality.

## Relationship to machine learning and artificial intelligence

The idea that degrees of semantics are expressed by the structure of spacetime itself, together with an account of the scaling of such semantics, offers a new perspective on cognition, and its use in machine learning and artificial reasoning. The semantic spacetime argument posits that phenomena one normally associates with cognition and reasoning are rooted, in fact, in the recognition of basic spacetime geometry, which could provide (speculatively) evolutionary route for the emergence of intelligence in adaptive systems. The semantics of learning and reasoning were developed, with examples, and illustrating software.

## Lamport’s relativity in computer science

The view of time as a relative transition system goes back to the work of Leslie Lamport, in computer science of distributed systems[7]. Inspired by Einstein's observations about the relativity of local clocks to communicating computers in special relativity, Lamport rediscovered the idea that time can at best be understood as a precedence relation, in a discrete spacetime context. The discrete nature of space, in a computer network, accentuates the issues of relativity, which are almost unobservable on the scale of human activity, in the apparent continuum of physical spacetime.

In discrete network transition system, there is only a single speed: the speed of transitions. Unless agents have localised internal clocks, they cannot make transitions at different rates. Each agent's transitions (including message send and receive) counts as ticks of its clock, and lead to the advancement of time as a distinguishable state. Because there are many independent processes in a computer network, there are many different interpretations of time too. This often leads to semantic difficulties, such as when software engineers assume local semantics of non-local interactions. This is related to the difficulties surrounding the CAP conjecture.

## Renormalisation and scaling of semantics

An aspect of physical description, which is not evident in computer science models, is the notion of scaling in space, time, and dynamical processes. Computer science currently lacks an well developed technique for discussing the rescaling of system changes and dynamical variables. This problem is quite well understood in the physics of renormalisation, dimensional analysis, and dynamical similarity.

The scaling of semantic interpretations and functional behaviours is a key goal for semantic spacetime. A key finding is about the need for index or directory information in coarse grains in order to probe interior properties. Thus one may quantify the exact loss of semantics by scaling. In technology, this poses very practical questions:

- Does a computer program running on a single computer behave the same as the same computer program running on 10 computers?
- Does a gas of one atom behave the same as a gas of 10 atoms?
- How does the functional behaviour of a cup handle change when it is scaled to twice the size?

## Quantum Mechanics and Field Theory as semantic spacetime

The highly successful physical model of a quantum field has some similarities with semantic spacetime, as a special case with homogeneous semantics. A quantum field is a function that is defined across a spacetime region, with a status which is somewhere between space and matter. It has semantics associated with discrete excitations, usually called particles (though these particles do not have the same semantics as the billiard ball particles of classical mechanics), charge, and conservation, etc. Emission and absorption of information are complementary and have the same structure as cooperation in a semantic spacetime.

The propagation of influence in semantic spacetime has structural similarities with classical and quantum field theory. Information theoretically, matter is a property of a spacetime region. A viewpoint of a field as a labelled spacetime bridges the separate formulations of quantum field theory due to Feynman and Schwinger[8]. Feynman's spacetime paths are timeline trajectories of causal transitions through spacetime, in which sequences of interactions describe `cognitive' semantics. In Schwinger's view, these paths are formed from individual spacetime elements (agent excitations) in a field that pervades all the points, and only the endpoints (the source and the sink of an interaction) determine the observable semantics. The timeline `cognitive semantics' can be applied as operations to the generating functional for the quantum field, to generate the equivalent of Feynman diagrams by introducing fictitious sources, analogous to fictitious interaction vertices in Feynman's approach.

The promises made by each infinitesimal spacetime region in a Schwinger viewpoint, are related to the promises made on edges and vertices in Feynman graphs. The attractive feature of the Feynman diagrams lies in explicating a cognitive measurement process (which is a human point of view), i.e. realtime causation, while the advantage of the Schwinger approach lies in explicating the invariances and ensemble properties. One may thus think of the approaches as representing different semantic decomposition of a process, whose ensemble sum is equivalent, pertaining to cognitive and ensemble views. Schwinger later introduced Source Theory, as an alternative more `cognitive' model in own search to understand the measurement problem[9].

## Memory space

The ability for spacetime to store memories, depends on what kinds of promises each element is capable of keeping. Simple quantum fields of force have to establish stable fixed point structures like atoms and molecules in order to become addressable. Such structures prove problematic for theoretical frameworks in which invariance are central to the formulation. Thus we can view higher structures as semantic properties of spacetime, rather than as singular places where invariances break down.

## Formal semantics in computer science

Formal semantics are given by equational specification, and grammatical typology, etc. The tools of formal semantics are not readily equipped to deal with the kind of invariances and transformations that characterise spacetime, so they are not directly applicable to discussing spacetime without an intermediate bridge. Neither are the formulations of spacetime readily equipped to deal with the broken symmetries that lead to semantics, hence the need for a formulation that bridges the gap between these two worlds. This gives semantic spacetime a role to play in formal methods of computer science too.

#### References

[1] M. Burgess, Semantic Spacetimes: Formalizing the semantics of space and time, for cognition and measurement (a route to knowledge representation)[2] M. Burgess. Spacetimes with semantics (i). http://arxiv.org/abs/1411.5563, (2014).

[3] M. Burgess. Spacetimes with semantics (ii). http://arxiv.org/abs/1505.01716, (2015).

[4] M. Burgess. Spacetimes with semantics (iii). http://arxiv.org/abs/1608.02193, (2016).

[5] M. Burgess. On the scaling of functional spaces, from smart cities to cloud computing http://arxiv.org/abs/1602.06091 (2016)

[6] R. Milner. The space and motion of communicating agents, Cambridge, (2009)

[7] L. Lamport. Time, Clocks, and the Ordering of Events in a Distributed System. Communications of the ACM, 21, p. 558 (1978)

[8] F.J. Dyson. The Radiation Theories of Tomonaga, Schwinger, and Feynman. Physical Review 75, p486. (1949)

[9] J Schwinger. Particles, Sources, and Fields, Addison Wesley. (1989)