The Proof Previous Proofs: Direct combinatorial constructions Resulting matrices have {0,1} entries, for which we have quasipolynomial upper bounds [Razborov '90]. Our Proof: Prove a new lifting theorem to reduce the lower bound to bounding a new algebraic query measure on search problems. Our matrices have entries in , and so we can avoid 76 the above obstacle.
The Proof Overview Rank Measure [Razborov '90] : 77
The Proof Overview Rank Measure [Razborov '90] : Associate with certain special functions f (like GEN 1 and ST-CONN) a search problem Search(f) 78
The Proof Overview Rank Measure [Razborov '90] : Associate with certain special functions f (like GEN 1 and ST-CONN) a search problem Search(f) ( Lift ) Reduce constructing a good matrix A 2 for f to lower bounding a complexity measure on Search(f) 79
The Proof Overview Rank Measure [Razborov '90] : Associate with certain special functions f (like GEN 1 and ST-CONN) a search problem Search(f) ( Lift ) Reduce constructing a good matrix A 2 for f to lower bounding a complexity measure on Search(f) Actually prove the query lower bounds against 3 80 Search(f)
The Proof Overview Rank Measure [Razborov '90] : Associate with certain special functions f (like GEN 1 and ST-CONN) a search problem Search(f) ( Lift ) Reduce constructing a good matrix A 2 for f to lower bounding a complexity measure on Search(f) Actually prove the query lower bounds against 3 81 Search(f)
The Proof Overview Rank Measure [Razborov '90] : Follows from [Raz-Mckenzie '97] [Goos-Pitassi '15] Associate with certain special functions f (like GEN 1 and ST-CONN) a search problem Search(f) ( Lift ) Reduce constructing a good matrix A 2 for f to lower bounding a complexity measure on Search(f) Actually prove the query lower bounds against 3 82 Search(f)
The Proof Overview Rank Measure [Razborov '90] : Associate with certain special functions f (like GEN 1 and ST-CONN) a search problem Search(f) ( Lift ) Reduce constructing a good matrix A 2 for f to lower bounding a complexity measure on Search(f) Actually prove the query lower bounds against 3 83 Search(f)
The Proof Lifting Theorem 84
The Proof Lifting Theorem (Communication Setting) 85
The Proof Lifting Theorem (Communication Setting) Search Problem S = Search(f) 86
The Proof Lifting Theorem (Communication Setting) Search Problem S = Search(f) Decision Tree Hard for Weak Complexity Measure 87
The Proof Lifting Theorem (Communication Setting) Search Problem S = Search(f) Decision Tree Hard for Weak Complexity Measure 88
The Proof Lifting Theorem (Communication Setting) Search Problem S = Search(f) Decision Tree Compose S with some two input function g Alice gets x inputs Bob gets y inputs Hard for Weak Complexity Measure 89
The Proof Lifting Theorem (Communication Setting) Search Problem S = Search(f) Decision Communication Tree Matrix Compose S with some two input function g Alice gets x inputs Bob gets y inputs Hard for Hard for Weak Complexity Strong Complexity Measure Measure 90
The Proof Lifting Theorem (Our Setting) 91
The Proof Lifting Theorem (Our Setting) Search Problem S = Search(f) 92
The Proof Lifting Theorem (Our Setting) Search Problem S = Search(f) Hard for Strong Complexity Measure 93
The Proof Lifting Theorem (Our Setting) Search Problem S = Search(f) ? Hard for Hard for Weak Complexity Strong Complexity Measure Measure 94
The Proof Lifting Theorem (Our Setting) Search Problem S = Search(f) ? ? Hard for Hard for Weak Complexity Strong Complexity Measure Measure 95
The Proof Lifting Theorem (Our Setting) Search Problem S = Search(f) Polynomial ? certifying a large algebraic gap for S Hard for Hard for Weak Complexity Strong Complexity Measure Measure 96
The Proof Lifting Theorem (Our Setting) Search Problem S = Search(f) Polynomial certifying a large Compose p with algebraic gap two-input function for S g instead! Hard for Hard for Weak Complexity Strong Complexity Measure Measure 97
Lifting Theorem (ST-CONN) Theorem: ( Lifting Theorem for Rank Measure ) Consider layered ST-CONN on the grid, and let k be the algebraic gap complexity of the ST-CONN search problem. There is a real matrix A such that 98
Lifting Theorem (ST-CONN) Theorem: ( Lifting Theorem for Rank Measure ) Consider layered ST-CONN on the grid, and let k be the algebraic gap complexity of the ST-CONN search problem. There is a real matrix A such that Proof: Intuition on previous slide, extension of the Pattern Matrix Method [Sherstov '08]. 99
The Proof Overview Rank Measure [Razborov '90] : Associate with certain special functions f (like GEN 1 and ST-CONN) a search problem Search(f) ( Lift ) Reduce constructing a good matrix A 2 for f to lower bounding a complexity measure on Search(f) Actually prove the query lower bounds against 3 100 Search(f)
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