Mycielski graphs and PR proofs Emre Yolcu Xinyu Wu Marijn J. H. - PowerPoint PPT Presentation

Mycielski graphs and PR proofs Emre Yolcu Xinyu Wu Marijn J. H. Heule 23rd International Conference on Theory and Applications of Satisfiability Testing eyolcu@cs.cmu.edu 1 / 29

Problem ◮ Mycielski graphs: Triangle-free graphs M k with arbitrarily high chromatic number. M k is not properly colorable using k − 1 colors. eyolcu@cs.cmu.edu 2 / 29

Problem ◮ Mycielski graphs: Triangle-free graphs M k with arbitrarily high chromatic number. M k is not properly colorable using k − 1 colors. ◮ Propagation redundancy ( PR ): “Interference-based” propositional proof system. Generalizes the commonly-used DRAT. eyolcu@cs.cmu.edu 2 / 29

Problem ◮ Mycielski graphs: Triangle-free graphs M k with arbitrarily high chromatic number. M k is not properly colorable using k − 1 colors. ◮ Propagation redundancy ( PR ): “Interference-based” propositional proof system. Generalizes the commonly-used DRAT. Asked by Donald Knuth: For a family of formulas encoding the colorability of the Mycielski graphs, are there small PR proofs without using new variables? eyolcu@cs.cmu.edu 2 / 29

Mycielski graphs Mycielski graph µ ( G ) of G = ( V , E ) is constructed as follows: v 1 v 2 1. Include G in µ ( G ) as a subgraph. eyolcu@cs.cmu.edu 3 / 29

Mycielski graphs Mycielski graph µ ( G ) of G = ( V , E ) is constructed as follows: v 1 v 2 1. Include G in µ ( G ) as a subgraph. 2. For each v i ∈ V , add a vertex u i ∈ U adjacent to all of N G ( v i ) . u 1 u 2 eyolcu@cs.cmu.edu 3 / 29

Mycielski graphs Mycielski graph µ ( G ) of G = ( V , E ) is constructed as follows: v 1 v 2 1. Include G in µ ( G ) as a subgraph. 2. For each v i ∈ V , add a vertex u i ∈ U adjacent to all of N G ( v i ) . u 1 u 2 3. Add a vertex w adjacent to all of U . w eyolcu@cs.cmu.edu 3 / 29

Mycielski graphs Mycielski graph µ ( G ) of G = ( V , E ) is constructed as follows: v 1 v 2 1. Include G in µ ( G ) as a subgraph. 2. For each v i ∈ V , add a vertex u i ∈ U adjacent to all of N G ( v i ) . u 1 u 2 3. Add a vertex w adjacent to all of U . w ◮ Unless G has a triangle µ ( G ) does not have a triangle. ◮ µ ( G ) has chromatic number one higher than G . eyolcu@cs.cmu.edu 3 / 29

Mycielski graphs M 2 M 3 M 4 − − − → − − − → µ µ M k has Θ( 2 k ) vertices and Θ( 3 k ) edges. eyolcu@cs.cmu.edu 4 / 29

Main references Marijn J. H. Heule, Benjamin Kiesl, and Armin Biere (2019) Strong extension-free proof systems Sam Buss and Neil Thapen (2019) DRAT and propagation redundancy proofs without new variables eyolcu@cs.cmu.edu 5 / 29

Proof complexity Interested in proofs (refutations) of unsatisfiable formulas F . eyolcu@cs.cmu.edu 6 / 29

Proof complexity Interested in proofs (refutations) of unsatisfiable formulas F . Definition A valid clausal proof of F is a sequence ( C 1 , ω 1 ) , . . . , ( C N , ω N ) where, defining F i := F ∧ � i j = 1 C j , ◮ each clause C i preserves satisfiability, i.e. is redundant wrt F i − 1 , ◮ the redundancy of C i is decidable in polynomial time given ω i , ◮ C N = ⊥ (empty clause). eyolcu@cs.cmu.edu 6 / 29

Reverse unit propagation (RUP) ◮ Partial assignments are conjunctions of literals. Example: x �→ 1 , y �→ 0 , z �→ 1 is denoted by x ∧ y ∧ z . eyolcu@cs.cmu.edu 7 / 29

Reverse unit propagation (RUP) ◮ Partial assignments are conjunctions of literals. Example: x �→ 1 , y �→ 0 , z �→ 1 is denoted by x ∧ y ∧ z . ◮ C denotes the assignment corresponding to � p ∈ C p . eyolcu@cs.cmu.edu 7 / 29

Reverse unit propagation (RUP) ◮ Partial assignments are conjunctions of literals. Example: x �→ 1 , y �→ 0 , z �→ 1 is denoted by x ∧ y ∧ z . ◮ C denotes the assignment corresponding to � p ∈ C p . ◮ Unit propagation: satisfy a unit clause, restrict formula, repeat. eyolcu@cs.cmu.edu 7 / 29

Reverse unit propagation (RUP) ◮ Partial assignments are conjunctions of literals. Example: x �→ 1 , y �→ 0 , z �→ 1 is denoted by x ∧ y ∧ z . ◮ C denotes the assignment corresponding to � p ∈ C p . ◮ Unit propagation: satisfy a unit clause, restrict formula, repeat. ◮ C is a RUP inference from F if unit propagation refutes F ∧ C . eyolcu@cs.cmu.edu 7 / 29

Reverse unit propagation (RUP) ◮ Partial assignments are conjunctions of literals. Example: x �→ 1 , y �→ 0 , z �→ 1 is denoted by x ∧ y ∧ z . ◮ C denotes the assignment corresponding to � p ∈ C p . ◮ Unit propagation: satisfy a unit clause, restrict formula, repeat. ◮ C is a RUP inference from F if unit propagation refutes F ∧ C . ◮ F ⊢ 1 H means each clause C ∈ H is a RUP inference from F . We say that F implies H by unit propagation . eyolcu@cs.cmu.edu 7 / 29

PR proof system Definition Let F be a formula, C a clause, and α = C . C is propagation redundant with respect to F if there exists an assignment ω such that ω satisfies C and F | α ⊢ 1 F | ω . eyolcu@cs.cmu.edu 8 / 29

PR proof system Definition Let F be a formula, C a clause, and α = C . C is propagation redundant with respect to F if there exists an assignment ω such that ω satisfies C and F | α ⊢ 1 F | ω . Intuitively, PR clauses allow us to argue that satisfying assignments can be assumed to have certain properties. This can be seen as capturing “without loss of generality” arguments. eyolcu@cs.cmu.edu 8 / 29

PR variants Formula F , proof π , clause–witness pair ( C i , ω i ) ◮ SPR: Require dom( ω i ) = dom( α i ) . ◮ PR − : Restrict C i to only include variables appearing in F . ◮ DPR: Allow deletion of a previous clause in π or F . eyolcu@cs.cmu.edu 9 / 29

PR variants Formula F , proof π , clause–witness pair ( C i , ω i ) ◮ SPR: Require dom( ω i ) = dom( α i ) . ◮ PR − : Restrict C i to only include variables appearing in F . ◮ DPR: Allow deletion of a previous clause in π or F . Definition For a PR inference, its discrepancy is | dom( ω ) \ dom( α ) | . Theorem (Buss and Thapen, 2019) A PR proof of length N and discrepancy ≤ δ can be converted into an SPR proof of length O ( 2 δ N ) without additional variables. eyolcu@cs.cmu.edu 9 / 29

Proof complexity of PR With new variables, PR can simulate Extended Resolution. eyolcu@cs.cmu.edu 10 / 29

Proof complexity of PR With new variables, PR can simulate Extended Resolution. Without new variables, SPR − was shown to admit short proofs of: ◮ pigeonhole principle ◮ bit pigeonhole principle ◮ parity principle ◮ clique-coloring principle ◮ Tseitin tautologies eyolcu@cs.cmu.edu 10 / 29

Proof complexity of PR With new variables, PR can simulate Extended Resolution. Without new variables, SPR − was shown to admit short proofs of: ◮ pigeonhole principle ◮ bit pigeonhole principle ◮ parity principle ◮ clique-coloring principle ◮ Tseitin tautologies Without new variables, strictly stronger than RAT: it is known that RAT − does not simulate SPR − . eyolcu@cs.cmu.edu 10 / 29

Proof complexity of PR With new variables, PR can simulate Extended Resolution. Without new variables, SPR − was shown to admit short proofs of: ◮ pigeonhole principle ◮ bit pigeonhole principle ◮ parity principle ◮ clique-coloring principle ◮ Tseitin tautologies Without new variables, strictly stronger than RAT: it is known that RAT − does not simulate SPR − . Currently, there are no known lower bounds for PR − (or SPR − ). eyolcu@cs.cmu.edu 10 / 29

Mycielski graph formulas Encoding graph coloring: Given graph G = ( V , E ) and k colors. � x c for each x ∈ V c ∈ [ k ] x c ∨ y c for each xy ∈ E , c ∈ [ k ] eyolcu@cs.cmu.edu 11 / 29

Mycielski graph formulas Encoding graph coloring: Given graph G = ( V , E ) and k colors. � x c for each x ∈ V c ∈ [ k ] x c ∨ y c for each xy ∈ E , c ∈ [ k ] MYC k ≡ “ M k is ( k − 1 ) -colorable.” eyolcu@cs.cmu.edu 11 / 29

Results Let N = Θ( 3 k k ) be the length of MYC k . Theorem MYC k has DSPR − proofs of length O ( N log N ) and constant width. eyolcu@cs.cmu.edu 12 / 29

Results Let N = Θ( 3 k k ) be the length of MYC k . Theorem MYC k has DSPR − proofs of length O ( N log N ) and constant width. Theorem MYC k has PR − proofs of length O ( N log 3 2 ( log N ) 2 ) , constant width, and maximum discrepancy Θ( 2 k ) . eyolcu@cs.cmu.edu 12 / 29

Proof outline eyolcu@cs.cmu.edu 13 / 29

Proof outline V eyolcu@cs.cmu.edu 13 / 29

Proof outline U eyolcu@cs.cmu.edu 13 / 29

Proof outline w eyolcu@cs.cmu.edu 13 / 29

Proof outline eyolcu@cs.cmu.edu 13 / 29

Proof outline u 1 u 2 v 1 v 2 eyolcu@cs.cmu.edu 13 / 29

Proof outline eyolcu@cs.cmu.edu 13 / 29

Proof outline V eyolcu@cs.cmu.edu 13 / 29

Proof outline U eyolcu@cs.cmu.edu 13 / 29

Proof outline w eyolcu@cs.cmu.edu 13 / 29

Proof outline eyolcu@cs.cmu.edu 13 / 29

DPR − proof Partition the vertices of M k into V ∪ U ∪ { w } . Let E k − 1 denote the edge set of the ( k − 1 ) th Mycielski graph. Let n k = | V | = | U | . eyolcu@cs.cmu.edu 14 / 29