In chess, controlling the center isn’t always a good idea. For example, when you have the opportunity to effectively sacrifice your bishop on h6, or when you’re at risk of getting checkmated, controlling the center ought to be far from your mind. It would seem that, in addition to the heuristic “Control the center,” a computer also needs heuristical methods for choosing which heuristics to apply in a given situation: i.e., “When the sacrifice on h6 doesn’t lead to a win, try to control the center” or “When you’re about to get checkmated, forget about controlling the center.” Notice how each of these second-order heuristics serve to locate the first-order heuristics in a particular context. By contextualizing essentially context-independent chess wisdom like “Control the center” second-order heuristics help raise chess playing machines to a much higher level of play than any collection of first-order heuristics would be capable of. The problem, however, should be evident: which second-order heuristic should be applied at any given time? For example, what if I’m about to get checkmated but I also have a potentially fruitful bishop sacrifice on h6? The answer is, it depends on the context. Does the bishop sacrifice put the other king in check? Just how close are you to getting checkmated anyway? So a third-order heuristic is necessary. Etc.

It may sound like I’m setting up an ad infinitum proof that computers cannot play chess. But that, of course, would be silly, thanks to Deep Blue, who proved such arguments to be deeply suspect. Instead, I’m trying to show that the act of stupefication involves, at least in part, the navigation of and formalization of a potentially infinite recursive stack of heuristics. That the layers of heuristical guidance are not piled infinitely deep and, indeed, that they obviously need not be piled infinitely deep is proof that there exists a discoverable stopping place amidst the recursive descent. After all, we know that a computer can, with a finite stack of recursive rules, accomplish a high level of chess play.

This should indicate to us that arguments which seek to predict “what computers can’t do” and which cite as evidence the infinitely recursive nature of a particular problem domain may need to be reevaluated — given that a finite number of recursive heuristics may be sufficient to emulate human-caliber performance. And any suppositions about a potentially infinite stack of heuristics may be wholly irrelevant.

I mention this because linguistic systems are known to possess potentially unbridled (and largely uncharted) recursive structures. And incidentally, linguistic systems happen to be an important frontier in AI research. So my question (to paraphrase Turing) is, can a machine be programmed with enough heuristics (of various orders) to function at an apparently high level of linguistic competence? If the answer is yes (which is a huge “if”), then perhaps one way to effect such a breakthrough would be to first begin by programming systems to play very simple languages games that have clearly defined winning conditions and clearly defined transition rules (as mentioned in this post.)

More to come regarding language games.

Simply put, heuristics are rules for governing how a formal system ought to move (where “move” is understood to mean “transition from one state to another according to valid transition rules.”) But for our purposes, we should understand heuristics to have a quality of being definable fully within the system itself. In other words, the heuristic “Control the center” would have been useless to Deep Blue if the idea of “control” and the idea of “center” were impossible to define syntactically. (Note: This should be reminiscent of our similar stipulation that winning conditions be definable within the system itself — an idea mentioned here.)

Furthermore, as we have already mentioned, heuristics that govern a formal system designed to play a game (like chess) can be understood to be much more context-independent than rules of the form, “Look up your position in an opening book and make the indicated move.” It is important to note, however, that context-dependence is not a binary quality — but rather a spectrum. Consider a simple game in which two players are competing to acquire the largest number of stones, which are initially residing in a central pile. The rules are that, on your turn, you may 1) take a stone from the central pile, or 2) put a stone back into the central pile. The best strategy in this (rather boring) game should be obvious: “On your turn, take a stone. Never put one back.”

Within this simple game, the winning heuristic can be considered context-free. It doesn’t matter how many stones are in the central pile, or how many stones your opponent has, or how many previous moves have been played — you always want to take one stone. This heuristic depends on no other game-related concerns.

In chess, most (if not all) rules are more dependent on context than in the previous example. Some rules are more context-dependent than others because they have their context built into them explicitly: i.e. “If your position looks like this (context), always play this move (transition).” But most of the truly interesting heuristics are more context-independent: like, “Control the center,” and “Don’t bring your queen out early.”

Ironically, the really tricky parts of stupefying a task involve trying to apply such more-or-less context-independent heuristics in their proper context. (We should certainly note that, “more-or-less context-independent heuristics” are a far cry from context-free heuristics.) When is it best to control the center? How early is too early with regard to bringing the queen out? Should one bring out one’s queen early if that leads to controlling the center? How does one decide between two context-independent heuristics? How does one decide just how context-independent a particular heuristic is? This will be the subject of the next post.