Non-uniform Depth of Polynomial Time and Space Simulations

We discuss some connections between polynomial time and non-uniform, small depth circuits. A connection is shown with simulating deterministic time in small space. The well known result of Hopcroft, Paul and Valiant [HPV77] showing that space is more powerful than time can be improved, by making an assumption about the connection of deterministic time computations and non-uniform, small depth circuits. To be more precise, we prove the following: If every linear time deterministic computation can be done by non-uniform circuits of polynomial size and sub-linear depth, then \(\mathcal{DTIME}(t) \subseteq \mathcal{DSPACE}(t^{1-\epsilon})\) for some constant e> 0. We can also apply the same techniques to prove an unconditional result, a trade-off type of theorem for the size and depth of a non-uniform circuit that simulates a uniform computation.