Parameterised Counting in Logspace

One of the most important open problems in finite model theory is the question whether there is a logic characterising efficient computation. While this question usually concerns Ptime, it can also be applied to other complexity classes, and in particular to Logspace which can be seen as a formalisation of efficient computation for big data. One of the strongest candidates for a logic capturing Ptime is Choiceless Polynomial Time (CPT). It is based on the idea of choiceless algorithms, a general model of symmetric computation over abstract structures (rather than their encodings by finite strings). However, there is currently neither a comparably strong candidate for a logic for Logspace, nor a logic transferring the idea of choiceless computation to Logspace. We propose here a notion of Choiceless Logarithmic Space which overcomes some of the obstacles posed by Logspace as a less robust complexity class. The resulting logic is contained in both Logspace and CPT, and is strictly more expressive than all logics for Logspace that have been known so far. Further, we address the question whether this logic can define all Logspace-queries, and prove that this is not the case.

We develop in the language of graph homomorphisms the connection between the Tutte polynomial and the state models of statistical physics. The Tutte polynomial and homomorphism numbers. Spin models and edge coloring models. Connection matrices and the characterization of graph invariants arising from spin models. Homomorphism numbers and invariants of the cycle matroid of a graph. Graph homomorphism numbers as evaluations of graph polynomials. Other graph polynomials from counting graph homomorphisms such as the independence polynomial, the Averbouch–Godlin–Makowsky polynomial, and the Tittmann–Averbouch–Makowsky polynomial.

Pseudo-Boolean constraints are omnipresent in practical applications, and thus a significant effort has been devoted to the development of good SAT encoding techniques for them. Some of these encodings first construct a Binary Decision Diagram (BDD) for the constraint, and then encode the BDD into a propositional formula. These BDD-based approaches have some important advantages, such as not being dependent on the size of the coefficients, or being able to share the same BDD for representing many constraints. We first focus on the size of the resulting BDDs, which was considered to be an open problem in our research community. We report on previous work where it was proved that there are Pseudo-Boolean constraints for which no polynomial BDD exists. We also give an alternative and simpler proof assuming that NP is different from Co-NP. More interestingly, here we also show how to overcome the possible exponential blowup of BDDs by \emph{coefficient decomposition}. This allows us to give the first polynomial generalized arc-consistent ROBDD-based encoding for Pseudo-Boolean constraints. Finally, we focus on practical issues: we show how to efficiently construct such ROBDDs, how to encode them into SAT with only 2 clauses per node, and present experimental results that confirm that our approach is competitive with other encodings and state-of-the-art Pseudo-Boolean solvers.

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasimodular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by Gottsche-Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the generating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, Gottsche, and Lange-Sernesi. Further evidence is provided by Donaldson-Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject.

Guess-and-verify versus unrestricted nondeterminism for OBDDs and one-way Turing machines

Exponential space complexity for OBDD-based reachability analysis

Parallel algorithms are given for the exact solution of the hidden-line problem. Most of the parallel algorithms proposed for visibility problems in the literature give approximate solutions. and thus cannot yield an upper bound on the complexity of the particular problem. The first algorithm proposed here is worth mentioning not only for its simplicity. but also from a practical point of view: a speed up of a factor P is achieved by using P processors. 1≤P≤N, where N is the number of edges used to describe a polygonal scene. Additionally. the problem of aliasing inherent with approximation methods is avoided. The significance of the second algorithm, which is based on the first one, is mainly on the theoretical level: it is used to establish the parallel complexity of the hidden-line problem. The sequential complexity of this problem has recently been proved to be Θ(N2), and now we can prove that in the parallel case the problem is in the complexity class NC, i.e., it can be solved in time polynomial in log N by using a number of processors polynomial in N, assuming any reasonable model of parallel computation. More particularly, an O (log N) parallel time solution is given which cannot be further improved even if arbitrarily many processors of a concurrent read, exclusive write parallel RAM model are available.

Due to their many advantages, modular structures commonly exist in artificial and natural systems, and the concept of modular product design has recently received extensive attention from the engineering research community. Although some work has been done on modularity, most of it is qualitative and exploratory in nature, and little is quantitative. One reason for this gap is the lack of a clear definition of modularity. This thesis begins with a detailed discussion on the concepts of “modularity” and “module.” Based on the background presented here, a mutual information-based method is proposed to quantify modularity. The method is based on the view that coupling is information flow instead of real physical interactions. Information flow can be quantified by mutual information, which is based on randomness (or uncertainty). Since most engineering products can be modeled as stochastic systems and therefore have randomness, the mutual information-based method can be applied in very general cases, and it is shown that the commonly existing linkage counting modularity measure is a special case of the mutual information-based modularity measure. The mutual information-based method is applicable to final design products. But at the early stage of the engineering design process, there are generally only function diagrams. To exploit the benefits of modularity as early as possible, a minimal description length principle-based modularity measure is proposed to determine the modularity of graph structures, which can represent function diagrams. The method is used as criteria to hierarchically decompose abstract graph structures and the real function structure of an HP printer by evolutionary computation. Due to the specialty of genome representations in evolutionary computation, new genetic operators are developed to determine optimal hierarchical decompositions. This quantitative modularity measure has been developed to synthesize modular engineering products, especially by evolutionary design. There are many factors affecting evolving modular structures, such as genome representation, fitness function, learning, and task structure. The thesis preliminarily studies the effects of the modularity of tasks on the modularity of products in evolutionary computation. Using feed-forward neural networks as examples, the results show that the effects are task-dependent and rely on the amount of resources available for the tasks.

Based on a survey released by the TomTom Traffic Index in 2018, Indonesia was ranked seventh in the category of the most congested country in the world. One of the factors affecting traffic congestion in Indonesia is an inflexible and conventional traffic management system. In this regard, it is necessary to have a better traffic management system such as a Smart Traffic Light. One way to implement a smart traffic light system is to make a vehicle detection and counting system on the traffic CCTV video automatically. The methods used in this research are Haar Cascade Classifiers and Convolutional Neural Network. Haar Cascade Classifiers have fast computation processes and CNN is applied to validate the detection results of the Haar Cascade method for better accuracy. The average level of accuracy achieved by the system on quiet test data is 82%, normal test data is 69%, and busy test data is 60%. Meanwhile, the average computation time needed by the system for the quiet test data is 0.63 seconds, the normal test data is 0.52 seconds, and the busy test data is 1.05 seconds.

Perspective on complexity measures targeting read-once branching programs

We investigate autoreducibility properties of complete sets for NEXP under different polynomial-time reductions. Specifically, we show that under some polynomial-time reductions there are complete sets for NEXP that are not autoreducible. We show that settling the question whether every complete set for NEXP under non-adaptative reduction is autoreducible under NOR-truth-table reduction either positively or negatively would lead to major results about the exponential time complexity classes.

In this thesis, we introduce and examine four new temporal logic formalisms that can be used as specification languages for the automated verification of the reliability of hardware and software designs with respect to a desired behavior. The work is organized in two parts. In the first one, we reason about two logics for computations, GCTL* and MCTL*, which are useful to describe a correct execution of monolithic closed systems. In the second one, instead, we focus on two logics for strategies, SL and mATL*, which are useful to formalize several interesting properties about interactive plays in multi-entities systems modeled as multi-agent games. In the “Logics for Computations” part, we first study the immersion of the idea of graded quantifications into the temporal-logic framework. In first order logic, existential and universal quantifiers express the concepts of the existence of at least one individual object satisfying a formula, or that all individual objects satisfy a formula. In other logics, these quantifiers have been generalized to express that, for a given non-negative integer n, at least n or all but n individuals satisfy a particular formula. Here, we consider GCTL, a temporal logic with graded path quantifiers, which allows to describe properties like “there exist at least n different classes of computational fluxes in which a system reaches a predetermined state”, where the classes over paths are computed by means of a predetermined equivalence relation. More precisely, we uniformly extend the classic concept of graded quantifiers from states to paths, through the use of a concept of path equivalence with respect to a given path formula. About this logic, in particular, we study the expressiveness and succinctness relationships with respect to GµCalculus and the complexity of the satisfiability problem, which results to be ExpTime-Complete. This research is partially based on the works [BMM09] “Graded Computation Tree Logic” and [BMM10] “Graded Computation Tree Logic with Binary Coding” published, respectively, in the proceedings of the “IEEE Symposium on Logic in Computer Science, 2009” and “EACSL Annual Conference on Computer Science Logic, 2010”. Preliminary results can be also found in [Mog07]. Furthermore, we consider special quantifiers over substructures, which allow to select, using parametric criteria, small critical parts of a system to be successively verified. In literature, there are some attempts to define a logic that allows to modify the underlying structure under exam and then to verify on it some assigned property. However, as far as we know, none of them is able to select minimal submodels of a given property describing the criteria on which then execute the verification process. Here, we base our work on the search of a new operator that merges the concept of quantifiers on structures with that one derived by a generalization of the concept of pruning. The results of this work, is a class of three different extensions of CTL* with minimal model quantifiers, which we name MCTL*. Regarding these logics, we study several reductions among them, as well as the satisfiability problem that we prove to be highly undecidability, i.e., Σ_1^1-Hard, for two out of the three cases. This research is partially based on the work [MM09] “Branching-Time Temporal Logics with Minimal Model Quantifiers” published in the proceedings of the “International Conference on Developments in Language Theory, 2009”. In the “Logics for Strategies” part, we first study the problem of defining a new specification language through which it is possible to express several important properties of multi-entities systems that are neither expressible using classical monolithic temporal logics, such as CTL*, nor using two-agent-teams temporal logics, such as ATL*. In literature, we can found some proposal of logics that try to achieve this goal, but unfortunately, none of them succeeds completely on all the aspects. Among them, one of the most important attempts is CHP-SL, a logics in which one can use variables over strategies. However, this logic has a deep weakness, since it does not allow to describe games with more than two players and even two-players concurrent games. Here, we introduce SL, a logic with a syntax similar in some aspects to the first order logic, in which the strategies of the agent building the game are treated as first order objects on which we can quantify. This logic generalizes CHP-SL, by allowing the specification of the correct behavior of multi-agent concurrent games. In SL, for example, we are able to express very complex but useful Nash equilibria that are not expressible with CHP-SL. We enlighten that Nash equilibrium is one of the most important concepts in game theory. For the introduced logic, we solve two problems left open in the work on CHP-SL. Precisely, we show that the related model-checking problem is 2ExpTime-Complete, thus not harder of the same problem for several subsumed logics, while we prove that its satisfiability problem is highly undecidable, i.e., Σ_1^1-Hard. This research is partially based on the work [MMV10a] “Reasoning About Strategies” published in the proceedings of the “IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, 2010”. Finally, we consider the concept of relentful strategic reasoning, i.e., a formalism that expresses the ability of a strategy to be used not only to achieve a first given goal, but also to change its final goal in dependence of the history of the play. In the context of planning, memoryful quantification, i.e., quantification over computations that does not lose information about the past along the time, is one of the principal way to express the fact that a system is able to achieve a desired result, and shift to a different goal if some event happens. However, this kind of quantification was not considered before in the context of multi-agent planning. Here, we introduce mATL*, a fusion of the classic alternating temporal logic ATL* with memoryful quantification, with the aim of covering the previous idea. About this logics, we prove that, although it is equivalent to ATL*, it is exponentially more succinct. Nevertheless, we prove that both the model-checking and the satisfiability problems remain 2ExpTime-Complete, as for ATL*. This research is partially based on the work [MMV10b] “Relentful Strategic Reasoning in Alternating-Time Temporal Logic” published in the proceedings of the “International Conference on Logic for Programming Artificial Intelligence and Reasoning, 2010”.

We investigate Ordered Binary Decision Diagrams (OBDDs) — a model for computing Boolean functions. It is known that OBDD’s complexity can extremely depend on the order of reading variables. There are techniques for constructing functions that do not allow choosing the optimal order for reading the input, one of which we use in this paper. A function “Shuffled Inequality” NEQS is presented, for which a lower and an upper bounds for the complexity of nondeterministic OBDD are proved. The upper bound is an improvement of a previously known result. A quantum measure-many non-deterministic OBDD is constructed that is more efficient than the classical one. The hierarchy of complexity classes defined on the basis of OBDD models is clarified.

We consider computing a longest palindrome in the streaming model, where the symbols arrive one-by-one and we do not have random access to the input. While computing the answer exactly using sublinear space is not possible in such a setting, one can still hope for a good approximation guarantee. Our contribution is twofold. First, we provide lower bounds on the space requirements for randomized approximation algorithms processing inputs of length n. We rule out Las Vegas algorithms, as they cannot achieve sublinear space complexity. For Monte Carlo algorithms, we prove a lower bounds of Omega(M log min {|Sigma|, M}) bits of memory; here M=n/E for approximating the answer with additive error E, and M= log n / log (1 + epsilon) for approximating the answer with multiplicative error (1 + epsilon). Second, we design three real-time algorithms for this problem. Our Monte Carlo approximation algorithms for both additive and multiplicative versions of the problem use O(M) words of memory. Thus the obtained lower bounds are asymptotically tight up to a logarithmic factor. The third algorithm is deterministic and finds a longest palindrome exactly if it is short. This algorithm can be run in parallel with a Monte Carlo algorithm to obtain better results in practice. Overall, both the time and space complexity of finding a longest palindrome in a stream are essentially settled.

This dissertation is concerned with the application of notions and methods from the field of information theory to the field of communication complexity. It consists of two main parts. In the first part of the dissertation, we prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae has attracted interest mainly in the decision tree model. In the communication complexity model many interesting results deal with specific read-once formulae, such as disjointness and tribes. In this dissertation we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f )/c^d(f ) , where n(f ) is the number of variables and d(f ) is the depth of the formula, and they are optimal up to the constant in the base of the denominator. In the second part of the dissertation, we explore the applicability of the information-theoretic method in the number-on-the-forehead model. The work of Bar-Yossef, Jayram, Kumar & Sivakumar [BYJKS04] revealed a beautiful connection between Hellinger distance and two-party randomized communication protocols. Inspired by their work and motivated by the open questions in the number-on-the-forehead model, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party protocols. We provide a small toolbox that allows one to manipulate several Hellinger volume terms and also to lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. Finally, we show how to apply the new tools to obtain a lower bound on the informational complexity of the ANDk function.