Facilitating modular property-preserving extensions of programming languages

We will explore an approach to modular programming language descriptions and extensions in a denotational style. Based on a language core, language features are added stepwise on the core. Language features can be described separated from each other in a self-contained, orthogonal way. We present an extension semantics framework consisting of mechanisms to adapt semantics of a basic language to new structural requirements in an extended language preserving the behaviour of programs of the basic language. Common templates of extension are provided. These can be collected in extension libraries accessible to and extendible by language designers. Mechanisms to extend these libraries are provided. A notation for describing language features embedding these semantics extensions is presented

A Context-theoretic Framework for Compositionality in Distributional Semantics

Techniques in which words are represented as vectors have proved useful in many applications in computational linguistics, however there is currently no general semantic formalism for representing meaning in terms of vectors. We present a framework for natural language semantics in which words, phrases and sentences are all represented as vectors, based on a theoretical analysis which assumes that meaning is determined by context. In the theoretical analysis, we define a corpus model as a mathematical abstraction of a text corpus. The meaning of a string of words is assumed to be a vector representing the contexts in which it occurs in the corpus model. Based on this assumption, we can show that the vector representations of words can be considered as elements of an algebra over a field. We note that in applications of vector spaces to representing meanings of words there is an underlying lattice structure; we interpret the partial ordering of the lattice as describing entailment between meanings. We also define the context-theoretic probability of a string, and, based on this and the lattice structure, a degree of entailment between strings. We relate the framework to existing methods of composing vector-based representations of meaning, and show that our approach generalises many of these, including vector addition, component-wise multiplication, and the tensor product.Comment: Submitted to Computational Linguistics on 20th January 2010 for revie

Modular Composition of Language Features through Extensions of Semantic Language Models

Today, programming or specification languages are often extended in order to customize them for a particular application domain or to refine the language definition. The extension of a semantic model is often at the centre of such an extension. We will present a framework for linking basic and extended models. The example which we are going to use is the RSL concurrency model. The RAISE specification language RSL is a formal wide-spectrum specification language which integrates different features, such as state-basedness, concurrency and modules. The concurrency features of RSL are based on a refinement of a classical denotational model for process algebras. A modification was necessary to integrate state-based features into the basic model in order to meet requirements in the design of RSL. We will investigate this integration, formalising the relationship between the basic model and the adapted version in a rigorous way. The result will be a modular composition of the basic process model and new language features, such as state-based features or input/output. We will show general mechanisms for integration of new features into a language by extending language models in a structured, modular way. In particular, we will concentrate on the preservation of properties of the basic model in these extensions

Fuzzy inequational logic

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if-then rules which is obtained as particular case of the general result

The Sigma-Semantics: A Comprehensive Semantics for Functional Programs

A comprehensive semantics for functional programs is presented, which generalizes the well-known call-by-value and call-by-name semantics. By permitting a separate choice between call-by value and call-by-name for every argument position of every function and parameterizing the semantics by this choice we abstract from the parameter-passing mechanism. Thus common and distinguishing features of all instances of the sigma-semantics, especially call-by-value and call-by-name semantics, are highlighted. Furthermore, a property can be validated for all instances of the sigma-semantics by a single proof. This is employed for proving the equivalence of the given denotational (fixed-point based) and two operational (reduction based) definitions of the sigma-semantics. We present and apply means for very simple proofs of equivalence with the denotational sigma-semantics for a large class of reduction-based sigma-semantics. Our basis are simple first-order constructor-based functional programs with patterns

Second-Order Algebraic Theories

Fiore and Hur recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding

Mastering Heterogeneous Behavioural Models

Heterogeneity is one important feature of complex systems, leading to the complexity of their construction and analysis. Moving the heterogeneity at model level helps in mastering the difficulty of composing heterogeneous models which constitute a large system. We propose a method made of an algebra and structure morphisms to deal with the interaction of behavioural models, provided that they are compatible. We prove that heterogeneous models can interact in a safe way, and therefore complex heterogeneous systems can be built and analysed incrementally. The Uppaal tool is targeted for experimentations.Comment: 16 pages, a short version to appear in MEDI'201

Issues about the Adoption of Formal Methods for Dependable Composition of Web Services

Web Services provide interoperable mechanisms for describing, locating and invoking services over the Internet; composition further enables to build complex services out of simpler ones for complex B2B applications. While current studies on these topics are mostly focused - from the technical viewpoint - on standards and protocols, this paper investigates the adoption of formal methods, especially for composition. We logically classify and analyze three different (but interconnected) kinds of important issues towards this goal, namely foundations, verification and extensions. The aim of this work is to individuate the proper questions on the adoption of formal methods for dependable composition of Web Services, not necessarily to find the optimal answers. Nevertheless, we still try to propose some tentative answers based on our proposal for a composition calculus, which we hope can animate a proper discussion

An Investigation on the Basic Conceptual Foundations of Quantum Mechanics by Using the Clifford Algebra

We review our approach to quantum mechanics adding also some new interesting results. We start by giving proof of two important theorems on the existence of the and Clifford algebras. This last algebra gives proof of the von Neumann basic postulates on the quantum measurement explaining thus in an algebraic manner the wave function collapse postulated in standard quantum theory. In this manner we reach the objective to expose a self-consistent version of quantum mechanics. We give proof of the quantum like Heisenberg uncertainty relations, the phenomenon of quantum Mach Zender interference as well as quantum collapse in some cases of physical interest We also discuss the problem of time evolution of quantum systems as well as the changes in space location. We also give demonstration of the Kocken-Specher theorem, and also we give an algebraic formulation and explanation of the EPR . By using the same approach we also derive Bell inequalities. Our formulation is strongly based on the use of idempotents that are contained in Clifford algebra. Their counterpart in quantum mechanics is represented by the projection operators that are interpreted as logical statements, following the basic von Neumann results. Using the Clifford algebra we are able to invert such result. According to the results previously obtained by Orlov in 1994, we are able to give proof that quantum mechanics derives from logic. We show that indeterminism and quantum interference have their origin in the logic.Comment: forthcoming papers; http://www.m-hikari.com/astp/forth/index.htm