Finding Irrefutable Certificates for $S_2^p$ via Arthur and Merlin

Venkatesan Chakaravarthy (1) , Sambuddha Roy (1)

1 India Research Laboratory [Bangalore]

We show that $S_2^p\subseteq P^{prAM}$, where $S_2^p$ is the symmetric alternation class and $prAM$ refers to the promise version of the Arthur-Merlin class $AM$. This is derived as a consequence of our main result that presents an $FP^{prAM}$ algorithm for finding a small set of ``collectively irrefutable certificates'' of a given $S_2$-type matrix. The main result also yields some new consequences of the hypothesis that $NP$ has polynomial size circuits. It is known that the above hypothesis implies a collapse of the polynomial time hierarchy ($PH$) to $S_2^p\subseteq ZPP^{NP}$ (Cai 2007, Köbler and Watanabe 1998). Under the same hypothesis, we show that $PH$ collapses to $P^{prMA}$. We also describe an $FP^{prMA}$ algorithm for learning polynomial size circuits for $SAT$, assuming such circuits exist. For the same problem, the previously best known result was a $ZPP^{NP}$ algorithm (Bshouty et al. 1996).

Mots clés: Symmetric alternation; promise-AM; Karp-Lipton theorem; learning circuits

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Venkatesan Chakaravarthy, Sambuddha Roy. Finding Irrefutable Certificates for $S_2^p$ via Arthur and Merlin. STACS 2008, Feb 2008, Bordeaux, France. pp.157-168. ⟨hal-00214303⟩