Automated Conjecturing for Proof Discovery
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1 Automated Conjecturing for Proof Discovery Craig Larson (joint work with Nico Van Cleemput) Virginia Commonwealth University Ghent University CombinaTexas Texas A&M University 8 May 2016
2 Kiran Chilakamarri On Conjectures
3 Goal To tell you about a new idea for using our conjecture-generating program:
4 Goal To tell you about a new idea for using our conjecture-generating program: generating sketches of proofs (or proof ideas);
5 Goal To tell you about a new idea for using our conjecture-generating program: generating sketches of proofs (or proof ideas); and a new proof of the Friendship Theorem.
6 Our Program Black box.
7 Our Program Black box. Main heuristic idea from Fajtlowicz s Graffiti.
8 Our Program Black box. Main heuristic idea from Fajtlowicz s Graffiti. Ingredients: Objects, Invariants, Properties, Choice of Invariant or property of interest, choice of upper or lower bounds.
9 The Program Open-source
10 The Program Open-source Written for Sage
11 The Program Open-source Written for Sage Python
12 The Program Open-source Written for Sage Python Lots of standard mathematical packages: GAP, R, GLPK, CVXOPT, NumPy, SciPy, L A TEX, matlabplot
13 The Program Open-source Written for Sage Python Lots of standard mathematical packages: GAP, R, GLPK, CVXOPT, NumPy, SciPy, L A TEX, matlabplot Lots of Graph Theory: graphs, invariants, properties, constructors,...
14 The Program Open-source Written for Sage Python Lots of standard mathematical packages: GAP, R, GLPK, CVXOPT, NumPy, SciPy, L A TEX, matlabplot Lots of Graph Theory: graphs, invariants, properties, constructors,... Growing
15 The Program
16 Properties Example: pairs have unique common neighbor: Every pair of vertices has exactly one common neighbor.
17 Properties Example: pairs have unique common neighbor: Every pair of vertices has exactly one common neighbor. Figure: A flower F 4 with four petals.
18 Sufficient Condition Conjectures 164 graphs in the main database 87 properties 5 propositional operators: and, or, implies, not, xor
19 Sufficient Condition Conjectures 164 graphs in the main database 87 properties 5 propositional operators: and, or, implies, not, xor
20 What you get
21 What you get Conjectures that are true for all input objects.
22 What you get Conjectures that are true for all input objects. Conjectures that say something not implied by any previously output conjecture.
23 Theory Idea: You want conjectures that are an improvement on existing theory.
24 Theory Idea: You want conjectures that are an improvement on existing theory. property = pairs_have_unique_common_neighbor theory = [is_k3] propertybasedconjecture(graph_objects, properties, property, theory = theory, sufficient = True, precomputed = precomputed)
25 Theory Idea: You want conjectures that are an improvement on existing theory. property = pairs_have_unique_common_neighbor theory = [is_k3] propertybasedconjecture(graph_objects, properties, property, theory = theory, sufficient = True, precomputed = precomputed) Conjecture: (((is_eulerian)&(is_planar))&(is_gallai_tree))-> (pairs_have_unique_common_neighbor)
26 Proof Sketch Generating Idea Prove: P = Q. Run necessary condition conjectures for P, using Q as the theory. The generated conjectures must be better than Q for at least one graph conjectures. Get: P = C 1 P = C 2 By the truth test, each object x that has property P has properties C 1 and C 2. Thus x is in the intersection of the set of graphs having properties C 1 and C 2. If there is x in the graph database with x in C 1 C 2 but x Q then program wouldn t stop as conjectures could be improved.
27 Proof Sketch Generating Idea Prove: P = Q. Run necessary condition conjectures for P, using Q as the theory. Lemma 1: Lemma 2: P = C 1 P = C 2 Lemma 3: C 1 C 2 Q Then lemmas imply Theorem: (semantic proof) P = Q
28 The Friendship Theorem If every pair of people in a group have exactly one friend in common, then there is a person in the group that is friends with all of them.
29 The Friendship Theorem If every pair of people in a group have exactly one friend in common, then there is a person in the group that is friends with all of them. If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices (Erdős, Rényi, Sós, 1966).
30 The Friendship Theorem If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices.
31 The Friendship Theorem If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices. Useful Observation: Can t have any four-cycles.
32 Conjectured Lemmas Investigate: (pairs_have_unique_common_neighbor)->(has_star_center)
33 Conjectured Lemmas Investigate: (pairs_have_unique_common_neighbor)->(has_star_center) Investigation: property = pairs_have_unique_common_neighbor theory = [has_star_center]
34 Conjectured Lemmas Investigate: (pairs_have_unique_common_neighbor)->(has_star_center) Investigation: property = pairs_have_unique_common_neighbor theory = [has_star_center] Conjectures: (pairs_have_unique_common_neighbor)->(is_eulerian) (pairs_have_unique_common_neighbor)->(is_circular_planar) (pairs_have_unique_common_neighbor)->(is_gallai_tree)
35 Conjectured Lemma 1 (pairs_have_unique_common_neighbor)->(is_eulerian) Euler s criterion: A connected graph is eulerian if and only if every degree is even.
36 Conjectured Lemma 1 (pairs_have_unique_common_neighbor)->(is_eulerian)
37 Conjectured Lemma 1 (pairs_have_unique_common_neighbor)->(is_eulerian)
38 Conjectured Lemma 1 (pairs_have_unique_common_neighbor)->(is_eulerian)
39 Conjectured Lemma 2 (pairs_have_unique_common_neighbor)->(is_circular_planar) Theorem (Chartrand & Harary, 1967) A graph is outerplanar if and only if it does not contain a subdivision of K 4 or K 3,3.
40 Conjectured Lemma 2 (pairs_have_unique_common_neighbor)->(is_circular_planar)
41 Conjectured Lemma 3 (pairs_have_unique_common_neighbor)->(is_gallai_tree)
42 Conjectured Lemma 3 Proof Main Ideas: Two-connected Gallai-trees are complete graphs or odd cycles.
43 Conjectured Lemma 3 Proof Main Ideas: Two-connected Gallai-trees are complete graphs or odd cycles. Outerplanar and Gallai-tree means the blocks are odd cycles.
44 Conjectured Lemma 3 Proof Main Ideas: Two-connected Gallai-trees are complete graphs or odd cycles. Outerplanar and Gallai-tree means the blocks are odd cycles. If cycle has degree two vertices then the neighbors of these vertices must be adjacent.
45 Conjectured Lemma 3 Proof Main Ideas: Two-connected Gallai-trees are complete graphs or odd cycles. Outerplanar and Gallai-tree means the blocks are odd cycles. If cycle has degree two vertices then the neighbors of these vertices must be adjacent
46 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian.
47 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4.
48 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4. Then the unique neighbor condition yields a contradiction.
49 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4. Then the unique neighbor condition yields a contradiction. So the graph is a triangle.
50 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4. Then the unique neighbor condition yields a contradiction. So the graph is a triangle. Apply induction.
51 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4. Then the unique neighbor condition yields a contradiction. So the graph is a triangle. Apply induction. Assume the graph has a cut vertex v.
52 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4. Then the unique neighbor condition yields a contradiction. So the graph is a triangle. Apply induction. Assume the graph has a cut vertex v. Each block is two-connected and must have the property that every pair of vertices has a unique common neighbor.
53 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4. Then the unique neighbor condition yields a contradiction. So the graph is a triangle. Apply induction. Assume the graph has a cut vertex v. Each block is two-connected and must have the property that every pair of vertices has a unique common neighbor. So each block is a triangle.
54 Conjectured Lemma 3 Proof Main Ideas: Assume the graph is two-connected. It is outerplanar and eulerian. If the graph is not a cycle, there is a vertex of degree at least 4. Then the unique neighbor condition yields a contradiction. So the graph is a triangle. Apply induction. Assume the graph has a cut vertex v. Each block is two-connected and must have the property that every pair of vertices has a unique common neighbor. So each block is a triangle. All components of G v are Gallai trees.
55 The Friendship Theorem If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices. Proof Ideas: Assume all pairs have a unique common neighbor.
56 The Friendship Theorem If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices. Proof Ideas: Assume all pairs have a unique common neighbor. Then the graph is eulerian, outerplanar, and a Gallai-tree.
57 The Friendship Theorem If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices. Proof Ideas: Assume all pairs have a unique common neighbor. Then the graph is eulerian, outerplanar, and a Gallai-tree. If its two-connected, its a triangle.
58 The Friendship Theorem If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices. Proof Ideas: Assume all pairs have a unique common neighbor. Then the graph is eulerian, outerplanar, and a Gallai-tree. If its two-connected, its a triangle. If it has two or more cut-vertices, diameter is at least 3, violating the common neighbor condition.
59 The Friendship Theorem If every pair of vertices in a graph have a unique common neighbor, then there is a vertex in the graph that is adjacent to all the other vertices. Proof Ideas: Assume all pairs have a unique common neighbor. Then the graph is eulerian, outerplanar, and a Gallai-tree. If its two-connected, its a triangle. If it has two or more cut-vertices, diameter is at least 3, violating the common neighbor condition. So it has at most one cut vertex, and all blocks are triangles.
60 Using Conjectures to Investigate Lemma 2 Investigate: (pairs_have_unique_common_neighbor)->(is_circular_planar) Investigation:
61 Using Conjectures to Investigate Lemma 2 Investigate: (pairs_have_unique_common_neighbor)->(is_circular_planar) Investigation: property = pairs_have_unique_common_neighbor theory = [is_circular_planar]
62 Using Conjectures to Investigate Lemma 2 Investigate: (pairs_have_unique_common_neighbor)->(is_circular_planar) Investigation: property = pairs_have_unique_common_neighbor theory = [is_circular_planar] Conjectures: (pairs_have_unique_common_neighbor)-> ((is_regular)->(is_planar_transitive)) (pairs_have_unique_common_neighbor)->(is_interval) (pairs_have_unique_common_neighbor)->(is_factor_critical) (pairs_have_unique_common_neighbor)->(is_kite_free)
63 New Project Graph Theory 164 graphs with precomputed data
64 New Project Graph Theory 164 graphs with precomputed data 145 graphs with some missing data
65 New Project Graph Theory 164 graphs with precomputed data 145 graphs with some missing data 87 properties
66 New Project Graph Theory 164 graphs with precomputed data 145 graphs with some missing data 87 properties 78 invariants
67 New Project Graph Theory 164 graphs with precomputed data 145 graphs with some missing data 87 properties 78 invariants Open-source, on GitHub anyone can use these definitions or add to them.
68 Human s Can t Make Better Conjectures There are no simpler statements that are true and significant.
69 The Dream & Kiran Code all published graph theory concepts and examples.
70 The Dream & Kiran Code all published graph theory concepts and examples. This will be a huge project.
71 The Dream & Kiran Code all published graph theory concepts and examples. This will be a huge project. What you d get:
72 The Dream & Kiran Code all published graph theory concepts and examples. This will be a huge project. What you d get: Conjectures that are true for all published examples.
73 The Dream & Kiran Code all published graph theory concepts and examples. This will be a huge project. What you d get: Conjectures that are true for all published examples. The simplest conjectures using published concepts.
74 The Dream & Kiran Code all published graph theory concepts and examples. This will be a huge project. What you d get: Conjectures that are true for all published examples. The simplest conjectures using published concepts. Kiran would have liked that.
75 Thank You! Automated Conjecturing in Sage: C. E. Larson and N. Van Cleemput, Automated Conjecturing I: Fajtlowicz s Dalmatian Heuristic Revisited, Artificial Intelligence 231 (2016) clarson@vcu.edu
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