Showing papers by "Erich Grädel published in 1991"

On Transitive Closure Logic

Abstract: We present Ehrenfeucht-Fraisse games for transitive closure logic (FO + TC) and for quantifier classes in (FO + TC). With this method we investigate the fine structure of positive transitive closure logic (FO + pos TC), and identify an infinite quantifier hierarchy inside (FO + pos TC), formed by interleaving universal quantifiers and TC-operators.

The expressive power of second order Horn logic

Abstract: We investigate second order Horn logic, the restriction of second order logic to formulae whose first order part is a universal Horn formula. It is shown that this logic collapses to its existential fragment. In the presence of a successor relation, second order Horn logic has the same expressive power as fixpoint logic and therefore captures precisely the class of polynomial time computable queries. In the absence of the successor relation this logic is strictly weaker than fixed point logic.

Capturing complexity classes by fragments of second order logic

Abstract: The expressive power of certain fragments of second-order logic on finite structures is investigated. The fragments are second-order Horn logic, second-order Krom logic, and a symmetric and a deterministic version of the latter. It is shown that all these logics collapse to their existential fragments. In the presence of a successor relation they provide characterizations of polynomial time, deterministic and nondeterministic logspace and of the complement of symmetric logspace. Without a successor relation these logics can still express certain problems that are complete in the corresponding complexity classes, but they are strictly weaker than previously known logics for these classes and fail to express some very simple properties. >

Simple sentences that are hard to decide

Abstract: It was considered to be “typical for first order theories” that a restriction to sentences with only a limited number of quantifier alternations leads to an exponential decrease of complexity. Using domino games, which were treated in a previous paper to describe computations of alternating Turing machines, we prove that this is not always true. We present a list of theories, all of them decidable in ⌣ c>0 ATIME (2 cn , n) , for which the subclasses with bounded quantifier alternations still have alternating exponential time complexity. In particular this yields non-deterministic exponential time lower bounds for very simple prefix classes (with 2 or 3 alternations). Theories with such behaviour are the theory of Boolean algebras, the theory of polynomial rings over finite fields, the theory of idempotent rings, the theory of finite sets with inclusion, the theory of semilattices, the theory of Stone algebras, the theory of distributive p-algebras in the Lee-class B n, and the theories of natural numbers with divisibility or coprimeness.