As spatial locality leads to advantages of computation speed-up and sequence reuse in molecular computing, molecular walkers that exhibit localized reactions are of interest for implementing logic computations. We use molecular spiders, which are a type of molecular walkers, to implement logic circuits. We develop an extended multi-spider model with a dynamic environment where signal transmission is triggered locally, and use this model to implement three basic gates (AND, OR, NOT) and a mechanism to cascade the gates. We use a kinetic Monte Carlo algorithm to simulate gate computations, and we analyze circuit complexity: our design scales linearly with formula size and has a logarithmic time complexity.