Complexity theory is the part of theoretical computer science that attempts to prove that certain transformations from input to output are impossible to compute using a reasonable amount of resources. Theorem 1 below illustrates the type of ‘‘impossibility’’ proof that can sometimes be obtained (1); it talks about the problem of determining whether a logic formula in a certain formalism (abbrev...

I would like to say that I am thrilled to take over as editor of this column. But I am more \daunted" than \thrilled". Juris Hartmanis started this column back in issue # 31 of the Bulletin, and it has been an extremely successful run. I'm not eager to invite comparison. In upcoming editions of this column, you should expect to nd concise and readable accounts of some exciting developments and ...

McAllester, Given, and Fatima 10] have developed a procedure for infering tax-onomic relationships between classes deened by predicates and relations. Their decision procedure runs in O(n 3) time and O(n 2) space on a sequential random-access machine (RAM). I have investigated the computational complexity of this inference problem with a view to seeing whether faster sequential algorithms or go...

This expository paper describes some of the results of two recent research papers [GOP05,GPSZ05]. The first of these papers proves that every NP-complete set is many-one autoreducible. The second paper proves that every many-one autoreducible set is many-one mitotic. It follows immediately that every NP-complete set is many-one mitotic. Hence, we have the compelling result that every NP-complet...

Complexity is one of the crucial scientific phenomena of our times. There are many possible ways tomeasure the “complexity” of a given finite object taken from a known underlying set of objects. In the information-theoretic sense, an object is treated as “complex”, if many binary bits are necessary to encode the object, that is, many bits are necessary to distinguish it from the remaining objec...

We introduce a quantum lambda calculus inspired by Lafont’s Soft Linear Logic and capturing the polynomial quantum complexity classes EQP, BQP and ZQP. The calculus is based on the ‘‘classical control and quantum data’’ paradigm. This is the first example of a formal systemcapturing quantumcomplexity classes in the spirit of implicit computational complexity — it is machine-free and no explicit...

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Ra...

In this paper we will introduce well motivated definitions of computational complexity on type-2 objects. Similar to the model of Kawamura and Cook (and earlier models) our models will be both based on classical Turing machines and the notion of representations. The main differences to the model of Kawamura and Cook is that we give up type-2-polynomials and, additionally, use parameter based ty...

The complexity classes Nearly-BPP and Med DisP had been recently proposed as limits of eecient computation Yam1 96, Schi 96]. For both classes, a polynomial time bounded algorithm with bounded probabilistic error has to compute correct outputs for at least 1 ? 1 n !(1) of inputs of length n. We generalize this notion to general error probabilities and arbitrary complexity classes. For proving t...