What do you mean by "energy metric"? If you're asking for a conserved quantity whose conservation is violated, then you're not going to get that by definition.
What I mean by having no 2nd law of thermodynamics is that it's possible to construct a Universal Turing machine that can operate indefinitely without using up any irreplaceable resources.
What do you mean by "energy metric"?
Something that works as a measure over the state variables for the purposes of Lyapunov's stability theorems. That is, take a set of state variables that completely define the system, and take some measure that is always non-negative and is an increasing function of every variable. (Lyapunov's theorem -- one version -- says that if you can find such a measure, and if it's strictly non-increasing with time, the system is stable -- but this isn't guaranteed from the definition.)
Maybe you can find one that in...
Many experts suspect that there is no polynomial-time solution to the so-called NP-complete problems, though no-one has yet been able to rigorously prove this and there remains the possibility that a polynomial-time algorithm will one day emerge. However unlikely this is, today I would like to invite LW to play a game I played with with some colleagues called what-would-you-do-with-a-polynomial-time-solution-to-3SAT? 3SAT is, of course, one of the most famous of the NP-complete problems and a solution to 3SAT would also constitute a solution to *all* the problems in NP. This includes lots of fun planning problems (e.g. travelling salesman) as well as the problem of performing exact inference in (general) Bayesian networks. What's the most fun you could have?