A probabilistic anytime algorithm for the halting problem

Abstract

The Halting Problem, the most (in)famous undecidable problem, has important applications in theoretical and applied computer science and beyond, hence the interest in its approximate solutions.Experimental results reported on various models of computation suggest that halting programs are not uniformly distributedrunning times play an important role. A reason is that a program which eventually stops but does not halt "quickly", stops at a time which is algorithmically compressible.In this paper we work with running times to define a class of computable probability distributions on the set of halting programs in order to construct an anytime algorithm for the Halting problem with a probabilistic evaluation of the error of the decision.

A Review of Methods for Estimating Algorithmic Complexity: Options, Challenges, and New Directions

“…This is, again, one of the values of approaching algorithmic complexity from a experimental mathematics standpoint, as attested to by Gregory Chaitin himself when officially evaluating CTM [ 54 ] to his own satisfaction. Calude found empirical evidence from CTM that could provide pointers to new results on halting probabilities [ 63 ], as well as confirm and re-examine previous theoretical results. For the first time, access to an empirical output distribution (and ranking) of small strings from a large set of computer programs was possible (see Figure 4 ), despite the challenges and the assumptions made (see Table 2 ).…”

Exploring Halting Times for Unconventional Halting Schemes

“…Per Definition 1 of halting and the TM enumeration schematic used in this study, where a halt state is defined in the TM rule set, 2,2 TMs have a 43.3% chance of halting. A given halting TM is more likely to halt at the first step than any other; that is, the halting probability of a TM decreases with time and will have a smaller chance of halting at every step it progresses, reflecting previous observations such as in [4] or [7]. Furthermore, the distribution of these halting times likely approximated the Levin universal semimeasure distribution.…”

“…As can be easily noted, the number of possible TM rules increases at a great rate, so although TMs reaching up to 6,6 TMs have been studied, the amount of data that would have to be analyzed in order to obtain meaningful results has slightly limited the extent to which these and other more complicated TMs could be analyzed. A TM will generally either halt relatively quickly or never halt, as proven by Calude and Stay [3], though there exist a few exceptions where a TM will only halt after an "algorithmically compressible" amount of time [4]. Within this paper, not only is this theoretical expectation examined numerically, but also the problem of reachability is explored, both in this context and that of a fixed, unchanging point of the machine tape.…”

“…This will be further discussed in Section 3. The time at which a TM halts is a function of its algorithmic (or Kolmogorov-Chaitin) complexity [1,[3][4][5][6], and it is known that with time, the halting probability of a TM that has not halted yet approaches zero [4,7].…”

Spurious, Emergent Laws in Number Worlds

“…These limits can be mitigated from a practical point of view with various methods; for example, the halting problem can be solved probabilistically with arbitrarily high precision[16].…”