Infinite Versions of Some Problems From Finite Complexity Theory

Recently, several authors have explored the connections between NP-complete problems for finite objects and the complexity of their analogs for infinite objects. In this paper, we will categorize infinite versions of several problems arising from finite complexity theory in terms of their recursion theoretic complexity and proof theoretic strength. These infinite analogs can behave in a variety of unexpected ways. Startling parallels exist between the computational complexity of certain graph theoretic problems and the recursion theoretic complexity and proof theoretic strength of their infinite analogs. For example, the problem of deciding which finite graphs have an Euler path is known to be P-time computable [9], and Beigel and Gasarch [4] have shown that the problem of deciding which infinite recursive graphs have an Euler path is arithmetical. By contrast, the problem of deciding which finite graphs have Hamilton paths is NP-complete [8], and Harel [6] has shown that the problem of deciding which infinite recursive graphs have a Hamilton graph is Σ11 complete. Thus, the possibly greater computational complexity is paralleled by a demonstrable increase in recursion theoretic complexity. This pattern can also be seen through an application of the techniques of reverse mathematics. The 1991 Mathematics Subject Classification. 03F35, 03D45.