# Simulation Theorems via Pseudorandom Properties

## 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.

## Key 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].