Logical Definability of Counting Functions

Kevin J. Compton, Erich Grädel
1996 Journal of computer and system sciences (Print)  
The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is *P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logic L, *L is the class of functions on finite structures counting the tuples (T , cÄ ) satisfying a given formula (T , cÄ ) in L. Saluja, Subrahmanyam, and Thakur showed that on classes of ordered structures *FO=*P (where
more » ... denotes first-order logic) and that every function in * 1 has a fully polynomial randomized approximation scheme. We give a probabilistic criterion for membership in * 1 . A consequence is that functions counting the number of cliques, the number of Hamilton cycles, and the number of pairs with distance greater than two in a graph, are not contained in * 1 . It is shown that on ordered structures * 1 1 captures the previously studied class spanP. On unordered structures *FO is a proper subclass of *P and * 1 1 is a proper subclass of spanP; in fact, no class *L contains all polynomial-time computable functions on unordered structures. However, it is shown that on unordered structures every function in *P is identical almost everywhere with some function *FO, and similarly for * 1 1 and spanP. Finally, we discuss the closure properties of *FO under arithmetical operations. ]
doi:10.1006/jcss.1996.0069 fatcat:fksc6h5r2jdg5nohcko3pxeawq