There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

Handbook of logic in computer science.

The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of `name-abstraction' and `fresh name' that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science.

/pdf/a-new-approach-to-abstract-syntax-with-variable-binding-44i2k31trz.pdf

A New Approach to Abstract Syntax with Variable Binding

Structure and interpretation of computer programs

p-calculus in (Co)inductive-type theory

We present a logical framework γ for reasoning on a very general class of languages featuring binding operators, called nominal algebras, presented in higher-order abstract syntax (HOAS). γ is based on an axiomatic syntactic standpoint and it consists of a simple types theory a la Church extended with a set of axioms called the Theory of Contexts, recursion operators and induction principles. This framework is rather expressive and, most notably, the axioms of the Theory of Contexts allow for a smooth reasoning of schemata in HOAS. An advantage of this framework is that it requires a very low mathematical and logical overhead. Some case studies and comparison with related work are briefly discussed.

/pdf/an-axiomatic-approach-to-metareasoning-on-nominal-algebras-4jynl59f8a.pdf

An Axiomatic Approach to Metareasoning on Nominal Algebras in HOAS

In this paper we survey some well-known approaches proposed as general models for calculi dealing with names (like for example process calculi with name-passing). We focus on (pre)sheaf categories, nominal sets, permutation algebras and named sets, studying the relationships among these models, thus allowing techniques and constructions to be transferred from one model to the other.

/pdf/about-permutation-algebras-pre-sheaves-and-named-sets-160sjsihct.pdf

About permutation algebras, (pre)sheaves and named sets

We investigate a framework for representing and reasoning about syntactic and semantic aspects of typed languages with variable binders.First, we introduce typed binding signatures and develop a theory of typed abstract syntax with binders. Each signature is associated to a category of "presentation" models, where the language of the typed signature is the initial model.At the semantic level, types can be given also a computational meaning in a (possibly different) semantic category. We observe that in general, semantic aspects of terms and variables can be reflected in the presentation category by means of an adjunction. Therefore, the category of presentation models is expressive enough to represent both the syntactic and the semantic aspects of languages.We introduce then a metalogical system, inspired by the internal languages of the presentation category, which can be used for reasoning on both the syntax and the semantics of languages. This system is composed by a core equational logic tailored for reasoning on the syntactic aspects; when a specific semantics is chosen, the system can be modularly extended with further "semantic" notions, as needed.

/pdf/a-framework-for-typed-hoas-and-semantics-1m96t6b9po.pdf

A framework for typed HOAS and semantics

In type theory based logical frameworks, recursive and co-recursive definitions are subject to syntactic restrictions that ensure their termination and productivity. These restrictions however greately decrease the expressive power of the language. In this work we propose a general approach for systematically defining fixed points for a broad class of well given recursive definition. This approach unifies the ones based on well-founded order to the ones based on complete metrics and contractive functions, thus allowing for mixed recursive/corecursive definitions. The resulting theory, implemented in the Coq proof assistant, is quite simple and hence it can be used broadly with a small, sustainable overhead on the user.

Marino Miculan

Papers

p-calculus in (Co)inductive-type theory

An Axiomatic Approach to Metareasoning on Nominal Algebras in HOAS

About permutation algebras, (pre)sheaves and named sets

A framework for typed HOAS and semantics

A unifying approach to recursive and Co-recursive definitions