Relativizations of the P =? DNP Question for the BSS Model

We consider the uniform BSS model of computation where the machines can perform additions, multiplications, and tests of the form $xgeq 0$. The oracle machines can also check whether a tuple of real numbers belongs to a given oracle set ${cal O}$ or not. We construct oracles such that the classes P and DNP relative to these oracles are equal or not equal

Logics for complexity classes

A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered structures, and another form to define them on unordered non-Aristotelian structures. Using the canonical forms, logics are developed for complete problems in various complexity classes. Evidence is shown that there cannot be any complete problem on Aristotelian structures for several complexity classes. Our approach is extended beyond complete problems. Using a similar form, a logic is developed to capture the complexity class $NP\cap coNP$ which very likely contains no complete problem.Comment: This article has been accepted for publication in Logic Journal of IGPL Published by Oxford University Press; 23 pages, 2 figure

On the Impossibility of Probabilistic Proofs in Relativized Worlds

We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or nondeterministic computations that make queries to the oracle. This question is intimately related to a recent line of work that seeks to improve the efficiency of probabilistic proofs for computations that use functionalities such as cryptographic hash functions and digital signatures, by instantiating them via constructions that are "friendly" to known constructions of probabilistic proofs. Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it. We prove several impossibility results for probabilistic proofs relative to natural oracles. Our results provide strong evidence that tailoring certain natural functionalities to known probabilistic proofs is inherent

Strong Turing Degrees for Additive BSS RAM's

For the additive real BSS machines using only constants 0 and 1 and order tests we consider the corresponding Turing reducibility and characterize some semi-decidable decision problems over the reals. In order to refine, step-by-step, a linear hierarchy of Turing degrees with respect to this model, we define several halting problems for classes of additive machines with different abilities and construct further suitable decision problems. In the construction we use methods of the classical recursion theory as well as techniques for proving bounds resulting from algebraic properties. In this way we extend a known hierarchy of problems below the halting problem for the additive machines using only equality tests and we present a further subhierarchy of semi-decidable problems between the halting problems for the additive machines using only equality tests and using order tests, respectively

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)