Research Statement. Donald M. Stull. December 29, 2016

Transcription

1 Research Statement Donald M. Stull December 29, Overview My research is focused on the interaction between analysis and theoretical computer science. Typically, theoretical computer science is concerned with discrete objects such as the natural numbers or finite graphs. However, there is a deep connection with classical analysis and the theory of computing. I am primarily interested in the ways theoretical computer science can be used to study the structures and objects of analysis. An example of this is the field of algorithmic randomness. Algorithmic randomness uses the theory of computing to give a meaningful definition of a real number being random. Classically, such a notion would seem paradoxical, as any individual point has probability zero. A growing body of work has shown that there is a deep connection between algorithmic randomness and measure-theoretic analysis. Researchers have used classical theorems from analysis to characterize various notions of algorithmic randomness. My work has contributed to this area by focusing on this connection in the polynomial space setting. In joint work with Xiang Huang [4], we introduced a new notion of polynomial space randomness, weak PSPACE randomness. We then characterized this notion using the Lebesgue Differentiation Theorem. Kolmogorov complexity is a closely related subject that interests me. Kolmogorov complexity makes essential use of theoretical computer science to define the randomness of an individual binary string. Mayordomo [9] and Lutz [6] have given a definition of computable Hausdorff dimension using Kolmogorov complexity. With this definition, tools from theoretical computer science can be used to study a variety of mathematical fields, including fractal geometry and geometric measure theory. In joint work with Neil Lutz, we use Kolmogorov complexity to study the computable Hausdorff dimension of lines in Euclidean space [8]. We show that the dimensions of the points on a line are highly dependent on the dimension of the slope and intercept of 1

2 a line. As a consequence, we improve on a bound of Molter and Rela for Furstenberg sets [12]. The emerging field of molecular programming also interests me. Molecular programming uses techniques from computer science and molecular biology to achieve fine grain control of molecules. Recent work in this area has come to view chemical reaction networks (CRNs) as a programming language for bio-molecular systems. Many of the promising applications of molecular programming are in the bio-medical field. It is, therefore, vital to have necessary tools to analyze the behavior of CRNs. Reachability analysis is a common tool in analyzing safety critical distributed systems. In joint work with Adam Case and Jack Lutz, we study the computational complexity of reachability problems in continuous chemical reaction networks [2]. We prove that for continuous CRNs, the reachability problem can be computed in polynomial time. Another of my research directions focuses on computable analysis. This field studies computability theory in the continuous domains common in analysis and topology. I am interested in studying a classical area of computability theory, computable categoricity, in the context of computable analysis. Computable categoricity has historically been studied for discrete structures such as N and Q. In joint work with Tim McNicholl, we have shown that the degree of categoricity for l p is characterized by the c.e. degrees [11]. 2 Algorithmic Randomness and Analysis Algorithmic randomness uses computability theory to study the intrinsic randomness of real numbers (infinite binary sequences). Jack Lutz initiated the study of resource-bounded algorithmic randomness using martingales [5]. This allows for a definition of a randomness relative to a computationally limited observer. Kolmogorov complexity theory is a closely related area, which studies the intrinsic randomness of finite binary strings. The connection between these fields and analysis intrigues me. 2.1 Resource-bounded Randomness and Analysis A recent line of investigation has shown a deep connection between algorithmic randomness and measure-theoretic analysis, mediated by computable analysis. In measure-theoretic analysis, theorems state that a property P holds for almost every real number; i.e., P holds except for an exceptional set of measure zero. By adding computability restrictions, researchers have been able to characterize various notions of algorithmic randomness using classical 2

3 theorems from measure-theoretic analysis. As a theoretical computer scientist, I am interested in whether this connection holds in the resource-bounded setting. That is, can we characterize polynomial time or space randomness using measure theoretic analysis? While there have been some results showing this connection does exist, the question is still not as understood as in the computable setting. In joint work with Xiang Huang, we address this question in the context of polynomial space randomness [4]. We first define a new notion of polynomial space randomness, weak PSPACE randomness. We show that Lutz PSPACE randomness implies weak PSPACE randomness. We then study the connection between weak PSPACE randomness and the Lebesgue Differentiation Theorem. In the computable setting, the Lebesgue Differentiation Theorem has been used to characterize Schnorr randomness. We prove that this connection persists in the polynomial space setting by showing that the Lebesgue Differentiation Theorem characterizes weak PSPACE randomness. 2.2 Kolmogorov Complexity and Geometric Measure Theory The Kolmogorov complexity of a finite string w is defined to be the shortest program π such that, when π is run on a universal Turing Machine, it produces x on its output tape. The Hausdorff dimension of a set E R n, dim H (E), gives a notion of the size of E. It has proven highly useful in many areas of mathematics, particularly fractal geometry and geometric measure theory. In 2000, Jack Lutz developed the computable Hausdorff dimension of points, dim, using s-gales. Mayordomo [9] and Lutz [6] subsequently characterized computable Hausdorff dimension using Kolmogorov complexity. They showed that dim(x) = lim inf r K r(x) r, where, roughly, K r (x) is the Kolmogorov complexity of the first r bits of x. Jack and Neil Lutz used this characterization to show that the classical Hausdorff dimension of a set is determined by the computable Hausdorff dimension of its points [7]. Formally, they show that, for any set E R n, dim H (E) = min sup A N x E dim A (x). (1) With this characterization, tools for theoretical computer science can be used to understand the Hausdorff dimension of sets. Many longstanding conjectures of geometric measure theory are on the Hausdorff dimension of sets involving lines. The two most prominent being the Kakeya and Nikodym conjectures, which are still open for dimensions 3

4 n > 2. In joint work with Neil Lutz, we study the computable Hausdorff dimension of lines in the Euclidean plane [8]. We prove that the computable Hausdorff dimensions of the points on a line are highly dependent on the dimensions of the slope and intercept. With this result, we are able to resolve (in the negative) an open problem asked by Joseph Miller. We are also able to give new proofs of the Kakeya and Nikodym problems for R 2. Furthermore, through the characterization of (1), we are able to improve the bounds of Furstenberg sets due to Molter and Rela [12]. 3 CRNs and Molecular Programming Chemical reaction networks (CRNs) model the behavior of molecules in a well-mixed solution. The emerging field of molecular programming uses CRNs not only as a descriptive tool, but as a programming language for chemical computation. Recently, Chen, Doty and Soloveichik introduced rate-independent continuous CRNs (CCRNs) to study the chemical computation of continuous functions [3]. In joint work with Adam Case and Jack Lutz, we study the reachability problem for this model of chemical kinetics [2]. The reachability problem asks whether, given two states c and d, if there is a path in the network from c to d. We show that this problem for continuous CRNs, CCRN-REACH, is decidable in polynomial time. We also study a closely related problem, Sub- CCRN-REACH. In Sub-CCRN-REACH, we are given two states c, d and an integer k. In this problem, the question is whether d is reachable from c using at most k reactions of the network. In contrast to the computational ease of the reachability problem, we show that Sub-CCRN-REACH is NP-complete. 4 Computable Analysis Computable analysis allows for the study of computation over continuous domains. This allows for well defined notions of computable real numbers, computable functions f : R R and other objects typically studied in classical analysis. In computable structure theory, an important area of research is computable categoricity. This subject studies the computability of isomorphisms between discrete structures. A typical question is, if A and B are isomorphic structures, are they computably isomorphic? Structures where isomorphic structures are computably isomorphic are called categorical. For example, Q is categorical, whereas N is not. Recent work has studied categoricity in the context of computable analysis. Tim McNicholl showed that, for every computable real p 2, l p is not computably categorical [10]. In 4

5 joint work with Tim McNicholl, we study the isometry degree of l p [11]. The isometry degree is the least powerful oracle that computes an isomorphism between any two of its computable copies. We show that, when p is a computable real so that p 1 and p 2, the isometry degrees of the computable copies of l p are precisely the c.e. degrees. 4.1 Complexity Theory Also among my interests is the field of computational complexity. In particular, I enjoy thinking about the connection between derandomization and circuit lower bounds. Circuit lower bounds have been the object of intense study in complexity theory. Apart from their intrinsic interest, circuit lower bounds have a deep connection with uniform complexity classes. Indeed, proving that NP does not have polynomial sized boolean circuits implies that P NP. Circuit lower bounds are also highly relevant for derandomization. My work in this area gives a downward separation for Σ 2 -time classes. Specifically, I prove that, if Σ 2 E does not have polynomial size non-deterministic circuits, then Σ 2 SubEXP does not have fixed polynomial size non-deterministic circuits. To achieve this result, I use Santhanam s technique [13] on augmented Arthur-Merlin protocols defined by Aydinlioğlu and van Melkebeek [1]. In proving this result, I give an unconditional circuit lower bound which may be of independent interest. Specifically, I show that augmented Arthur-Merlin protocols with one bit of advice do not have fixed polynomial size non-deterministic circuits. Furthermore, I also give a weak unconditional derandomization of a certain type of promise Arthur-Merlin protocols. Using Williams easy hitting set technique [14], we show that Σ 2 - promise AM problems can be decided in Σ 2 SubEXP with n c advice, for some fixed constant c. 5 Future Directions I am highly interested in continuing the lines of research described in the previous sections. I would also like to investigate new research directions in a variety of topics in theoretical computer science. My previous work has opened up many intriguing questions for future research. In this section, I will highlight several questions on which I plan to focus in the immediate future. 5

6 5.1 Resource-bounded Randomness The new notion of polynomial space randomness, weak PSPACE randomness, that Xiang Huang and I have defined leaves many directions for future research. One of the most promising is its connection to complexity theory. Lutz originally defined his notion resource-bounded randomness in order to study complexity theory [5]. This direction has proved very fruitful, and is the basis of the Lutz hypothesis in complexity theory. I would like to pursue an analogous direction in the context of weak PSPACE randomness. A closely related direction is to extend weak PSPACE randomness to other computational resources. For example, I would like to define and investigate weak polynomial time randomness. Additionally, I am interested in continuing my research into the connection between resource-bounded randomness and measure theoretic analysis. In tandem with defining randomness for various resources, I would like to characterize these notions with classical theorems of analysis. One of the theorems which as been most studied in this direction for the computable setting is the celebrated Birkhoff Ergodic Theorem. I am currently investigating the resource-bounded randomness notions characterized by this theorem. 5.2 Kolmogorov Complexity and Geometric Measure Theory The point-to-set theorem of Jack and Neil Lutz [7] allowed for theoretical computer science to make contributions to fractal geometry and geometric measure theory. I wish to continue my work in this area. Specifically, I would like to continue the work I have started with Neil Lutz of investigating the dimension of points on lines in Euclidean space. I am currently working with Neil Lutz on giving a complete characterization of the dimension spectra for the dimensions of lines in two dimensions. I believe a deeper understanding of the computable dimension of points on lines would help make progress on several longstanding problems in geometric measure theory. 5.3 CRNs and Molecular Programming I am very interested in further contributing to the field of molecular programming. Chemical reaction networks open up many avenues of research for theoretical computer science to make an impact in molecular programming. I would like to study the computational power of different models of CRNs, and their connection to computable analysis. 6

7 5.4 Computable Analysis The extension of computable categoricity to continuous domains opens many new avenues of research. Of immediate interest is the question of which structures of computable analysis are computably categorical. I am currently investigating this question for L p spaces with Tim McNicholl and Joseph Clanin. As a theoretical computer scientist, I am also interested in these questions in the context of resource-bounded computable analysis. I intend to study polynomial time and space categoricity for a variety of continuous structures. References [1] Baris Aydinlioğlu and Dieter van Melkebeek. Nondeterministic circuit lower bounds from mildly de-randomizing arthur-merlin games. In Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, June 26-29, 2012, pages IEEE Computer Society, [2] Adam Case, Jack H. Lutz, and D. M. Stull. Reachability problems for continuous chemical reaction networks. In Unconventional Computation and Natural Computation - 15th International Conference, UCNC 2016, Manchester, UK, July 11-15, 2016, Proceedings, pages 1 10, [3] Ho-Lin Chen, David Doty, and David Soloveichik. Rate-independent computation in continuous chemical reaction networks. In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS 14, Princeton, NJ, USA, January 12-14, 2014, pages ACM, [4] Xiang Huang and D. M. Stull. Polynomial space randomness in analysis. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, Kraków, Poland, volume 58 of LIPIcs, pages 86:1 86:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, [5] Jack H. Lutz. Almost everywhere high nonuniform complexity. J. Comput. Syst. Sci., 44(2): , [6] Jack H. Lutz. The dimensions of individual strings and sequences. Inf. Comput., 187(1):49 79,

8 [7] Jack H. Lutz and Neil Lutz. Algorithmic information and plane kakeya sets. CoRR, abs/ , [8] Neil Lutz and D. M. Stull. Bounding the dimension of points on a line. submitted, [9] Elvira Mayordomo. Kolmogorov complexity characterization of constructive hausdorff dimension. Inf. Process. Lett., 84(1):1 3, [10] Timothy H McNicholl. Computable copies of l p. arxiv preprint arxiv: , [11] Timothy H McNicholl and D. M. Stull. The isometry degree of a computable copy of l p. arxiv preprint arxiv: , [12] Ursula Molter and Ezequiel Rela. Furstenberg sets for a fractal set of directions. Proceedings of the American Mathematical Society, 140(8): , [13] Rahul Santhanam. Circuit lower bounds for merlin arthur classes. SIAM J. Comput., 39(3): , [14] R. Ryan Williams. Natural proofs versus derandomization. SIAM J. Comput., 45(2): ,