Relations between diagonalization, proof systems, and complexity gaps

AbstractIn this paper we study diagonal processes over time bounded computations of one-tape Turing machines by diagonalizing only over those machines for which there exist formal proofs that they operate in the given time bound. This replaces the traditional “clock” in resource bounded diagonalization by formal proofs about running times and establishes close relations between properties of proof systems and existence of sharp time bounds for one-tape Turing machine complexity classes. These diagonalization methods also show that the Gap Theorem for resource bounded computations can hold only for those complexity classes which differ from the corresponding provable complexity classes. Furthermore, we show that there exist recursive time bounds T(n) such that the class of languages for which we can formally prove the existence of Turing machines which accept them in time T(n) differs from the class of languages accepted by Turing machines for which we can prove formally that they run in time T(n). We also investigate the corresponding problems for tape bound computations and discuss the difference time and tapebounded computations

On the Structure of Solutions of Computable Real Functions

The relationship between the structure of a domain and the complexity of computing over that domain is a fundamental question of computer science. This paper studies how the structure of the real numbers constrains the behavior of computable real functions. In particular, we uncover a close correlation between the structure of the zero set of a computable real function, and the complexity of the zeros. We show that computable real functions with hard solutions perforce have many solutions. Furthermore, as the complexity of solutions increases, the number of solutions increases. We prove that computable real functions with nonrecursive, nonarithmetical, or random zeros have solution sets that are, respectively, infinite,“˜ uncountable, or of positive measure. In addition, we show that the computational complexity of the zero set of a computable real function is limited by its topological complexity. These results suggest an emerging paradigm-the inability of machines to name complex strings can serve as the basis of powerful proof techniques in computational complexity theory

On Uniformity and Circuit Lower Bounds

Abstract—We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1. Lower Bounds Against Medium-Uniform Circuits. Informally, a circuit class is “medium uniform ” if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium uniform circuit classes, including: • For all k, P is not contained in P-uniform SIZE(n k). That is, for all k there is a language Lk ∈ P that does not have O(n k)-size circuits constructible in polynomial time. This improves Kannan’s lower bound from 1982 that NP is not in P-uniform SIZE(n k) for any fixed k

A Conversation with Juris Hartmanis

Juris Hartmanis is video taped in a far-reaching conversation (70 minutes) with colleague David Gries. They discuss Hartmanis’ childhood and family background and his immigration to the United States. Next they trace his extraordinary career at the GE Research Laboratory, where he collaborated with Richard Stearns on pioneering research that eventually was recognized by ACM’s prestigious, highest honor – the Turing Award. After having served earlier as an Instructor in Cornell’s Mathematics Department, Juris returned to Cornell as a full professor and the founding chair of a new department of Computer Science. This Department was embedded in two colleges, Engineering and Arts and Sciences. Cornell was among the first Universities to establish a Department of Computer Science. His pioneering work on computational complexity blossomed into a new field and under his leadership the Computer Science department matured into a robust, national leader with a strong theoretical emphasis. After a successful stint at the National Science Foundation leading the transition of the academic research network NSFnet to become the Internet, he returned to Cornell. At Cornell he continues an active program of research and maintains a leadership role in developing information technologies that have become a ubiquitous element across the entire Cornell academic scene.Juris Hartmanis joined Cornell in 1965 as the founding chair of the new Department of Computer Science. One of the first CS departments (the first started in 1964), CS was embedded in two colleges, Engineering and Arts & Sciences. Under his leadership, CS matured into a robust, national leader with a strong theoretical emphasis. Juris came from GE, where he collaborated with Richard Stearns on pioneering research that was later recognized by ACM’s prestigious, highest honor: the Turing Award. Fittingly, Juris is known as “the father of computational complexity”. He is a member of the NAE and NAS, has honorary doctorates, and received the Grand Medal of the Latvian Academy of Sciences. Like most of the CS faculty, Juris spent time in the service of the CS community. He chaired a National Research Council Study, resulting in the book “Computing the Future”. In 1996-1998, he was Assistant Director of the NSF Directorate of Computer and Information Science and Engineering (CISE). In this conversation (70 minutes), Juris and David talk about his childhood, his family background, his immigration to the US, and his career. Running Time: 70 min. http://hdl.handle.net/1813/149341_u9odoqp

Some embedding theorems for lattices

In this thesis we give a general definition of a geometry on a set S and consider the lattices of the subspaces of these geometries. First, we show that all such geometries on a fixed set S form a lattice and we investigate its properties. Secondly, we show that the lattice of all geometries on a fixed set S is isomorphic to the lattice of subspaces of some geometry and we characterize all such geometries. Finally, we show that every finite lattice can be embedded in the lattice of all geometries on some finite set S. This reduces the unsolved problem of embedding every finite lattice into a finite partition lattice to the problem of embedding every finite lattice of geometries into a finite partition lattice

Computational Complexity of Formal Translations

The purpose of this paper is to define a mathematical model for the study of quantitative problems about translations between universal languages and to investigate such problems. The results derived in this paper deal with the efficiency of the translated algorithms, the optimality of translations and the complexity of the translation process between different languages. Keywords: universal languages, Goedel numberings, translations, complexity of translations, optimality, length of translated programs

On the Succintness of Different Representations of Languages

The purpose of this paper is to give simple new proofs of some interesting recent results about the relative succintness of different representations of regular, deterministic and unambiguous context-free languages and to derive some new results about how the relative succintness of representations change when the representations contain a formal proof that the languages generated are in the desired subclass of languages

Generalized Kolmogorov Complexity and the Structure of Feasible Computations

In this paper we define a generalized, two-parameter, Kolmogorov complexity of finite strings which measures how much and how fast a string can be compressed and we show that this string complexity measure is an efficient tool for the study of computational complexity. The advantage of this approach is that it not only classifies strings as random or not random, but measures the amount of randomness detectable in a given time. This permits the study how computations change the amount of randomness of finite strings and thus establish a direct link between computational complexity and generalized Kolmogorov complexity of strings. This approach gives a new viewpoint for computational complexity theory, yields natural formulations of new problems and yields new results about the structure of feasible computations

On Non-Isomorphic NP Complete Sets

In this note we show that if the satisfiability of Boolean formulas of low Kolmogorov complexity can be determined in polynomial-time then there exist NP complete sets that are not polynomial-time isomorphic. Keywords: NP complete sets, isomorphism, Kolmogorov complexity