Simulation theorems via pseudo-random properties

We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999), to any gadget which satisfies a certainhitting property. We prove that inner product and gap-Hammingsatisfy this property, and as a corollary, we obtain a deterministic simulationtheorem for these gadgets, where the gadget’s input size is logarithmicin the input size of the outer function. This yields the firstdeterministic simulation theorem with a logarithmic gadget size, answeringan open question posed by Göös, Pitassi & Watson (in: Proceedingsof the 56th FOCS, 2015). Our result also implies the previous results for the indexing gadget, withbetter parameters than was previously known. Moreover, a simulationtheorem with logarithmic-sized gadget implies a quadratic separationin the deterministic communication complexity and the logarithm ofthe 1-partition number, no matter how high the 1-partition number iswith respect to the input size—something which is not achievable by previous results of Göös, Pitassi & Watson (2015).