Complexity of optimizing over the integers

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and "complexity" in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of {\em mixed-integer convex optimization}, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition. We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities

Combined decision procedures for nonlinear arithmetics, real and complex

We describe contributions to algorithmic proof techniques for deciding the satisfiability of boolean combinations of many-variable nonlinear polynomial equations and inequalities over the real and complex numbers. In the first half, we present an abstract theory of Grobner basis construction algorithms for algebraically closed fields of characteristic zero and use it to introduce and prove the correctness of Grobner basis methods tailored to the needs of modern satisfiability modulo theories (SMT) solvers. In the process, we use the technique of proof orders to derive a generalisation of S-polynomial superfluousness in terms of transfinite induction along an ordinal parameterised by a monomial order. We use this generalisation to prove the abstract (“strategy-independent”) admissibility of a number of superfluous S-polynomial criteria important for efficient basis construction. Finally, we consider local notions of proof minimality for weak Nullstellensatz proofs and give ideal-theoretic methods for computing complex “unsatisfiable cores” which contribute to efficient SMT solving in the context of nonlinear complex arithmetic. In the second half, we consider the problem of effectively combining a heterogeneous collection of decision techniques for fragments of the existential theory of real closed fields. We propose and investigate a number of novel combined decision methods and implement them in our proof tool RAHD (Real Algebra in High Dimensions). We build a hierarchy of increasingly powerful combined decision methods, culminating in a generalisation of partial cylindrical algebraic decomposition (CAD) which we call Abstract Partial CAD. This generalisation incorporates the use of arbitrary sound but possibly incomplete proof procedures for the existential theory of real closed fields as first-class functional parameters for “short-circuiting” expensive computations during the lifting phase of CAD. Identifying these proof procedure parameters formally with RAHD proof strategies, we implement the method in RAHD for the case of full-dimensional cell decompositions and investigate its efficacy with respect to the Brown-McCallum projection operator. We end with some wishes for the future

LDS - Labelled Deductive Systems: Volume 1 - Foundations

Traditional logics manipulate formulas. The message of this book is to manipulate pairs; formulas and labels. The labels annotate the formulas. This sounds very simple but it turned out to be a big step, which makes a serious difference, like the difference between using one hand only or allowing for the coordinated use of two hands. Of course the idea has to be made precise, and its advantages and limitations clearly demonstrated. `Precise' means a good mathematical definition and `advantages demonstrated' means case studies and applications in pure logic and in AI. To achieve that we need to address the following: \begin{enumerate} \item Define the notion of {\em LDS}, its proof theory and semantics and relate it to traditional logics. \item Explain what form the traditional concepts of cut elimination, deduction theorem, negation, inconsistency, update, etc.\ take in {\em LDS}. \item Formulate major known logics in {\em LDS}. For example, modal and temporal logics, substructural logics, default, nonmonotonic logics, etc. \item Show new results and solve long-standing problems using {\em LDS}. \item Demonstrate practical applications. \end{enumerate} This is what I am trying to do in this book. Part I of the book is an intuitive presentation of {\em LDS} in the context of traditional current views of monotonic and nonmonotonic logics. It is less oriented towards the pure logician and more towards the practical consumer of logic. It has two tasks, addressed in two chapters. These are: \begin{itemlist}{Chapter 1:} \item [Chapter1:] Formally motivate {\em LDS} by starting from the traditional notion of `What is a logical system' and slowly adding features to it until it becomes essentially an {\em LDS}. \item [Chapter 2:] Intuitively motivate {\em LDS} by showing many examples where labels are used, as well as some case studies of familiar logics (e.g.\ modal logic) formulated as an {\em LDS}. \end{itemlist} The second part of the book presents the formal theory of {\em LDS} for the formal logician. I have tried to avoid the style of definition-lemma-theorem and put in some explanations. What is basically needed here is the formulation of the mathematical machinery capable of doing the following. \begin{itemize} \item Define {\em LDS} algebra, proof theory and semantics. \item Show how an arbitrary (or fairly general) logic, presented traditionally, say as a Hilbert system or as a Gentzen system, can be turned into an {\em LDS} formulation. \item Show how to obtain a traditional formulations (e.g.\ Hilbert) for an arbitrary {\em LDS} presented logic. \item Define and study major logical concepts intrinsic to {\em LDS} formalisms. \item Give detailed study of the {\em LDS} formulation of some major known logics (e.g.\ modal logics, resource logics) and demonstrate its advantages. \item Translate {\em LDS} into classical logic (reduce the `new' to the `old'), and explain {\em LDS} in the context of classical logic (two sorted logic, metalevel aspects, etc). \end{itemize} \begin{itemlist}{Chapter 1:} \item [Chapter 3:] Give fairly general definitions of some basic concepts of {\em LDS} theory, mainly to cater for the needs of the practical consumer of logic who may wish to apply it, with a detailed study of the metabox system. The presentation of Chapter 3 is a bit tricky. It may be too formal for the intuitive reader, but not sufficiently clear and elegant for the mathematical logician. I would be very grateful for comments from the readers for the next draft. \item [Chapter 4:] Presents the basic notions of algebraic {\em LDS}. The reader may wonder how come we introduce algebraic {\em LDS} in chapter 3 and then again in chapter 4. Our aim in chapter 3 is to give a general definition and formal machinery for the applied consumer of logic. Chapter 4 on the other hand studies {\em LDS} as formal logics. It turns out that to formulate an arbitrary logic as an {\em LDS} one needs some specific labelling algebras and these need to be studied in detail (chapter 4). For general applications it is more convenient to have general labelling algebras and possibly mathematically redundant formulations (chapter 3). In a sense chapter 4 continues the topic of the second section of chapter 3. \item [Chapter 5:] Present the full theory of {\em LDS} where labels can be databases from possibly another {\em LDS}. It also presents Fibred Semantics for {\em LDS}. \item [Chapter 6:] Presents a theory of quantifers for {\em LDS}. The material for this chapter is still under research. \item [Chapter 7:] Studies structured consequence relations. These are logical system swhere the structure is not described through labels but through some geometry like lists, multisets, trees, etc. Thus the label of a wff $A$ is implicit, given by the place of $A$ in the structure. \item [Chapter 8:] Deals with metalevel features of {\em LDS} and its translation into two sorted classical logic. \end{itemlist} Parts 3 and 4 of the book deals in detail with some specific families of logics. Chapters 9--11 essentailly deal with substructural logics and their variants. \begin{itemlist}{Chapter10:} \item [Chapter 9:] Studies resource and substructural logics in general. \item [Chapter 10:] Develops detailed proof theory for some systems as well as studying particular features such as negation. \item [Chapter 11:] Deals with many valued logics. \item [Chapter 12:] Studies the Curry Howard formula as type view and how it compres with labelling. \item [Chapter 13:] Deals with modal and temporal logics. \end{itemlist} Part 5 of the book deals with {\em LDS} metatheory. \begin{itemlist}{Chapter15:} \item [Chapter 14:] Deals with labelled tableaux. \item [Chapter 15:] Deals with combining logics. \item [Chapter 16:] Deals with abduction. \end{itemlist

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Mathematics & Statistics 2017 APR Self-Study & Documents

UNM Mathematics & Statistics APR self-study report, review team report, response report, and initial action plan for Spring 2017, fulfilling requirements of the Higher Learning Commission