Communication Cost for Updating Linear Functions when Message Updates are Sparse: Connections to Maximally Recoverable Codes

We consider a communication problem in which an update of the source message needs to be conveyed to one or more distant receivers that are interested in maintaining specific linear functions of the source message. The setting is one in which the updates are sparse in nature, and where neither the source nor the receiver(s) is aware of the exact {\em difference vector}, but only know the amount of sparsity that is present in the difference-vector. Under this setting, we are interested in devising linear encoding and decoding schemes that minimize the communication cost involved. We show that the optimal solution to this problem is closely related to the notion of maximally recoverable codes (MRCs), which were originally introduced in the context of coding for storage systems. In the context of storage, MRCs guarantee optimal erasure protection when the system is partially constrained to have local parity relations among the storage nodes. In our problem, we show that optimal solutions exist if and only if MRCs of certain kind (identified by the desired linear functions) exist. We consider point-to-point and broadcast versions of the problem, and identify connections to MRCs under both these settings. For the point-to-point setting, we show that our linear-encoder based achievable scheme is optimal even when non-linear encoding is permitted. The theory is illustrated in the context of updating erasure coded storage nodes. We present examples based on modern storage codes such as the minimum bandwidth regenerating codes.Comment: To Appear in IEEE Transactions on Information Theor

Extending the Finite Domain Solver of GNU Prolog

International audienceThis paper describes three significant extensions for the Finite Domain solver of GNU Prolog. First, the solver now supports negative integers. Second, the solver detects and prevents integer overflows from occurring. Third, the internal representation of sparse domains has been redesigned to overcome its current limitations. The preliminary performance evaluation shows a limited slowdown factor with respect to the initial solver. This factor is widely counterbalanced by the new possibilities and the robustness of the solver. Furthermore these results are preliminary and we propose some directions to limit this overhead

Executing Gödel’s programme in set theory

The study of set theory (a mathematical theory of infinite collections) has garnered a great deal of philosophical interest since its development. There are several reasons for this, not least because it has a deep foundational role in mathematics; any mathematical statement (with the possible exception of a few controversial examples) can be rendered in set-theoretic terms. However, the fruitfulness of set theory has been tempered by two difficult yet intriguing philosophical problems: (1.) the susceptibility of naive formulations of sets to contradiction, and (2.) the inability of widely accepted set-theoretic axiomatisations to settle many natural questions. Both difficulties have lead scholars to question whether there is a single, maximal Universe of sets in which all set-theoretic statements are determinately true or false (often denoted by ‘V ’). This thesis illuminates this discussion by showing just what is possible on the ‘one Universe’ view. In particular, we show that there are deep relationships between responses to (1.) and the possible tools that can be used in resolving (2.). We argue that an interpretation of extensions of V is desirable for addressing (2.) in a fruitful manner. We then provide critical appraisal of extant philosophical views concerning (1.) and (2.), before motivating a strong mathematical system (known as‘Morse-Kelley’ class theory or ‘MK’). Finally we use MK to provide a coding of discourse involving extensions of V , and argue that it is philosophically virtuous. In more detail, our strategy is as follows: Chapter I (‘Introduction’) outlines some reasons to be interested in set theory from both a philosophical and mathematical perspective. In particular, we describe the current widely accepted conception of set (the ‘Iterative Conception’) on which sets are formed successively in stages, and remark that set-theoretic questions can be resolved on the basis of two dimensions: (i) how ‘high’ V is (i.e. how far we go in forming stages), and (ii) how ‘wide’ V is (i.e. what sets are formed at successor stages). We also provide a very coarse-grained characterisation of the set-theoretic paradoxes and remark that extensions of universes in both height and width are relevant for our understanding of (1.) and (2.). We then present the different motivations for holding either a ‘one Universe’ or ‘many universes’ view of the subject matter of set theory, and argue that there is a stalemate in the dialectic. Instead we advocate filling out each view in its own terms, and adopt the ‘one Universe’ view for the thesis. Chapter II (‘G¨odel’s Programme’) then explains the Universist project for formulating and justifying new axioms concerning V . We argue that extensions of V are relevant to both aspects of G¨odel’s Programme for resolving independence. We also identify a ‘Hilbertian Challenge’ to explain how we should interpret extensions of V , given that we wish to use discourse that makes apparent reference to such nonexistent objects. Chapter III (‘Problematic Principles’) then lends some mathematical precision to the coarse-grained outline of Chapter I, examining mathematical discourse that seems to require talk of extensions of V . Chapter IV (‘Climbing above V ?’), examines some possible interpretations of height extensions of V . We argue that several such accounts are philosophically problematic. However, we point out that these difficulties highlight two constraints on resolution of the Hilbertian Challenge: (i) a Foundational Constraint that we do not appeal to entities not representable using sets from V , and (ii) an Ontological Constraint to interpret extensions of V in such a way that they are clearly different from ordinary sets. 5 Chapter V (‘Broadening V ’s Horizons?’), considers interpretations of width extensions. Again, we argue that many of the extant methods for interpreting this kind of extension face difficulties. Again, however, we point out that a constraint is highlighted; a Methodological Constraint to interpret extensions of V in a manner that makes sense of our naive thinking concerning extensions, and links this thought to truth in V . We also note that there is an apparent tension between the three constraints. Chapter VI (‘A Theory of Classes’) changes tack, and provides a positive characterisation of apparently problematic ‘proper classes’ through the use of plural quantification. It is argued that such a characterisation of proper class discourse performs well with respect to the three constraints, and motivates the use of a relatively strong class theory (namely MK). Chapter VII (‘V -logic and Resolution’) then puts MK to work in interpreting extensions of V . We first expand our logical resources to a system called V -logic, and show how discourse concerning extensions can be thereby represented. We then show how to code the required amount of V -logic usingMK. Finally, we argue that such an interpretation performs well with respect to the three constraints. Chapter VIII (‘Conclusions’) reviews the thesis and makes some points regarding the exact dialectical situation. We argue that there are many different philosophical lessons that one might take from the thesis, and are clear that we do not commit ourselves to any one such conclusion. We finally provide some open questions and indicate directions for future research, remarking that the thesis opens the way for new and exciting philosophical and mathematical discussion

Hidden Markov Models

Hidden Markov Models (HMMs), although known for decades, have made a big career nowadays and are still in state of development. This book presents theoretical issues and a variety of HMMs applications in speech recognition and synthesis, medicine, neurosciences, computational biology, bioinformatics, seismology, environment protection and engineering. I hope that the reader will find this book useful and helpful for their own research