Operator axiom systems (6) 

MultiwayOperatorSystem supports all forms of operator axiom systems, including standard Boolean algebra:

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Wolfram's minimal axiom system for Boolean algebra:

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Ring theory:

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Modified Presburger arithmetic:

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Tarski's axioms of Euclidean geometry:

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Note that, in the above, one of the axioms (the axiom of continuity) is actually an axiom schema, which here we represent using a second-order axiom involving operator patterns phi_ and psi_:

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Axiom schemas and higher-order logics (2) 

Axiom schemas or higher-order axioms can be represented using operator patterns, such as phi_ in the following specification of the axiom (schema) of induction in Peano arithmetic:

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For instance, use this axiom to prove by induction that prop[m] is true for all m:

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Rules and initial conditions (2) 

MultiwayOperatorSystem accepts both individual operator axioms and lists of operator axioms:

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MultiwayOperatorSystem accepts both individual initial conditions and lists of initial conditions:

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Event selection functions (2) 

Apply only the first possible event at each step:

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Apply the first and last possible events at each step:

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State node rendering (3) 

By default, states are labeled by their contents:

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Use no labeling for states:

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"StatesGraphStructure" yields the same result:

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Use raw state names as node labels:

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Use a named shape as each state label:

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Event node rendering (2) 

By default, both states and events are labeled by their contents:

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Use no labels for states or events:

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"EvolutionEventsGraphStructure" yields an equivalent result:

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Use raw event expressions as their labels:

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Initialization events (2) 

By default, "AllEventsList" does not include initialization events:

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The option "IncludeInitializationEvents" allows one to override this default:

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Initialization events have special default rendering:

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Graph layout options (2) 

Place arrows in the middle of edges:

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Force the initial state node to be at the top:

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Step numbers and state IDs (4) 

By default, equivalent states are merged across all time steps:

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Merging of equivalent states across different time steps can be prevented by including step numbers:

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List the states:

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Merging of equivalent states at the same time step can be prevented by also including state IDs:

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List the states:

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Step numbers and IDs also apply to events:

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See the events:

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Event instances (2) 

By default, multiple instances of equivalent updating events are merged in the states graph:

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Merging of equivalent events can be prevented by including event instances:

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State (path) weights (2) 

Vertices of a states graph can be weighted by their relative rate of occurrence at each time step:

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Vertices can also be weighted by the number of distinct evolution paths that lead to them:

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MaxItems (2) 

By default, "CausalGraphInstances" returns all possible causal graphs:

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The number of causal graphs returned can be limited using MaxItems:

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Predecessors and resolvents (2) 

By default, "BranchPairsList" returns only a list of branch pairs:

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Common predecessor states can be shown using "GivePredecessors":

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Similarly, "BranchPairResolutionsList" by default lists only resolved and unresolved branch pairs:

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Common resolvents of resolved branch pairs can be shown using "GiveResolvents":

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Show both common predecessors and common resolvents, where appropriate:

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Full branchial space (2) 

By default, non-branch pair states are not shown as part of the branchial graph:

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They can be shown by using "IncludeFullBranchialSpace":

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