Recursion-Theoretic Alternation

Abstract

We introduce four recursion schemes, which, operating on a tree-like data structure, capture different models of computation based on alternating bounded quantifiers. By encoding inputs as paths, we recover and expand characterizations of complexity classes between deterministic linear time and polynomial space; by encoding them as balanced trees, we recover characterizations of alternating logarithmic time and polylogarithmic space.

We propose recursion-theoretic characterizations of logarithmic and polylogarithmic time, as defined via Turing machines with random access to the input, and show that the classes of functions obtained capture, at least, the desired classes, and, at most, their alternating versions.

Should the proposed characterizations be precise, we show that characterizations of linear and polynomially bounded alternating classes can be adapted to alternating classes with logarithmic and polylogarithmic resource bounds, simply by changing the way in which inputs are encoded. We discuss how, from these characterizations, some open problems in complexity theory can be obtained from known results by making alterations to recursion schemes.

Appendix

We have used the work of Bloch, in [6], to justify the lower bound for our characterization of LIN (Theorem 1), and in [7], to justify the upper bound for our characterization of ALOG and POLYLOGSPACE. In this Appendix, we discuss the relationship between our setting and his, to facilitate the comparison between the two perspectives.

Regarding LIN, all the base functions in Definition 7 of [6] have a corresponding function in our Definition 1, when inputs are encoded as paths. For example, the function Half$(;x) = \lfloor \frac{x}{2}\rfloor$ there corresponds to our L, when inputs are encoded as paths. The difference then is that Bloch uses simultaneous recursion instead of our regular recursion scheme. However, since we have $*$, L and R as initial functions, we can replace a system of equations defining $f_1,\dots ,f_k$ by a single recursion scheme, defining $f = f_1*\dots *f_k$ with a step function which uses L and R to extract each $f_i$ from $f_1*\dots *f_k$, applies to it its respective step function, and encodes the result again.

Regarding ALOG and POLYLOGSPACE, rewriting from our tree-like data structure to binary inputs, all the functions from our Definition 1 are in the base class of [7] (Definition 1 there). For example, when inputs are encoded as balanced trees, our $*$ corresponds to the concatenation function and our $\textsf{L}$ and $\textsf{R}$ to the “front half” and “back half” functions. Definitions 6 and 7 of [7] contain the schemes of safe and very safe divide and conquer recursion, which allow a function to call itself on the back and front halves of the input word. In our data structure, this corresponds to, without a path parameter, allowing a function on input $u*v$ to call itself on input u and v, which generalizes our path-recursion schemes, where the function has to choose which path to follow. This means that, when inputs are encoded as balanced trees, our schemes pSR and vpSR can be simulated by the respective safe and very safe forms of divide and conquer recursion. Thus, our classes $[\mathcal {I};\textsf{SC},\textsf{vpSR}]$ and $[\mathcal {I};\textsf{SC},\textsf{pSR}]$ characterize, at most, ALOG and APOLYLOG, respectively (Theorems 14 and 24 of [7]). As AR introduces only the power of alternation in a class (see Theorem 4), and ALOG and APOLYLOG are already alternating classes, we get that $[\mathcal {I};\textsf{SC},\textsf{vpSR},\textsf{AR}]$ and $[\mathcal {I};\textsf{SC},\textsf{pSR},\textsf{AR}]$ also characterize, at most, ALOG and APOLYLOG, respectively, which is the desired upper bound.