Universal Algebra and Computational Complexity Lecture 3

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Variety Membership Problem (VAR-MEM) INPUT: two finite algebras A, B in the same signature. QUESTION: Is A ∈ V(B)? The obvious algorithm (J. Kalicki, ‘52): determine whether the identity map on A extends to a homomorphism F V(B) (A) → A. Theorem (C. Bergman & G. Slutzki, ‘00) The obvious algorithm puts VAR-MEM in 2-EXPTIME. 2-EXPTIME def = ∞⋃ TIME(2 (2O(Nk )) ) k=1 · · · NEXPTIME ⊆ EXPSPACE ⊆ 2-EXPTIME ⊆ N(2-EXPTIME) · · · Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 21 / 31
What is the “real” complexity of VAR-MEM? Theorem (Z. Székely, thesis ‘00) VAR-MEM is NP-hard (i.e., 3SAT ≤ P VAR-MEM). Theorem (M. Kozik, thesis ‘04) VAR-MEM is EXPSPACE-hard. Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 22 / 31
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Universal Algebra and Computational
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Summary of Lecture 2 Recall from Tu
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Encoding finite algebras: size matt
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Encoding finite algebras: size matt
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Size of an algebra Define some para
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Some decision problems involving al
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Categories of answers Suppose D is
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Categories of answers Suppose D is
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An easy problem: Subalgebra Members
Page 25 and 26: An obvious upper bound for SUB-MEM
Page 27 and 28: An obvious upper bound for SUB-MEM
Page 29 and 30: The Complexity of SUB-MEM So SUB-ME
Page 35 and 36: A variation: 1-SUB-MEM 1-SUB-MEM Th
Page 37 and 38: A variation: 1-SUB-MEM 1-SUB-MEM Th
Page 39 and 40: Some tractable problems about algeb
Page 49 and 50: Clone Membership Problem (CLO) INPU
Page 51 and 52: Clone Membership Problem (CLO) INPU
Page 53 and 54: The Primal Algebra Problem (PRIMAL)
Page 55 and 56: PRIMAL But testing primality of alg
Page 57 and 58: MALTSEV INPUT: a finite algebra A.
Page 59 and 60: Similar characterizations give EXPT
Page 61 and 62: Freese & Valeriote’s theorem For
Page 67 and 68: Freese & Valeriote’s theorem Proo
Page 69 and 70: Open Problem 2. Are the following e
Page 71 and 72: Open Problem 2. Are the following e
Page 73 and 74: Surprisingly, the previous problems
Page 75: Variety Membership Problem (VAR-MEM
Page 79 and 80: The Equivalence of Terms problem (E
Page 81 and 82: Obviously INEQUIV -TERM is in NP. (
Page 83 and 84: Obviously INEQUIV -TERM is in NP. (
Page 85 and 86: But WAIT!!!! There’s more!!!! For
Page 91 and 92: There are a huge number of publicat
Page 93 and 94: There are a huge number of publicat
Page 95 and 96: An outrageous scandal Theorem (G. H
Page 97 and 98: Equivalence of Terms Problem (corre
Page 99 and 100: Equivalence of Terms Problem (corre
Page 101 and 102: Two problems for relational structu
Page 103 and 104: Fix a finite relational structure B
Page 105 and 106: Fix a finite relational structure B
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Universal Algebra and Computational Complexity Lecture 3

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