Splitting the Computational Universe

Transcription

1 Splitting the Computational Universe

2 s Theorem Definition Existential Second Order Logic (ESO) is the collection of all statements of the form: ( S )Φ(S, S )

3 s Theorem Definition Existential Second Order Logic (ESO) is the collection of all statements of the form: ( S )Φ(S, S ) Theorem N P = ESO

4 Datalog Definition (Datalog - Take 1) Logic programs without function symbols.

5 Datalog Definition (Datalog - Take 1) Logic programs without function symbols. Definition (Datalog - Take 2) Extension of conjunctive queries by recursion.

6 Datalog Definition (Datalog - Take 1) Logic programs without function symbols. Definition (Datalog - Take 2) Extension of conjunctive queries by recursion. Definition (Datalog - Take 3) Negation-free, function-free Horn Clauses.

7 s Theorem Theorem P = Datalog + {negation on input variables} + {total order on domain}.

8 s Theorem Theorem A Boolean CSP is N P-Complete unless it is equivalent to a problem in one of the following forms, in which case it is in P Valid problem 2. 1-Valid problem 3. Horn Clauses 4. Anti-Horn Clauses 5. 2-SAT 6. Linear equations mod 2.

9 s Theorem Theorem If P N P, then there is an incomplete set in N P \ P.

10 Feder and Vardi

11 How far can we push?

12 First Steps Definition (SNP) SNP is the class of ESO sentences with universal first order part: ( S )( x)φ(x, S, S ), where Φ is quantifier free.

13 First Steps Definition (SNP) SNP is the class of ESO sentences with universal first order part: ( S )( x)φ(x, S, S ), where Φ is quantifier free. Theorem SNP is equivalent to N P under polynomial time reductions.

14 Hobbling SNP Can place restrictions on first order part of SNP sentence: 1. Monotone 2. Monadic 3. Inequality Free

15 Hobbling SNP Can place restrictions on first order part of SNP sentence: 1. Monotone 2. Monadic 3. Inequality Free Theorem SNP with any 2 of the above properties is equivalent to NP under polynomial time reductions.

16 Hobbling SNP Can place restrictions on first order part of SNP sentence: 1. Monotone 2. Monadic 3. Inequality Free Theorem SNP with any 2 of the above properties is equivalent to NP under polynomial time reductions. Definition (MMSNP) MMSNP is the class of SNP sentences satisfying all of the above conditions

17 Open Problem 1 Question Is MMSNP polynomially equivalent to NP?

18 Enter Constraints Theorem CSP is equivalent to MMSNP under randomised polynomial reductions.

19 Open Problem 2 Question Is it possible to derandomise the reduction from CSP to MMSNP? That is, give a deterministic polynomial procedure for constructing graphs of arbitrary girth and chromatic number (Erdős).

20 Definition A CSP problem has bounded width if its complement can be expresed in Datalog

21 Definition A CSP problem has bounded width if its complement can be expresed in Datalog Theorem All bounded width problems are in P.

22 Definition A CSP problem has bounded width if its complement can be expresed in Datalog Theorem All bounded width problems are in P. Almost There 0-Valid, 1-Valid, Horn, Anti-Horn and 2-SAT problems all have bounded width.

23 How to Count Definition ((Intuitive)) A CSP has the ability to count if it contains constraints of the form: C(x, y, z) : x + y + z = 1 Z (x) : x = 0 (With a condition to avoid contradiction).

24 How to Count Theorem If a CSP has the ability to count, then it does not have bounded width. Corollary Linear equations mod 2 does not have bounded width

25 Generalising Lemma Any abelian group has the ability to count

26 Generalising Lemma Any abelian group has the ability to count Actually, can deal with nonabelian groups too...

27 Dealing with Linear Equations Problem A General Subgroup Problem is a CSP whose domain is a group G and whose relations are subgroups and cosets of G k.

28 Dealing with Linear Equations Problem A General Subgroup Problem is a CSP whose domain is a group G and whose relations are subgroups and cosets of G k. Theorem The General Subgroup Problem is in P.

29 Dealing with Linear Equations Problem A General Subgroup Problem is a CSP whose domain is a group G and whose relations are subgroups and cosets of G k. Theorem The General Subgroup Problem is in P. For nonabelian group, need to consider near subgroups - dealt with by Aschbacher.

30 Closing the Deal Theorem Adding a subset that is not a near subgroup to the general subgroup problem makes the problem N P-Complete.

31 Closing the Deal Theorem Adding a subset that is not a near subgroup to the general subgroup problem makes the problem N P-Complete. Adding any number of near subgroups does not seem to do this!