1. Eric Allender Rutgers University Graph Automorphism & Circuit Size Joint work with Joshua A. Grochow and Cristopher Moore (SFI) Simons Workshop, September 29, 2015

2. Eric Allender: Graph Automorphism & Circuit Size < 2 >< 2 > The Context  The Minimum Circuit Size Problem (MCSP) = {(f,i) : f is the truth-table of a function that has a circuit of size ≤ i}.  In NP, but not known (or widely believed) to be NP-complete. [Kabanets, Cai], [Murray, Williams], [A, Holden, Kabanets]  Lots of reasons to believe it’s not in P.

3. Eric Allender: Graph Automorphism & Circuit Size < 3 >< 3 > More Context  Factoring is in ZPP MCSP.  Graph Isomorphism is in RP MCSP.  Every promise problem in SZK is in (Promise) BPP MCSP.  A motivating question: Is Graph Isomorphism in ZPP MCSP ?

4. Eric Allender: Graph Automorphism & Circuit Size < 4 >< 4 > More Context  Factoring is in ZPP MCSP.  Graph Isomorphism is in RP MCSP.  Every promise problem in SZK is in (Promise) BPP MCSP.  An obstacle: EACH of these reductions follows the same route….

5. Eric Allender: Graph Automorphism & Circuit Size < 5 >< 5 > Yet More Context  The well-trodden path:  MCSP is a wonderful test to distinguish random from pseudorandom distributions.  Thus, via [HILL], MCSP is an oracle that allows a probabilistic algorithm to invert poly- time functions with high probability.  Note that this approach can’t show a result like “A is in ZPP MCSP ” unless we already know that A is in NP∩coNP.

6. Eric Allender: Graph Automorphism & Circuit Size < 6 >< 6 > Context, Context, Context  MCSP is more like a family of problems, than a single problem.  For instance “size” could mean “# of wires” or “# of gates”, or “# of bits to describe the circuit”, etc.  None of these is known to be reducible to any other – but all can stand in for “MCSP”.  One more such variant: MKTP = {(x,i) : KT(x) ≤ i}

7. Eric Allender: Graph Automorphism & Circuit Size < 7 >< 7 > What Can We Show?  Graph Automorphism is in ZPP MKTP.  As observed on an earlier slide, this involves a different type of reduction than all earlier reductions to MKTP or MCSP (since Graph Automorphism is not known to be in NP∩coNP).  We are unable to extend this, to show Graph Automorphism is in ZPP MCSP. – This is a new phenomenon; all other reductions to MKTP carried over to MCSP.

8. Eric Allender: Graph Automorphism & Circuit Size < 8 >< 8 > What Can We Show?  Graph Automorphism is in ZPP MKTP.  We are also unable to extend this, to ZPP- reduce Graph Isomorphism to MKTP.  …although we can adapt our framework to show that Graph Isomorphism is in BPP MKTP.  …which is weaker than the previously-known result: Graph Isomorphism is in RP MKTP – but it may be useful to build up some tools for providing reductions to MKTP and MCSP.

9. Eric Allender: Graph Automorphism & Circuit Size < 9 >< 9 > The Proof (1)  Theorem: Graph Automorphism is in ZPP MKTP.  It suffices to solve the Graph Isomorphism problem, restricted to rigid graphs. [KST]  That is: Consider the promise problem with – YES Instances: {(G 0,G 1 ) : G 0 ≡ G 1 }. – NO Instances: {(G 0,G 1 ) : G 0 and G 1 are rigid and are not isomorphic}  Might seem odd to impose rigidity only on the NO instances – but this stronger result holds with the same proof.

10. Eric Allender: Graph Automorphism & Circuit Size The Proof (1)  Theorem: Graph Automorphism is in ZPP MKTP.  It suffices to solve the Graph Isomorphism problem, restricted to rigid graphs. [KST]  That is: Consider the promise problem with – YES Instances: {(G 0,G 1 ) : G 0 ≡ G 1 }. – NO Instances: {(G 0,G 1 ) : G 0 and G 1 are rigid and are not isomorphic}  We already have GI in RP MKTP. Thus we need to solve this promise problem in coRP MKTP.

11. Eric Allender: Graph Automorphism & Circuit Size The Proof (2)  On input (G 0,G 1 ) – Randomly pick a bit string w=w 1 w 2 …w t. – Pick random permutations π 1 …π t. – Let z= π 1 (G w 1 )π 2 (G w 2 )…π t (G w t )  If G 0 and G 1 are not isomorphic, then z allows us to reconstruct w and π 1 …π t, so that z has (non-time-bounded) K-complexity around t+ts (where s = log n!), whp. Hence KT(z) > t+ts.  Otherwise, KT(z) is around n 2 +ts.

12. Eric Allender: Graph Automorphism & Circuit Size The Proof (2)  On input (G 0,G 1 ) – Randomly pick a bit string w=w 1 w 2 …w t. – Pick random permutations π 1 …π t. – Let z= π 1 (G w 1 )π 2 (G w 2 )…π t (G w t )  If G 0 and G 1 are not isomorphic, then z allows us to reconstruct w and π 1 …π t, so that z has (non-time-bounded) K-complexity around t+ts (where s = log n!), whp. Hence KT(z) > t+ts.  Otherwise, KT(z) is around n 2 +ts. QED

13. Eric Allender: Graph Automorphism & Circuit Size Why does this break for MCSP?  The easiest way to answer this, is to present a version of MCSP where the proof does go through (and also to explain a bit about how KT complexity is defined).  KT(x) = min{|d|+t : U can construct x from description d in time t}

14. Eric Allender: Graph Automorphism & Circuit Size Why does this break for MCSP?  The easiest way to answer this, is to present a version of MCSP where the proof does go through (and also to explain a bit about how KT complexity is defined).  KT(x) = min{|d|+t : U d (i) = the i th bit of x, and runs in time t}  A multiplexer circuit has AND, OR and NOT gates, and also INDEX gates (of fan-in log m) that access a (fixed) array of size m.  They can simulate U efficiently. [Store d in the array.]

15. Eric Allender: Graph Automorphism & Circuit Size Why does this break for MCSP?  The easiest way to answer this, is to present a version of MCSP where the proof does go through (and also to explain a bit about how KT complexity is defined).  KT(x) = min{|d|+t : U d (i) = the i th bit of x, and runs in time t}  In contrast, implementing a multiplexer using AND and OR requires size O(m) = O(|d|) for each query that U makes to d.  …which kills the argument!

16. Eric Allender: Graph Automorphism & Circuit Size Gap amplification  The way to modify this approach, to make it work for MCSP, would be to “amplify” the gap between KT ≥ t(s+1) and KT ≤ ts, to something like KT ≥ b and KT ≤ b ε for some ε > 0.  This would pay other dividends. It would give a way to reduce some of the different variants of MCSP to each other.  This would provide a notion of “robustness” for MCSP – which is currently lacking. (Different versions of the problem might have different complexity.)

17. Eric Allender: Graph Automorphism & Circuit Size Different versions of MCSP  For instance, is it possible that one version of MCSP is NP-complete, while another version is in P?

18. Eric Allender: Graph Automorphism & Circuit Size Open Questions  Is Graph Automorphism in ZPP MCSP ?  How about Graph Isomorphism?  …or all of SZK?  Is there some way to relate the complexities of MKTP and (the many versions of) MCSP?  …through Gap Amplification, or via some other means?  And there are tons of other important questions about MCSP.

19. Eric Allender: Graph Automorphism & Circuit Size Thank you!