co-NP

Abstract: It is a well-known result of Fagin that the complexity class NP coincides with the class of problems expressible in existential second-order logic (Σ11). Monadic NP is the class of problems expressible in monadic Σ11, i.e., Σ11 with the restriction that the second-order quantifiers range only over sets (as opposed to ranging over, say, binary relations). We prove that connectivity of finite graphs is not in monadic NP, even in the presence of arbitrary built-in relations of moderate degree (that is, degree (log n)o(1)). This extends earlier results of Fagin and de Rougemont. Our proof uses a combination of three techniques: (1) an old technique of Hanf for showing that two (infinite) structures agree on all first-order sentences, under certain conditions, (2) a recent new approach to second-order Ehrenfeucht-Fraisse games by Ajtai and Fagin, and (3) playing Ehrenfeucht-Fraisse games over random structures (this was also used by Ajtai and Fagin). Regarding (1), we give a version of Hanf′s result that is better suited for use as a tool in inexpressibility proofs for classes of finite structures. The power of these techniques is further demonstrated by using them (actually, using just the first two techniques) to give a very simple proof of the separation of monadic NP from monadic co-NP without the presence of built-in relations.

Abstract: Many models in theoretical computer science allow for computations or representations where the answer is only slightly biased in the right direction. The best-known of these is the complexity class PP, for "probabilistic polynomial time". A language is in PP if there is a randomized polynomial-time Turing machine whose acceptance probability is greater than 1/2 if, and only if, its input is in the language. Most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PPcc, and a version with unrestricted bias called UPPcc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PPcc is strictly included in UPPcc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials. We will discuss some complexity theoretical consequences of this separation. This presentation is bases on joined work with Nikolay Vereshchagin and Ronald de Wolf.

Abstract: We analyze the computational complexity of various two-dimensional platform games. We state and prove several meta-theorems that identify a class of these games for which the set of solvable levels is NP-hard, and another class for which the set is even PSPACE-hard. Notably COMMANDERKEEN is shown to be NP-hard, and PRINCE OF PERSIA is shown to be PSPACE-complete. We then analyze the related game Lemmings, where we construct a set of instances which only have exponentially long solutions. This shows that an assumption by Cormode in [3] is false and invalidates the proof that the general version of the LEMMINGS decision problem is in NP. We then augment our construction to only include one entrance, which makes our instances perfectly natural within the context of the original game.