If execution of logic is always performed in the same order, then the partially updated world state is not a problem for determinism (though it may be for correctness). Double buffering may let you reorder the execution of logic in different ways and still keep it deterministic. This would likely be needed for logic that executes in parallel threads, which is increasingly important for high-performance game engines.

    diff = differential(state)
    state.update(diff) # or state = update(state,diff)
    # or state += ∂_∂t(state) if you're feeling overly mathy

I've been tempted lately to model application state as the integral over time of all events (deltas) that have occurred. For example, imagine a simple game state:

  type GameState = {
    x: float; // x coordinate
    t: float; // current time
    v: float; // velocity variable
  }

  { x: 0,
    t: 0,
    v: 0, };

  [{t: 1, v:1}, {t:2, v:-1 /*this is deltaV, so back to v: 0*/}, {t: 3, v:2}, {t:5, v:-2}]

I've found this approach kinda sucks when you add things like collision detection. Your application would have to emit velocity deltas when objects would collide. If you've got a point that bounces around in a box with perfectly elastic collisions, then over time you'd have an infinite number of these collision/velocity delta events. But as you receive new events, all your precomputed collision events are for nothing (if your player logs out). So the traditional update loop simulate each tick as it occurs seems to work best.

Is there a more general way of thinking about this integration? Maybe with respect to another variable? Perhaps it would address this problem I'm having where I feel forced to quantize my game state onto ticks.

[1] Elliott, Conal; Hudak, Paul (1997), "Functional Reactive Animation" <http://conal.net/papers/icfp97/>