Universal Algebra and Computational Complexity Lecture 3

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Some decision problems involving algebras INPUT: a finite algebra A. 1 Is A simple? Subdirectly irreducible? Directly indecomposable? 2 Is A primal? Quasi-primal? Maltsev? 3 Is V(A) congruence distributive? Congruence modular? INPUT: two finite algebras A, B. 4 Is A ∼ = B? 5 Is A ∈ V(B) Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 5 / 31
Some decision problems involving algebras INPUT: a finite algebra A. 1 Is A simple? Subdirectly irreducible? Directly indecomposable? 2 Is A primal? Quasi-primal? Maltsev? 3 Is V(A) congruence distributive? Congruence modular? INPUT: two finite algebras A, B. 4 Is A ∼ = B? 5 Is A ∈ V(B) INPUT: A finite algebra A and two terms s(⃗x), t(⃗x). 6 Does s = t have a solution in A? 7 Is s ≈ t an identity of A? Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 5 / 31
Page 1 and 2: Universal Algebra and Computational
Page 3 and 4: Summary of Lecture 2 Recall from Tu
Page 5 and 6: Encoding finite algebras: size matt
Page 7 and 8: Encoding finite algebras: size matt
Page 9 and 10: Size of an algebra Define some para
Page 11 and 12: Some decision problems involving al
Page 13: Some decision problems involving al
Page 17 and 18: Some decision problems involving al
Page 19 and 20: Categories of answers Suppose D is
Page 21 and 22: Categories of answers Suppose D is
Page 23 and 24: 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 63 and 64: Freese & Valeriote’s theorem For
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Freese & Valeriote’s theorem For
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Freese & Valeriote’s theorem Proo
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Open Problem 2. Are the following e
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Open Problem 2. Are the following e
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Surprisingly, the previous problems
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Variety Membership Problem (VAR-MEM
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What is the “real” complexity o
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The Equivalence of Terms problem (E
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Obviously INEQUIV -TERM is in NP. (
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Obviously INEQUIV -TERM is in NP. (
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But WAIT!!!! There’s more!!!! For
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Page 89 and 90:
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There are a huge number of publicat
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There are a huge number of publicat
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An outrageous scandal Theorem (G. H
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Equivalence of Terms Problem (corre
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Equivalence of Terms Problem (corre
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Two problems for relational structu
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Fix a finite relational structure B
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Fix a finite relational structure B
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Universal Algebra and Computational Complexity Lecture 3

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