Define the Hypergraph Class

Think (DeepSeek): [Thought for 223 seconds]

Okay, so I need to figure out how to implement the Wolfram Physics Project's approach using Pharo Smalltalk and the Glamorous Toolkit (GT). Let me start by understanding the problem. The core idea is to Model the Universe as a Hypergraph where abstract elements are connected through relations. These hypergraphs evolve over time based on certain rules, and from their structure, we might derive physical laws.

First, I need to represent a hypergraph in Pharo. The user mentioned that in simple cases, it's a graph, but hypergraphs can have edges connecting more than two nodes. So, a hypergraph is a collection of hyperedges, each being a set of nodes. For example, {{1,2}, {2,3}, {3,4}, {2,4}} is a hypergraph with four hyperedges. Each hyperedge is an unordered collection of nodes, but the order within each hyperedge matters in the sense that the relation is specified between those elements. Wait, the user said the order within each relation does matter. Hmm, so maybe the hyperedges are actually ordered tuples? Or are they sets where the order doesn't matter? Wait, the example given uses curly braces, which in mathematics denote sets (unordered), but the user says "the order within each relation does matter". That seems contradictory. Wait, maybe the hyperedges are ordered? Let me check the original text again. The user wrote: "All we're specifying here are the relations between elements (like {2,3}). The order in which we state the relations doesn't matter (although the order within each relation does matter)." Oh, so the order in which the relations are stated doesn't matter, meaning that the collection of hyperedges is a set (unordered). But within each hyperedge, the order of elements does matter. Wait, but the example hyperedge is {2,3}, which is a set (unordered). But if the order within the relation matters, maybe it's actually an ordered tuple? So perhaps the hyperedges are ordered lists, not sets. That would make sense if, for example, the relation is directed. So maybe the hypergraph is a collection of ordered tuples, which in the example are written with curly braces but actually ordered. But in standard math notation, ordered tuples are with parentheses. Hmm. This is a bit confusing. Wait, the user says, "the order within each relation does matter." So for example, the relation {2,3} is different from {3,2} if the order matters. But in set notation, {2,3} and {3,2} are the same. So perhaps the hyperedges are actually ordered pairs (or tuples) but written with curly braces here. That might be a notation issue. So in code, perhaps we need to represent each hyperedge as an ordered collection (like an array or ordered collection) where the order of elements matters. So in the example, {2,3} is an ordered pair, and {3,2} would be a different hyperedge if order matters. That would align with the statement that the order within each relation matters. So in Pharo, I can model each hyperedge as an OrderedCollection or an Array. Then, the hypergraph is a set (or a collection where duplicates are not allowed) of these ordered hyperedges. But in Pharo, sets are unordered and do not allow duplicates. So if I have a hypergraph as a Set of Arrays, each array being an ordered hyperedge. That makes sense. So first step: create a Hypergraph class that contains a collection of hyperedges, each hyperedge being an ordered collection of nodes. But nodes themselves are abstract elements. So perhaps nodes can be represented as objects. But for simplicity, maybe just use integers as node identifiers. So the example given uses numbers 1,2,3,4 as elements. So perhaps the nodes are just integers, and hyperedges are ordered collections of integers. But in the Wolfram model, the actual labels (numbers) don't matter; it's the connectivity that's important. So the structure is defined up to isomorphism. But for implementation, we need concrete representations. So the Hypergraph class could have an instance variable 'hyperedges' which is a Set of Arrays. Each array is an ordered collection of node identifiers (integers). The nodes themselves can be inferred from the hyperedges. Now, the next part is evolving the hypergraph according to certain rules. The Wolfram Physics Project uses substitution rules that determine how the hypergraph changes over time. For example, a rule might specify that a particular pattern of hyperedges is replaced by another set of hyperedges. Implementing rules would involve pattern matching on the hypergraph. In Pharo, we can model rules as objects that have a left-hand side (pattern) and a right-hand side (replacement). Applying a rule would involve finding matches for the left-hand side in the hypergraph and replacing them with the right-hand side. But pattern matching in hypergraphs can be complex, especially since hyperedges can have arbitrary arity. Let's consider a simple rule. For example, a rule that takes a hyperedge {a, b} and replaces it with two hyperedges {a, c} and {c, b}, where c is a new node. This is similar to a graph rewriting rule where an edge is split into two with a new node in between. To implement this, we need to: 1. Find all occurrences of a hyperedge matching the left-hand side pattern. In this case, any hyperedge with two elements. 2. For each match, create a new node (c), then replace the original hyperedge with the two new hyperedges. But in the Wolfram model, the rules might be more complicated. The rules could be based on causal invariance, and the specific rules determine the possible universes (hypergraphs) that can emerge. Another example from the project is the use of multiway systems, where multiple possible updates are considered, leading to a branching structure of possible hypergraphs. However, implementing the full scope might be too ambitious. Let's start with a basic example: creating a hypergraph, applying a simple rule, and visualizing it. First, create a Hypergraph class. In Pharo, a class can be defined as: Object subclass: #Hypergraph instanceVariableNames: 'hyperedges' classVariableNames: '' package: 'WolframPhysics' Then, the initialize method: Hypergraph>>initialize hyperedges := Set new. Add methods to add hyperedges, remove them, etc. Hypergraph>>addHyperedge: anOrderedCollection hyperedges add: anOrderedCollection asArray. But wait, in Pharo, Sets use equality to check for duplicates. Arrays are equal if their contents are equal and in the same order. So if you add two arrays that are the same, they will be considered duplicates and only one will be kept. That's correct, since the user said "the order in which we state the relations doesn't matter" implying that the collection of hyperedges is a set. Now, to create the example hypergraph {{1,2}, {2,3}, {3,4}, {2,4}}: hypergraph := Hypergraph new. hypergraph addHyperedge: #(1 2). hypergraph addHyperedge: #(2 3). hypergraph addHyperedge: #(3 4). hypergraph addHyperedge: #(2 4). But in Pharo, #(1 2) is a literal array, which is ordered. So that's correct. Next, visualization. Using Glamorous Toolkit (GT), which has tools for visualizing graphs and structures. GT allows creating morphs that represent graphs. However, hypergraphs are more complex because hyperedges connect multiple nodes. Standard graph visualizations typically handle edges between two nodes, so hyperedges (connecting more than two) need a different representation. One approach is to represent each hyperedge as a node connected to all its elements. For example, a hyperedge {1,2,3} could be represented as a central node (the hyperedge) connected to 1, 2, 3. But this might complicate the visualization. Alternatively, use a different visual cue for hyperedges, like colored regions or shapes encompassing the connected nodes. But implementing this in GT might require custom morphs. Alternatively, for simplicity, represent the hypergraph as a bipartite graph where nodes are either elements or hyperedges, and connections are between elements and their hyperedges. This is a common way to represent hypergraphs as graphs. For example, each hyperedge becomes a node, and edges connect the hyperedge node to each element in the hyperedge. Then, using GT's graph visualization tools, we can display this bipartite graph. So, to visualize the hypergraph, we can transform it into a bipartite graph and then render that. But let's see. The original example hypergraph has hyperedges as {1,2}, {2,3}, etc. Transforming into a bipartite graph would create nodes for each hyperedge (maybe labeled e1, e2, etc.) and connect each hyperedge node to its elements. But the user might want to visualize the hypergraph directly. Alternatively, in simple cases where hyperedges are binary (edges), it's a regular graph. So for the example, it's a regular graph with edges 1-2, 2-3, 3-4, and 2-4. So maybe for visualization, when all hyperedges are binary, we can treat them as edges. For hyperedges with more elements, we need a different approach. But given that the user's example is a graph (all hyperedges are pairs), perhaps we can proceed by treating the hypergraph as a standard graph for visualization purposes. So, using GT's graph viewer, we can create nodes and edges. First, create a Roassal2 view (since GT uses Roassal for visualizations). For each node in the hypergraph (extracted from the hyperedges), create a node element. For each hyperedge (which is an ordered collection), create an edge from the first element to the second. Wait, but if hyperedges can have more than two elements, this approach won't work. However, in the example, they are pairs. So for the example hypergraph, which is a standard graph, the visualization would show nodes 1,2,3,4 with edges between them as specified. In code: hypergraph := Hypergraph new. hypergraph addHyperedge: #(1 2). hypergraph addHyperedge: #(2 3). hypergraph addHyperedge: #(3 4). hypergraph addHyperedge: #(2 4). nodes := hypergraph nodes. "This method would collect all unique elements from hyperedges" edges := hypergraph hyperedges. "Each edge is a pair" Then, using Roassal: view := RTView new. nodes := (hyperedges flatCollect: #yourself) asSet sorted. elements := RTEllipse new color: Color blue; size: 20; elementsOn: nodes. edges := hyperedges collect: [:edge | RTEdge from: (edge at: 1) to: (edge at: 2)]. view addAll: elements. view addAll: edges. RTGridLayout on: view. view open. But this is a simplistic approach. However, in the case where hyperedges have more than two elements, this method wouldn't work. So for general hypergraphs, we need a different visualization. Alternatively, for a hyperedge with three elements, say #(1 2 3), we could represent it as a triangle connecting 1-2-3-1. But that might not be accurate. Another approach is to create a central point connected to all elements, but that could get cluttered. Given the complexity, perhaps starting with the assumption that all hyperedges are binary (i.e., standard graph edges) for visualization, and later extend to hyperedges. But according to the user's example, they have a graph, so starting with that is okay. Now, the next step is to implement a rule that evolves the hypergraph. Let's take a simple rule: any edge {a, b} is replaced by two edges {a, c} and {c, b}, where c is a new node. This is akin to inserting a new node in the middle of each edge. Implementing this requires: 1. Iterating over all hyperedges in the current hypergraph. 2. For each hyperedge of size 2 (i.e., an edge), remove it and add two new edges with a new node. 3. Generating new unique node identifiers. But in the Wolfram model, the rules might be more complex, involving pattern matching and conditional rewriting. However, this simple rule serves as an example. In Pharo, the code could look like: applyRule: aRule to: hypergraph | newHyperedges newNodes | newHyperedges := Set new. newNodes := 0. hypergraph hyperedges do: [:edge | edge size = 2 ifTrue: [ newNodes := newNodes + 1. newC := hypergraph nextNodeId + newNodes. "Assuming hypergraph keeps track of node IDs" newHyperedges add: {edge first. newC}. newHyperedges add: {newC. edge second}. ] ifFalse: [ newHyperedges add: edge. ] ]. hypergraph hyperedges: newHyperedges. But this requires the Hypergraph to manage node IDs. Alternatively, track the maximum node ID in the hypergraph and increment from there. But in the Wolfram model, nodes are abstract and their labels don't matter, so perhaps we can generate new unique symbols each time. In code, using integers and keeping track of the highest used integer. So the Hypergraph class can have an instance variable 'maxNode' which is the highest node ID used. Then, when a new node is needed, increment maxNode and use that. So adding to the Hypergraph class: Hypergraph>>initialize hyperedges := Set new. maxNode := 0. Hypergraph>>addHyperedge: anArray self checkNodes: anArray. hyperedges add: anArray asArray. Hypergraph>>checkNodes: anArray anArray do: [:node | (node isInteger) ifTrue: [ maxNode := maxNode max: node. ] ifFalse: [ self error: 'Nodes must be integers' ] ] Hypergraph>>nextNode maxNode := maxNode + 1. ^maxNode Then, the rule application would use nextNode to generate new nodes. So, modifying the applyRule method: Hypergraph>>applyEdgeSplittingRule | newHyperedges | newHyperedges := Set new. self hyperedges do: [:edge | edge size = 2 ifTrue: [ newC := self nextNode. newHyperedges add: {edge first. newC}. newHyperedges add: {newC. edge second}. ] ifFalse: [ newHyperedges add: edge. ] ]. self hyperedges: newHyperedges. Wait, but in Pharo, Sets are immutable when you add elements. Wait, no: in Pharo, you can modify a Set by adding or removing elements. But in this code, we're creating a new Set 'newHyperedges' and then replacing the hypergraph's hyperedges with this new set. That's safe. So, after applying this rule once to the initial hypergraph, each edge is replaced by two edges with a new node in between. For example, applying the rule to the initial hypergraph: Original edges: {1,2}, {2,3}, {3,4}, {2,4} After applying the rule: Each edge is split. Let's see: Edge {1,2} becomes {1,5} and {5,2} (assuming 5 is the next node). Edge {2,3} becomes {2,6} and {6,3}. Edge {3,4} becomes {3,7} and {7,4}. Edge {2,4} becomes {2,8} and {8,4}. So the new hyperedges would be {1,5}, {5,2}, {2,6}, {6,3}, {3,7}, {7,4}, {2,8}, {8,4}. Visualizing this new hypergraph would show nodes 1,2,3,4,5,6,7,8 with edges as above. But this is a simple example. The Wolfram Physics Project likely uses more complex rules, possibly matching larger patterns or involving different replacement strategies. Another aspect is multiway systems, where multiple rule applications are possible, leading to branching structures. Implementing this would require managing multiple hypergraphs simultaneously, which complicates things. However, for a basic example, applying a single rule in a deterministic way is manageable. Now, integrating this with Glamorous Toolkit for visualization. GT provides a live coding environment, so we can create a morph that displays the hypergraph and updates as rules are applied. To create a GT visualization, we can use the Grapher tool in GT. Alternatively, use the interactive morphs to build a custom view. For example, in GT, you can create a presenter for the hypergraph: hypergraph := Hypergraph new. hypergraph addHyperedge: #(1 2); addHyperedge: #(2 3); addHyperedge: #(3 4); addHyperedge: #(2 4). view := GTView new. graphPresenter := view add: hypergraph asMorph. view open. But we need to define how to convert the hypergraph into a morph. This would involve creating a custom morph that uses Roassal2 to render the graph. Alternatively, use the GTGraphView from GT's toolkit: nodes := hypergraph nodes. "Implement a method to collect all nodes" edges := hypergraph hyperedges. "Assuming hyperedges are pairs" graph := GTGraph new. nodes do: [:each | graph addNode: each ]. edges do: [:each | graph addEdgeFrom: each first to: each second ]. graphView := graph visualize. graphView open. But this requires that the hypergraph can provide nodes and edges in a way GT understands. So, the Hypergraph class should have methods to extract nodes and edges. Implementing: Hypergraph>>nodes ^ self hyperedges flatCollect: #yourself asSet asSortedCollection Hypergraph>>edges ^ self hyperedges select: [:each | each size = 2 ] Then, for visualization: hypergraph := Hypergraph new. hypergraph addHyperedge: #(1 2); addHyperedge: #(2 3); addHyperedge: #(3 4); addHyperedge: #(2 4). graph := GTGraph new. hypergraph nodes do: [:node | graph addNode: node asString ]. hypergraph edges do: [:edge | graph addEdgeFrom: edge first asString to: edge second asString ]. graphView := graph visualize. graphView open. This would display a graph with nodes 1,2,3,4 and edges between them. When the rule is applied, the hypergraph changes, and the visualization can be updated. However, in GT, updating the view automatically requires observable models. So perhaps making the Hypergraph an observable object that notifies the view when it changes. Alternatively, after modifying the hypergraph, regenerate the view. But for simplicity, after applying the rule, we can re-render the graph. So putting it all together: Create a Hypergraph class with methods to add hyperedges, apply rules, and extract nodes and edges. Then, use GT to visualize the graph. Apply the rule and update the visualization. Another aspect is handling hyperedges with more than two elements. For example, a hyperedge {1,2,3}. How to visualize this? One approach is to represent it as a hyperedge node connected to all its elements. For example: For hyperedge {1,2,3}, create a node 'e1' and edges from e1 to 1, e1 to 2, e1 to 3. This converts the hypergraph into a bipartite graph where hyperedges are nodes connected to their elements. So in code, when extracting edges for visualization: For each hyperedge in hyperedges: If the hyperedge has size > 2, create a hyperedge node (like 'e1', 'e2', etc.) and edges from the hyperedge node to each element in the hyperedge. But this complicates the visualization, but it's necessary for accurate representation. So modifying the Hypergraph>>edges method is not sufficient. Instead, need to process all hyperedges, creating virtual nodes for hyperedges and connecting them. Alternatively, use a different visualization strategy for hyperedges. However, given time constraints, maybe start with binary edges and handle hyperedges as a future extension. In conclusion, the steps to implement a basic version are: 1. Define a Hypergraph class in Pharo that manages hyperedges as ordered collections (arrays) and node IDs. 2. Implement a rule application method that modifies the hypergraph (e.g., splitting edges by adding new nodes). 3. Use Glamorous Toolkit to visualize the hypergraph as a standard graph (for binary edges) and update the visualization after rule application. 4. Test with the example hypergraph and observe the evolution after applying rules. This is a simplified approach, but it captures the essence of the Wolfram Physics Project's starting point: simple hypergraphs evolving via rules, visualized in a live environment like GT.