The complexity of tensor calculus

Tensor calculus over semirings is shown relevant to complexity theory in unexpected ways. First, evaluating well-formed tensor formulas with explicit tensor entries is shown complete for $\bigoplusP$, for NP, and for #P as the semiring varies. Indeed the permanent of a matrix is shown expressible as the value of a tensor formula in much the same way that Berkowitzs theorem expresses its determinant. Second, restricted tensor formulas are shown to capture the classes LOGCFL and NL, their parity counterparts $\bigoplusLOGCFL$ and $\bigoplusL$, and several other counting classes. Finally, the known inclusions $\NP/\poly \subseteq \bigoplusP/\poly$, $\LOGCFL/\poly \subseteq \bigoplusLOGCFL/\poly$, and $\NL/\poly \subseteq \bigoplusL/\poly$, which have scattered proofs in the literature (Valiant & Vazirani 1986; Gál & Wigderson 1996), are shown to follow from the new characterizations in a single blow. As an intermediate tool, we define and make use of the natural notion of an algebraic Turing machine over a semiring $ \mathcal{S}$.     

Hardness Hypotheses, Derandomization, and Circuit Complexity

We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences:

Space Complexity Vs. Query Complexity

Combinatorial property testing deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x satisfies the property or is “far” from satisfying it. The main focus of property testing is in identifying large families of properties that can be tested with a certain number of queries to the input. In this paper we study the relation between the space complexity of a language and its query complexity. Our main result is that for any space complexity s(n) ≤ log n there is a language with space complexity O(s(n)) and query complexity 2^Ω(s(n)). Our result has implications with respect to testing languages accepted by certain restricted machines. Alon et al. [FOCS 1999] have shown that any regular language is testable with a constant number of queries. It is well known that any language in space o(log log n) is regular, thus implying that such languages can be so tested. It was previously known that there are languages in space O(log n) that are not testable with a constant number of queries and Newman [FOCS 2000] raised the question of closing the exponential gap between these two results. A special case of our main result resolves this problem as it implies that there is a language in space O(log log n) that is not testable with a constant number of queries. It was also previously known that the class of testable properties cannot be extended to all context-free languages. We further show that one cannot even extend the family of testable languages to the class of languages accepted by single counter machines.      

Inverse NP Problems

The inverse problem relative to a verifier V of proofs of membership for a NP language is the problem of deciding, given a set π of proofs, whether or not there exists a string x having exactly π as its set of proofs. In this paper, we study the complexity of inverse problems. We develop a new notion of reduction which allows one to compare the complexity of inverse problems. Using this notion, we classify as coNP-complete the inverse problems for the “natural” verifiers of many NP-complete problems. We also show that the inverse complexity of a verifier for a language L cannot be predicted solely from the complexity of L, but rather, is highly dependent upon the choice of verifier used to accept L. In this context, a verifier with a Σ₂ ^p -complete inverse problem is exhibited, giving a new and natural example of a Σ₂ ^p -complete problem.