Universal Algebra and Computational Complexity Lecture 3

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The Complexity of SUB-MEM So SUB-MEM ∈ TIME(N 2 ), or maybe TIME(N 4+ɛ ), or surely in TIME(N 55 ), and so we get the “obvious” upper bound: SUB-MEM ∈ P. Next questions: Can we obtain P as a lower bound for SUB-MEM? What was that P-complete problem again?. . . (CVAL or HORN-3SAT ) Can we show HORN-3SAT ≤ L SUB-MEM? Theorem (N. Jones & W. Laaser, ‘77) Yes. In other words, SUB-MEM is P-complete. Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 9 / 31
A variation: 1-SUB-MEM 1-SUB-MEM This is the restriction of SUB-MEM to unary algebras (all fundamental operations are unary). I.e., INPUT: A unary algebra A, a set S ⊆ A, and b ∈ A. QUESTION: Is b ∈ Sg A (S)? Ross Willard (Waterloo) Algebra and Complexity Třešť, September 2008 10 / 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 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 31 and 32: The Complexity of SUB-MEM So SUB-ME
Page 33: The Complexity of SUB-MEM So SUB-ME
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 and 76: Variety Membership Problem (VAR-MEM
Page 77 and 78: What is the “real” complexity o
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
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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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