Multicomputation

Documentation

Multi Object

Multi is a lazy non-deterministic container with tensor product semantics. It represents expressions with multiple possible values and supports multiway rewriting systems.

Basic Usage

(* Multiple alternatives *)
Multi[{1, 2, 3}]                        (* 3 alternatives *)

(* Tensor product - lazy combination *)
Multi[{1, 2}] + Multi[{10, 20}]         (* 4 combinations: {11, 21, 12, 22} *)

(* Rule-based binding *)
Multi[5, x_ :> {x, x+1, x^2}]           (* 5 → {5, 6, 25} *)

(* Automatic chaining via UpValues *)
f[Multi[{a, b}]]                        (* → Multi with {f[a], f[b]} *)

Constructors

Syntax	Description
Multi[{a, b, c}]	Alternatives from list
Multi[expr, rule]	Apply replacement rule to expression
Multi[expr, {rules...}]	Apply multiple rules
Multi[assoc]	Create from association (internal)

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Properties Reference

Access properties via multi["PropertyName"] or multi["PropertyName", args...].

Core Data Properties

Property	Description
"Expression"	The underlying expression with placeholders
"HoldExpression"	Expression wrapped in HoldComplete
"Data"	Full internal association
"ValidQ"	Whether the Multi is well-formed
"Values"	Association mapping positions to alternative values
"Matches"	Association mapping positions to rule match results
"Rules"	List of rewrite rules used
"Keys"	Combined keys from Values and Matches
"Positions"	Position specifications for all alternatives
"Placeholders"	Association of placeholder symbols to values
"Bindings"	Pattern variable bindings from rule matches
"Size"	Number of alternative positions
"ValueCount"	Product of alternative counts (total combinations)
"MatchCount"	Total number of rule matches

Values & Tuples

Property	Description
"ListValues"	List of possible values at each position
"HoldListValues"	Same, wrapped in HoldForm
"Tuples"	All combinations (Cartesian product) of values
"Tuples", n	First n tuples
"HoldTuples"	Tuples with held values
"MatchBindings"	Pattern bindings for each match

Evaluation Properties

These evaluate the expression by substituting alternatives:

Property	Description
"EvaluateList"	List of all evaluated alternatives
"EvaluateList", n	First n alternatives
"HoldEvaluateList"	Evaluated alternatives, held
"EvaluateListOnce"	One step of evaluation per position
"HoldEvaluateListOnce"	One step, held

Multi-Returning Evaluation

These return new Multi objects:

Property	Args	Description
"Evaluate"	n	Apply rules n times, return Multi
"EvaluateOnce"	n, m, k	One-step eval, n iterations, m positions, k tuples
"HoldEvaluate"		Evaluate with held results
"HoldEvaluateOnce"		One step, held
"MultiEvaluate"	n	Return list of Multi (one per branch)
"HoldMultiEvaluate"	n	Same, held
"MultiEvaluateList"		Expressions from MultiEvaluate

List/Multi Hybrid Evaluation

Property	Description
"MultiList"	Split into list of Multis (one per list element)
"ListEvaluate"	Evaluate each list element's Multi separately
"HoldListEvaluate"	Same, held
"MultiListEvaluate"	Flatten ListEvaluate back to single Multi
"HoldMultiListEvaluate"	Same, held
"ListEvaluateWithKeys"	ListEvaluate with position keys

Transformation Properties

Property	Args	Description
"Apply"	f, pos	Apply function f at position(s)
"HoldApply"	f, pos	Apply without evaluating argument
"ListApply"	f	Apply to expression as list
"DeleteDuplicates"		Remove duplicate alternatives
"ReplacePart"	rules	Replace parts of expression
"RematchRules"		Re-apply stored rules
"ModifyData"	f	Transform internal data with function

Graph Properties

Generate visualizations of multiway evolution:

Property	Args	Description
"Graph"	n, opts	Evolution graph for n steps
"HoldGraph"	n, opts	Evolution graph with held expressions
"CausalGraph"	n, opts	Causal relationships between events
"BranchialGraph"	n, opts	Connections between concurrent states
"TokenEventGraph"	n, opts	Token-event relationships
"EvolutionCausalGraph"	n, opts	Combined evolution + causal
"CausalBranchialGraph"	n, opts	Combined causal + branchial
"CausalStatesGraph"	n, opts	Causal graph of states

Event & Edge Properties

Property	Description
"Events"	Tagged events from evaluation
"HoldEvents"	Events with held expressions
"Edges"	Directed edges for graph construction
"HoldEdges"	Edges with held expressions
"Branches"	Output states from each event
"HoldBranches"	Branches with held expressions
"BranchPairs"	Pairs of concurrent branches
"HoldBranchPairs"	Branch pairs, held
"Foliations"	Layer-by-layer evolution with counts
"HoldFoliations"	Foliations with held expressions

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Examples

Tensor Product (Lazy Cartesian Product)

m = Multi[{1, 2, 3}] + Multi[{10, 20}];
m["Expression"]      (* $MultiPlaceholder$1 + $MultiPlaceholder$2 *)
m["Tuples"]          (* {{1,10}, {1,20}, {2,10}, {2,20}, {3,10}, {3,20}} *)
m["EvaluateList"]    (* {11, 21, 12, 22, 13, 23} *)

Rule-Based Binding

(* Each x produces variable alternatives based on value *)
m = Multi[{1, 2, 3}, x_ :> Range[x]];
m["EvaluateList"]    (* 1×2×3 = 6 combinations *)

(* Conditional branching *)
m = Multi[Range[5], {
  x_ /; EvenQ[x] :> {x, x/2},
  x_ /; OddQ[x] :> {x, 3x + 1}
}];
m["EvaluateList"]    (* 2^5 = 32 combinations *)

Multiway Evolution Graph

m = Multi[1, x_ :> {x + 1, x * 2}];
m["Graph", 4]        (* Visualize 4 steps of evolution *)

Subset Sum Algorithm

(* Each element: include (element) or exclude (0) *)
subsetSums[nums_] := Total[Multi[{0, #}] & /@ nums]

ss = subsetSums[{3, 7, 1, 8, 4}];
Pick[ss["Tuples"], ss["EvaluateList"], 12]  (* Subsets summing to 12 *)

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Automatic Chaining

Multi has UpValues that propagate through expressions:

m = Multi[{1, 2, 3}];
m + 10                 (* → Multi, evaluates to {11, 12, 13} *)
Sin[m]                 (* → Multi, evaluates to Sin values *)
f[g[m]]                (* → Multi, expression f[g[$placeholder]] *)

(* Rule-based Multi chains too *)
m2 = Multi[5, x_ :> {x, x+1}] * 10;
m2["EvaluateList"]     (* {50, 60} *)

No explicit ["EvaluateList"] needed for intermediate steps—just use Multi in expressions naturally.