Simulation Theorems via Pseudorandom Properties
Arkadev Chattopadhyay, Michal Kouck\'y, Bruno Loff, Sagnik, Mukhopadhyay
TL;DR
This paper extends the deterministic simulation theorem to a broader class of gadgets satisfying a hitting property, including inner-product and gap-Hamming, improving understanding of pseudorandomness in computational complexity.
Contribution
It generalizes the simulation theorem to new gadgets with a hitting property, answering an open question and improving parameters for the Indexing gadget.
Findings
Inner-product and gap-Hamming satisfy the hitting property.
Deterministic simulation theorem applies to these gadgets with logarithmic input size.
Improved parameters for the Indexing gadget.
Abstract
We generalize the deterministic simulation theorem of Raz and McKenzie [RM99], to any gadget which satisfies certain hitting property. We prove that inner-product and gap-Hamming satisfy this property, and as a corollary we obtain deterministic simulation theorem for these gadgets, where the gadget's input-size is logarithmic in the input-size of the outer function. This answers an open question posed by G\"{o}\"{o}s, Pitassi and Watson [GPW15]. Our result also implies the previous results for the Indexing gadget, with better parameters than was previously known. A preliminary version of the results obtained in this work appeared in [CKL+17].
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Simulation Theorems via Pseudo-random Properties
Arkadev Chattopadhyay Tata Institute of Fundamental Research, Mumbai, [email protected]
Michal Koucký Charles University, Prague, [email protected]
Bruno Loff INESC-TEC and University of Porto, Porto, [email protected]
Sagnik Mukhopadhyay Tata Institute of Fundamental Research, Mumbai, [email protected]