Arity: The number of elements in something, typically a hyperedge. "Binary" is "arity 2", "Ternary" is "arity 3", etc. "Binary hyperedges" are just ordinary edges, of the kind that occur in an ordinary graph.
Black Hole: In typical physics, a region of spacetime into which light can go, but not emerge. In our models black holes can be formulated as features of the causal graph. They have generalizations in branchial and rulial space, where they relate to qubits and pockets of computational reducibility respectively.
Branch Pair: A pair of edges from a single node in the multiway graph. The node represents a state of the system; the edges represent possible two updating events for the state. Branch pairs are what lead to multiple histories, and are related to quantum indeterminacy. In causal invariant systems, branched paths always eventually merge. Branch pairs are called critical pairs in the theory of term rewriting.
Branchial Graph: The graph formed by joining states from the same branch pairs that exist in a particular slice of the multiway graph. The branchial graph can be interpreted as a map of entanglements between quantum states. Branchial graphs are typically rendered in a pinkish color.
Branchial Space: The space of quantum states, in which entangled states are nearby. Branchial space has the same relation to the multiway graph that physical space has to the causal graph, defining extent in each slice in a foliation of the multiway system. Branchial space is a generic term for what exists in branchlike hypersurfaces.
Branchlike Hypersurface: A slice through branchtime that involves only branchlike separated points, i.e. points that are not connected in the multiway graph. A branchial graph exists in a particular branchlike hypersurface. A sequence of branchlike hypersurfaces defines a foliation of branchtime. The evolution of a quantum system can be thought of as corresponding to a succession of branchlike hypersurfaces.
Branchlike Separated: States obtained by following different branches in the multiway evolution graph, obtained as outputs from overlapping (i.e. not spacelike separated) updating events. Branchlike-separated states correspond to quantum states that can exist in superpositions. In a distributed computing interpretation, they correspond to different branches in a race condition.
Branchtime: The analog of spacetime, but for branchial space, and time. Branchtime is related to the continuum limit of the multiway graph.
Causal Graph: The graph of causal relationships between updating events. The nodes are updating events; the edges define causal relationships. An event "depends on" a previous event if its "input" contains output from the previous event. The flux of edges in the causal graph is related to physical energy and momentum.
Causal Invariance: A property of multiway graphs whereby all possible paths yield the isomorphic causal graphs. When causal invariance exists, every branch in the multiway system must eventually merge. Causal invariance is a core property associated with relativistic invariance, quantum objectivity, etc. In the theory of term rewriting, a closely related property is confluence. In a terminating system, causal invariance implies that whatever path is taken, the "answer" will always be the same.
Cellular Automaton: A simple computational system which consists of a discrete array of cells, each with a value that is updated in a series of steps according to a fixed rule that depends on values of neighboring cells in the grid. Even when cellular automata have very simple rules (e.g. one dimension, two possible values, nearest neighbor rules) their behavior can exhibit great complexity, including randomness in rule 30 and computation universality in rule 110. Cellular automata are good models for many kinds of systems, but their rigid treatment of space and time makes them unsuitable as models of fundamental physics.
Completion: An additional rule introduced into a multiway system that makes it closer to being causal invariant. Some multiway systems can be made causal invariant by adding a finite number of completions. Completions may be implicitly done by quantum observers in trying form a consistent interpretation of the world. Completions are commonly used in theorem-proving systems for forcing confluence.
Computational Equivalence: The equivalence of computational capabilities implied for a wide range of systems by the Principle of Computational Equivalence. Computation universality is a special case.
Computational Irreducibility: The phenomenon whereby the behavior of a system cannot be determined much more efficiently than explicit simulation of each step in the evolution of the system. Computational irreducibility follows from the fact that the Principle of Computational Equivalence implies that observers can be no more computationally sophisticated than the systems they are observing.
Computational Reducibility: Ability to predict the behavior of a system more efficiently than by explicitly running the computation associated with the evolution of the system. Traditional methods of analysis rely on computational reducibility.
Computational Universe: The universe of computational systems such as a simple programs. A New Kind of Science is an exploration of the computational universe and implications of the phenomena observed in it, such as the Principle of Computational Equivalence.
Confluence: A simplified form of causal invariance considered in term rewriting systems such as ones that reach fixed points.
Continuum Limit: The limit in which discrete elements are taken to be sufficiently small or nearby that variations can be considered continuous. As an example, after arbitrarily many steps the hyperedges in a spatial hypergraph that is viewed as having a fixed overall scale can be small enough that the hypergraph can potentially be viewed as having reached a continuum limit. The continuum may be something like a manifold, or it be much wilder, or may not exist. When a continuum limit is reached, methods from calculus and continuous mathematics can potentially be used. The existence of a continuum limit is potentially related to intrinsic randomness generation through computational irreducibility.
Critical Pairs: See Branch Pair.
Dimension: An attribute of a space which roughly defines the number of degrees of freedom for motion. For ordinary Euclidean space dimension must be an integer. For a graph or hypergraph dimension can be defined in terms of the growth rate of the volume of geodesic balls, and need not be an integer.
Edge: The connection between nodes in a graph. Sometimes "edge" is used as an alternative to "hyperedge" for a hypergraph.
Element: An individual object appearing in relations that define hyperedges in the spatial hypergraph. An element can be thought of as an "atom of space".
Entanglement: Quantum mechanical correlation of subsystems. In our models, entanglement is associated with different states in the multiway graph having common ancestors. The branchial graph gives a map of entanglements between quantum states.
Entanglement Cone: The analog of a light cone in multiway evolution. The entanglement cone includes quantum states that can be reached by entanglements after a certain time. The growth rate of the entanglement cone is determined by the maximum entanglement speed ζ.
Evaluation Front: The latest slice in a foliation that defines the order of evaluation for subexpression of a given expression, or for recursive evaluation of expressions.
Event: See Updating Event.
Event Horizon: A boundary in spacetime that causal effects do not cross. Across a cosmic event horizon neither side can affect the other. Across a black event horizon, events outside the black hole can affect those inside, but not the other way around. Generalizations of event horizons occur in branchial space and rulial space. Knowing whether an event horizon exists ultimately depends on tracing the behavior of a system to an infinite time in the future.
Event Selection Function: Criterion by which the sequence of updating events to perform are chosen. Different event selection functions effectively correspond to different foliations of the multiway graph.
Foliation: A way of dividing the time evolution of a system into successive slices in which events can be considered to be simultaneous. Events in successive slices of a foliation can be joined by edges in a causal graph, while ones in a given slice cannot. Foliations of a spacetime causal graph define successive configurations of the spatial hypergraph. Those of the multiway graph define successive configurations of the branchial graph. Foliations are associated with reference frames used by observers.
Function Repository: The Wolfram Function Repository is a public collection of contributed Wolfram Language functions, including many functions constructed for or of use to the Wolfram Physics Project.
Gauge Invariance: Invariance under an internal symmetry group. Local gauge invariance allows different transformations at different points in space. The Standard Model of particle physics is based on SU(3)*SU(2)*U(1) transformations. In Wolfram models, gauge transformations presumably correspond to local changes in updating order, so gauge invariance is potentially related to causal invariance.
Geodesic: The shortest path between two points. In flat space this is a straight line. A geodesic on a graph or hypergraph is the sequence of edges that gives a shortest path between two nodes. In general relativity, the paths of particles acted on only by gravity are geodesics in curved space.
Geodesic Ball: A region consisting of points that can be reached from a given point by following geodesics of no more than a certain length. In flat d-dimensional space, a geodesic ball is a d-dimensional sphere. On a graph or hypergraph, a geodesic ball is the set of nodes within a certain graph distance of a given node. If the graph or hypergraph approximates d-dimensional space, the leading term in the growth rate of geodesic balls with radius is rd.
Geodesic Bundle: A collection of nearby geodesics. Einstein's equations imply that in the vacuum the cross-sectional area of a geodesic bundle will not change. In Wolfram models, geodesic bundles in the multiway graph correspond to quantum-mechanical wave packets.
Graph Distance: The minimum number of edges that must be traversed to get from one node in a graph to another.
Hyperedge: The connection between points in a hypergraph. In Wolfram models, hyperedges define the connectivity of space. In Wolfram models, hyperedges are also sometimes thought of as defining relations between elements.
Hypergraph: The basic structure used to represent space in Wolfram models. In a graph, edges connect pairs of points. In a hypergraph, hyperedges connect any number of points. Wolfram models typically use a specific type of ordered hypergraph.
Hypersurface: A generalization of the notion of a 2D surface in 3D to (d–1)-dimensional surfaces in d-dimensional space. In standard (3+1)-dimensional spacetime, spacelike hypersurfaces are 3-dimensional "slices".
"The Language": The Wolfram Language. The full-scale computational language whose first version was released in 1988, and used as the basis for the Wolfram Physics Project and a great many other things.
Light Cone: The region in spacetime which can be reached by light signals initiated at a particular position and time. In flat space the radius of the light cone at time t is c t, where c is the speed of light. The interior (and boundary) of the light cone (often all referred to collectively as "the light cone") contains all spacetime points that can be causally affected by the event at the original point. In Wolfram models, all geodesics from a point in the causal graph lie within the light cone.
Manifold: A space which on a sufficiently small scale can be approximated by Euclidean space. The spatial hypergraph and other structures in Wolfram models may be manifold in continuum limits.
Multicomputation: A generalization of the concept of computation in which there are multiple interwoven threads of evaluation. A typical example of a multicomputational system is a multiway system. In multicomputation there are many threads of time, and to determine a state of the system requires a reference frame imposed by an observer. Multicomputation is the foundation for a broad new modeling paradigm.
Multispace: A multiway generalization of ordinary space obtained from a slice of a foliation of the multiway causal graph. Multispace represents both spatial and branchial extent. If the multiway system has only a single branch, then multispace is like ordinary space. However, if the multiway system has multiple branches, these yield "stacks" in multispace, in which a single region of space can have multiple different forms.
Multiway Causal Graph: A graph giving the causal relationships between events that generate states in a multiway system from other states. Edges in the multiway graph can connect both spacelike and branchlike separated events. In a causal invariant system, the multiway causal graph factors into many isomorphic spacetime causal graphs.
Multiway Graph: A graph in which states are connected to their possible immediate successors obtained by applying rules. Branchings in the multiway graph lead to branch pairs. Causal invariance implies that all branches will eventually merge. In Wolfram models, the multiway graph represents the evolution of quantum states. The limiting behavior of geodesics in the multiway graph is governed by the path integral.
Multiway System: A system in which rules define multiple possible successors for states. The behavior of a multiway system is represented by a multiway graph. Multiway systems were discussed in A New Kind of Science, though analogs of them have arisen many times, under names such as term rewriting systems, semi-Thue systems, Bohm trees and poscau sets.
Node: A single point or element in a graph or hypergraph. Nodes are sometimes also called vertices. In the spatial hypergraph, the nodes are points in space.
Normal Form: A fixed point or final result obtained from repeatedly applying rules. Often rules will not terminate, and no normal form will be obtained.
Oligons: Hypothetical particles associated with localized structures involves a small number of edges in the spatial hypergraph. Oligons potentially have masses such as 10-20 electron masses. They may be associated with dark matter.
Observer: An idealization of an entity which interprets the universe based on measurements. Observers embedded within a system and following its rules of evolution are normally sensitive only to causal relationships between events in the system. In a typical case, a convenient idealization of an observer can be given in terms of equivalence classes of states within a foliation. In spacetime, observers define which sets of events can be considered simultaneous. In branchtime, observers define which sets of states can be considered to exist as a superposition.
Partial Ordering: An ordering relation in which pairs of elements can be strictly ordered, but need not be. Events in spacetime are an example (timelike-separated events are strictly ordered, but spacelike-separated ones are not). The causal graph represents the partial order relations between such events.
Poset: A set of elements for which a partial ordering is defined. Events in spacetime are an example. The causal graph represents the partial order relations between these events.
Quantum Observation Frame (or QoF): A foliation of the multiway graph that represents the way an observer interprets the evolution of the space of quantum states.
Quantum Amplitude: A complex number associated with a quantum state, or with a transition between states. In Wolfram models, the magnitude of the quantum amplitude is associated with path weights for geodesics in the multiway graph, while the phase is associated with position in branchial space. According to the Born rule, the modulus squared of the quantum amplitude is associated with the probability of being in a certain state, or making a certain transition between states.
Reference Frame: A way of interpreting (and potentially assigning coordinates to) spacetime and related constructs. A reference frame typically corresponds to a choice of foliation made by an observer.
Relation: A collection of elements that define a hyperedge in the spatial hypergraph.
Relativistic Invariance: The phenomenon whereby measurable features of physics are unaffected by the choice of reference frame. In Wolfram models, relativistic invariance is a consequence of causal invariance.
Rule Signature: See Signature.
Rulial Space: The space defined by allowing all possible rules of a given class to be followed between states of a system. Different foliations of rulial space can be thought of as corresponding to different languages for describing behavior. The Principle of Computational Equivalence implies a fixed maximum speed ρ in rulial space. Rulial space for Turing machines is obtained by allowing all possible non-deterministic transitions.
Ruliology: The pure basic science of what simple rules do.
Self-Loop: A connection of an element or node to itself in a graph or hypergraph.
Short Code: A hash code for a rule, as used in the Registry of Notable Universes.
Signature (or Rule Signature): A characterization of the structure of a rule, based on the size of its input and output.
Spacelike Hypersurface: A slice through spacetime that involves only spacelike separated points, i.e. points that are not connected in the causal graph. A spacelike hypersurface is a surface that can consistently be assumed to contain points in spacetime that are "simultaneous". A sequence of spacelike hypersurfaces defines a foliation of spacetime. The evolution of a universe can be thought of as corresponding to a succession of spacelike hypersurfaces.
Spacelike Separated: Events in spacetime that cannot causally affect each other, because they lie outside each other's light cones. Spacelike separated events can be considered to occur simultaneously in time, and so can appear in a single slice in a foliation.
Spatial Hypergraph: The hypergraph that defines a state of the universe in Wolfram models. The nodes in the hypergraph can be viewed as "atoms of space", joined by hyperedges whose patterns of connections define both the structure of space and of matter and everything "in space". The spatial hypergraph for our universe might currently contain 10400 nodes. It is continually being changed by updating events.
State: The configuration of a system. In Wolfram models, multiple states can co-exist in a multiway system, corresponding to superpositions in quantum mechanics.
String Substitution System: A computational system in which strings of characters such as "ABBA" are repeatedly rewritten by rules for transformations of substrings such as "BA"→"AAB". String substitution systems are a convenient idealization of Wolfram models for studying such things as the construction and properties of multiway graphs.
Superposition: A phenomenon in quantum mechanics whereby the state of a quantum system can consist of a combination of multiple basic quantum states. In Wolfram models, superpositions are associated with combinations of states that lie on separate branches in the multiway graph, but for which the foliation of the multiway graph allows them to exist simultaneously. States that can exist together in a superposition are considered to be entanglement, and the map of such states corresponds to the branchial graph.
Tensor: A generalized matrix that consists of an array of values of any given rank. Notable tensors include the Riemann curvature tensor Rαβγδ and the energy-momentum tensor Tμν.
Termination: The phenomenon whereby no rules apply to a particular state of the system, so the evolution of the system can be said to have terminated. Termination can be thought of as producing a canonical or normal form that represents the "output" from a system. Termination in a causal graph is analogous to a spacelike singularity.
Timelike Hypersurface: A slice through spacetime that involves only timelike separated points, i.e. points that are connected in the causal graph. Timelike hypersurfaces are orthogonal to spacelike hypersurfaces. In Wolfram models, the flux of causal edges through timelike hypersurfaces is associated with momentum.
Timelike Vector: A vector in spacetime or its generalizations that joins points that can be related through the progression of time. Events that are causally related are connected by timelike vectors.
Turing Machine: A simple computational system often used as a minimal idealization of computational processes. A Turing machine has a "head" that reads and writes values from a "tape". The Wolfram 2,3 Turing machine is the simplest computation universal Turing machine.
Undecidability: The phenomenon whereby something may not guarantee to be decided by any finite computation. Undecidability can be viewed as being a consequence of computational irreducibility in which one is asking for infinite-time results which can only by obtained by tracing a potentially infinite number of steps in the evolution of a system.
Updating Event: A single application of a rule, typically on the spatial hypergraph. The same term is also used for applications of rules in substitution systems. Each updating event corresponds to a node in the causal graph, and an edge in the multiway graph.