Relativized circuit complexity

Circuit Lower Bounds Collapse Relativized Complexity Classes

Since the publication of Furst, Saxe, and Sipser's seminal paper connecting AC 0 with the polynomial hierarchy FSS84], it has been well known that circuit lower bounds allow you to construct oracles that separate complexity classes. We will show that similar circuit lower bounds allow you to construct oracles that collapse complexity classes. For example, based on H astad's parity lower bound, ...

Proof Complexity of Relativized Statements

The first-order formulas that do not have finite models give rise to uniform families of propositional unsatisfiable formulas, one for each finite cardinality. Motivated by the question of characterizing the class of such formulas whose propositional translations have short resolution refutations, we study their closure properties. Our starting point is the observation that the class of relativ...

Capturing Relativized Complexity Classes without Order

We consider the problem of obtaining logical characterisations of oracle complexity classes. In particular, we consider the complexity classes LOGSPACE NP and PTIME NP. For these classes, characterisations are known in terms of NP computable Lindstrr om quantiiers which hold on ordered structures. We show that these charac-terisations are unlikely to extend to arbitrary (unordered) structures, ...

Relativized Separation of Reversible and Irreversible Space-Time Complexity Classes

Reversible computing can reduce the energy dissipation of computation, which can improve cost-efficiency in some contexts. But the practical applicability of this method depends sensitively on the space and time overhead required by reversible algorithms. Time and space complexity classes for reversible machines match conventional ones, but we conjecture that the joint space-time complexity cla...

Relativized Separation of Reversible and Irreversible Space-Time Complexity Classes