Monday, June 23, 2008

More Hyperlogic

In the previous hyperlogic post I described loosely the arithmetical hierarchy, and a "hyperlogic" to deal with it. Although I didn't describe this in detail, the definition of this hyperlogic was inspired by a particular set of hypercomputers that can calculate functions in the arithmetical hierarchy.

The particular hypercomputers that I was thinking of are infinite-time turing machines, but restricted to change each output square only once during each infinite subcomputation. An infinite subcomputation is the size of the first ordinal infinity, whereas the computation as a whole may be logner (and thus have many infinite subcomputations); the number of infinite subcomputations is determined by the level on the hierarchy. (Additionally I need to stipulate that the output squares can start out either all black or all white depending on if we are calculating an existential (assumed false) or a universal (assumed true).)

So, to extend the hyperlogic further, it seems as if I need a more powerful hypercomputer to use as a basis.

After the arithmetical hierarchy comes the analytical hierarchy. The shift is from 1st-order arithmetic to 2nd-order arithmetic. (It is important to distinguish this from 1st-order logic and 2nd-order logic. 1st order logic is complete and doesn't need hypercomputers to decide its quantifiers, while 1st-order arithmetic is incomplete and does need hypercomputation.) In the arithmetical hierarchy, we are only quantifying over integers. In the analytic hierarchy, we quantify over sets of integers. The space being quantified over is infinitely larger, in a way that makes the previous methods fail.

The class of infinite-time turing machines I was considering uses only countable infinities of time. The analytic hierarchy requires quantification over an uncountable set, meaning it needs uncountable infinities of time.

An uncountable infinity of turing-machine steps may seem strange (or worse, incoherent), but if we ignore the details it works. It doesn't even matter what order everything happens in; since all I need is existential and universal quantification, I can just say that the machine stops and returns true (/false) if an example (/counterexample) is found, and otherwise keeps going until it has checked everything (at which point it returns the opposite truth value).

I do need to specify further how "checking an individual case" works. But this is not too difficult. Quantifying over sets of numbers gives us in each case some predicate that returns either true or false for each number. If there are more than one such quantifiers, we get more than one predicates. Ignoring the inner workings of the predicates, we just use them as given in what is otherwise a normal case of arithmetical hypercomputation.

Now, can we convert this to a finite process that in some sense captures the concept?

My tentative proposal is: look at a generic case of the problem, converting all set quantifiers to undefined predicates. Run the decision procedure for this arithmetical statement. Every time we need a particular value of one of the predicates, split the computation; run one version with the value of the predicate being "true" for the number it was given, and a second version with the value being "false". The result (after all versions of the arithmetical decision process terminate) will be a table with the number of dimensions determined by the number of set quantifiers, and the number of entries determined by how many versions of each quantifier ended up being needed. In any case we can attribute an approximate truth value to the statement by deciding each quantifier in order based on the examples examined.

Now, I am less sure about this method than I was about the arithmetical method. It seemed that improvements to the arithmetical method could never cause a serious increase in power, so that the method was within the realm of "the best we can do". However, for this method it is not clear that the machine is doing the best that can be done. In particular, I am not sure that adding direct 2nd-order theorem proving would not seriously increase the power.

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