Oracles Are Subtle But Not Malicious

Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low- degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it implies the first nonrelativizing separation of "traditional" complexity classes, as opposed to interactive proof classes such as MIP and MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does not have circuits of size n^k for any fixed k. We present an alternative proof of this fact, which shows that PP does not even have quantum circuits of size n^k with quantum advice. To our knowledge, this is the first nontrivial lower bound on quantum circuit size. Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be parallelized by any relativizing technique, by giving an oracle relative to which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand, we also show that the NP queries could be parallelized if P=NP. Thus, classes such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish between relativizing and black-box techniques. Our results on this subject have implications for computational learning theory as well as for the circuit minimization problem.Comment: 20 pages, 1 figur

Arguments against 'subject' and 'direct object' as viable concepts in Chinese

Thirty-one years ago Tsu-lin Mei (1961) argued against the traditional doctrine that saw the subject-predicate distinction in grammar as parallel to the particular- universal distinction in logic, as he said it was a reflex of an Indo-European bias, and could not be valid, as ‘Chinese ... does not admit a distinction into subject and predicate’ (p. 153). This has not stopped linguists working on Chinese from attempting to define ‘subject’ (and ‘object’) in Chinese. Though a number of linguists have lamented the difficulties in trying to define these concepts for Chinese (see below), most work done on Chinese still assumes that Chinese must have the same grammatical features as Indo-European, such as having a subject and a direct object, though no attempt is made to justify that view. This paper challenges that view and argues that there has been no grammaticalization of syntactic functions in Chinese. The correct assignment of semantic roles to the constituents of a discourse is done by the listener on the basis of the discourse structure and pragmatics (information flow, inference, relevance, and real world knowledge) (cf. Li & Thompson 1978, 1979; LaPolla 1990)

PSPACE Bounds for Rank-1 Modal Logics

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant proof-theoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way

Quantum vs Classical Proofs and Subset Verification

We study the ability of efficient quantum verifiers to decide properties of exponentially large subsets given either a classical or quantum witness. We develop a general framework that can be used to prove that QCMA machines, with only classical witnesses, cannot verify certain properties of subsets given implicitly via an oracle. We use this framework to prove an oracle separation between QCMA and QMA using an "in-place" permutation oracle, making the first progress on this question since Aaronson and Kuperberg in 2007. We also use the framework to prove a particularly simple standard oracle separation between QCMA and AM.Comment: 23 pages, presentation and notation clarified, small errors fixe

Cognitive constraints and island effects

Competence-based theories of island effects play a central role in generative grammar, yet the graded nature of many syntactic islands has never been properly accounted for. Categorical syntactic accounts of island effects have persisted in spite of a wealth of data suggesting that island effects are not categorical in nature and that nonstructural manipulations that leave island structures intact can radically alter judgments of island violations. We argue here, building on work by Paul Deane, Robert Kluender, and others, that processing factors have the potential to account for this otherwise unexplained variation in acceptability judgments. We report the results of self-paced reading experiments and controlled acceptability studies that explore the relationship between processing costs and judgments of acceptability. In each of the three self-paced reading studies, the data indicate that the processing cost of different types of island violations can be significantly reduced to a degree comparable to that of nonisland filler-gap constructions by manipulating a single nonstructural factor. Moreover, this reduction in processing cost is accompanied by significant improvements in acceptability. This evidence favors the hypothesis that island-violating constructions involve numerous processing pressures that aggregate to drive processing difficulty above a threshold, resulting in unacceptability. We examine the implications of these findings for the grammar of filler-gap dependencies