A Structural Approach to Reversible Computation

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of low-level machine models. By contrast, we develop a more structural approach. We show how high-level functional programs can be mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic---but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis.Comment: 30 pages, appeared in Theoretical Computer Scienc

Types, equations, dimensions and the Pi theorem

The languages of mathematical physics and modelling are endowed with a rich "grammar of dimensions" that common abstractions of programming languages fail to represent. We propose a dependently typed domain-specific language (embedded in Idris) that captures this grammar. We apply it to explain basic notions of dimensional analysis and Buckingham's Pi theorem. We hope that the language makes mathematical physics more accessible to computer scientists and functional programming more palatable to modelers and physicists.Comment: Submitted for publication in the "Journal of Functional Programming" in August 202

The Material Theory of Induction

The fundamental burden of a theory of inductive inference is to determine which are the good inductive inferences or relations of inductive support and why it is that they are so. The traditional approach is modeled on that taken in accounts of deductive inference. It seeks universally applicable schemas or rules or a single formal device, such as the probability calculus. After millennia of halting efforts, none of these approaches has been unequivocally successful and debates between approaches persist. The Material Theory of Induction identifies the source of these enduring problems in the assumption taken at the outset: that inductive inference can be accommodated by a single formal account with universal applicability. Instead, it argues that that there is no single, universally applicable formal account. Rather, each domain has an inductive logic native to it.The content of that logic and where it can be applied are determined by the facts prevailing in that domain. Paying close attention to how inductive inference is conducted in science and copiously illustrated with real-world examples, The Material Theory of Induction will initiate a new tradition in the analysis of inductive inference

On the logics of algebra.

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2008.We present and consider a number of logics that arise naturally from universal algebraic considerations, but which are ‘inherently unalgebraizable’ in the sense of [BP89a], essentially because they have no theo- rems. Of particular interest is the membership logic of a quasivariety, which is determined by its theorems, which are the relative congruence classes of the term algebra together with the empty-set in the case that the quasivariety is non-trivial. The membership logic arises by a more general technique developed in this text, for inducing deductive systems from closed systems on the free algebras of quasivarieties. In order to formalize this technique, we develop a theory of logics over constructs, where constructs are concrete categories. With this theory in place, we are able to view a closed system over an algebra as a logic, and in particular a structural logic, structural with respect to a suitable construct, typically the construct con- sisting of all algebras in a quasivariety and all algebra homomorphisms between these algebras. Of course, in such a case, none of these logics are generally sentential (i.e., structural and finitary deductive systems in the sense of [BP89a]), since the formulae of sentential logics arise from the terms of the absolutely free term algebra, which is generally not a member of the quasivariety under interest. In such cases, where the term algebra is not a member of a quasivariety, the free algebra of the quasivariety on denumerably countable free generators takes on the role played by the term algebra in sentential logics. Many of the logics that we encounter in this text arise most naturally as finitary logics on this free algebra of the quasivariety and generally are structural with respect to the quasivariety. We call such logics canons, and show how such structural canons induce sentential calculi, which we call the induced ideal ; the filters of the ideal on the free algebra are precisely the theories of the canon. The membership logic is the ideal of the cannon whose theories are the relative congruence classes on the free algebra. The primary aim of this thesis is to provide a unifying framework for logics of this type which extends the Blok-Pigozzi theory of elementarily algebraizable (and protoalgebraic) deductive systems. In this extension there are two parameters: a set of formulae and a variable. When the former is empty or consists of theorems, the Blok-Pigozzi theory is recovered, and the variable is redundant. For the membership logic, the appropriate variant of equivalent algebraic semantics encompasses the relatively congruence regular quasivarieties. These results have appeared in [BR03]. The secondary aim of this thesis is to analyse our theory of parameterized algebraization from a non- parameterized perspective. To this end, we develop a theory of protoalgebraic logics over constructs and equivalence between logics from different constructs, which we then use to explain the results we obtained in our parameterized theories of protoalgebraicity, algebraic semantics and equivalent algebraic semantics. We relate this theory to the theory of deductively equivalent -institutions [Vou03], and as a consequence obtain a number of improved and new results in the field of categorical abstract algebraic logic. We also use our theory of protoalgebraic logics over constructs to obtain a new and simpler characterization of structural finitary n-deductive systems, which we then use to close the program begun in [BR99], by extending those results for 1-deductive systems to n-deductive systems, and in particular characterizing the protoalgebraicity of the sentential n-deductive system Sn(K,N), which is the natural extension of the 1-deductive system S(K, ) introduce in [BR99], in terms of the quasivariety K having hK,Ni-coherent N-classes (we cannot see how to obtain this result from the standard characterization of protoalgebraic n- deductive systems of [Pal03], which is very complex). With respect to this program of completing [BR99], we also show that a quasivariety K is an equivalent algebraic semantics for a n-deductive system with defining equations N iff K is hK,Ni-regular; a notion of regularity that we introduce and characterize by a quasi-Mal’cev condition. The third aim of this text is to unify as many disparate arguments and notions in algebraic logic under the banner of continuous translations between closed systems, where our use of the term continuous is in the topological sense rather than in the order-theoretic sense, and, where possible, to give elementary, i.e. first order, definitions and proofs. To this end, we show that closed systems, closure operators and conse- quence relations can all be characterized elementarily over orders, and put into one-to-one correspondence that reflects exactly, the standard correspondences between the well-known concrete notions with the same name. We show that when the order is the complete power order over a set, then these elementary structures coincide with their well-known counterparts with the same name. We also introduce two other elementary structures over orders, namely the closed equivalence relation and something we term the proto-Leibniz relation; these elementary structures are also in one-to-one correspondence with the earlier mentioned structures; we have not seen concrete versions of these structures. We then characterize the structure homomorphisms between these structures, as well as considering galois relations between them; galois relations are pairs of order-preserving function in opposite directions; we call these translations, and they are elementary notions. We demonstrate how notions as disparate as structurality, semantics, algebraic semantics, the filter correspondence property, filters, models, semantic consequence, protoalge- braicity and even the logic S(K, ) of [BR99] and our logic Sn(K,N), all fall within this framework, as does much of our parameterized theory and much of the theory of -institutions. A brief summary of the standard theory of deductive systems and their algebraization is provided for the reader unfamiliar with algebraic logics, as well as the necessary background material, including construct and category theory, the theory of structures and algebras, and the model theory of structures with and without equality

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

On The Möbius Function Of Permutations Under The Pattern Containment Order

We study several aspects of the Möbius function, μ[σ, π], on the poset of permutations under the pattern containment order. First, we consider cases where the lower bound of the poset is indecomposable. We show that μ[σ, π] can be computed by considering just the indecomposable permutations contained in the upper bound. We apply this to the case where the upper bound is an increasing oscillation, and give a method for computing the value of the Möbius function that only involves evaluating simple inequalities. We then consider conditions on an interval which guarantee that the value of the Möbius function is zero. In particular, we show that if a permutation π contains two intervals of length 2, which are not order-isomorphic to one another, then μ[1, π] = 0. This allows us to prove that the proportion of permutations of length n with principal Möbius function equal to zero is asymptotically bounded below by (1−1/e) 2 ≥ 0.3995. This is the first result determining the value of μ[1, π] for an asymptotically positive proportion of permutations π. Following this, we use “2413-balloon” permutations to show that the growth of the principal Möbius function on the permutation poset is exponential. This improves on previous work, which has shown that the growth is at least polynomial. We then generalise 2413-balloon permutations, and find a recursion for the value of the principal Möbius function of these generalisations. Finally, we look back at the results found, and discuss ways to relate the results from each chapter. We then consider further research avenues

Adventures in Formalisation: Financial Contracts, Modules, and Two-Level Type Theory

We present three projects concerned with applications of proof assistants in the area of programming language theory and mathematics. The first project is about a certified compilation technique for a domain-specific programming language for financial contracts (the CL language). The code in CL is translated into a simple expression language well-suited for integration with software components implementing Monte Carlo simulation techniques (pricing engines). The compilation procedure is accompanied with formal proofs of correctness carried out in Coq. The second project presents techniques that allow for formal reasoning with nested and mutually inductive structures built up from finite maps and sets. The techniques, which build on the theory of nominal sets combined with the ability to work with isomorphic representations of finite maps, make it possible to give a formal treatment, in Coq, of a higher-order module system, including the ability to eliminate at compile time abstraction barriers introduced by the module system. The development is based on earlier work on static interpretation of modules and provides the foundation for a higher-order module language for Futhark, an optimising compiler targeting data-parallel architectures. The third project presents an implementation of two-level type theory, a version of Martin-Lof type theory with two equality types: the first acts as the usual equality of homotopy type theory, while the second allows us to reason about strict equality. In this system, we can formalise results of partially meta-theoretic nature. We develop and explore in details how two-level type theory can be implemented in a proof assistant, providing a prototype implementation in the proof assistant Lean. We demonstrate an application of two-level type theory by developing some results on the theory of inverse diagrams using our Lean implementation.Comment: PhD thesis defended in January 2018 at University of Copenhagen, Department of Computer Scienc

The Material Theory of Induction

The fundamental burden of a theory of inductive inference is to determine which are the good inductive inferences or relations of inductive support and why it is that they are so. The traditional approach is modeled on that taken in accounts of deductive inference. It seeks universally applicable schemas or rules or a single formal device, such as the probability calculus. After millennia of halting efforts, none of these approaches has been unequivocally successful and debates between approaches persist. The Material Theory of Induction identifies the source of these enduring problems in the assumption taken at the outset: that inductive inference can be accommodated by a single formal account with universal applicability. Instead, it argues that that there is no single, universally applicable formal account. Rather, each domain has an inductive logic native to it. The content of that logic and where it can be applied are determined by the facts prevailing in that domain