Graphic Modelling Approach as a Support for Event-B Modelling

The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, it is argued that the often stated difference, that second order logic has categorical characterizations of relevant mathematical structures, while set theory has non-standard models, amounts to no difference at all. Second order logic and set theory permit quite similar categoricity results on one hand, and similar non-standard models on the other hand.

7 - System-level modeling, analysis, and design

Certification of safety-critical systems requires formal verification of system properties and behaviour as well as quantitative demonstration of safety. Usually, formal modelling frameworks do not include quantitative assessment of safety. This has a negative impact on productivity and predictability of system development. In this paper we present an approach to integrating quantitative safety analysis into formal system modelling and verification in Event-B. The proposed approach is based on an extension of Event-B, which allows us to perform quantitative assessment of safety within proof-based verification of system behaviour. This enables development of systems that are not only correct but also safe by construction. The approach is demonstrated by a case study - an automatic railway crossing system.

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

Physics-based sound radiation estimation from multiple speakers by combined lumped parameter and reduced-order finite element modeling

In this article, we define an approach for model transformation.We use the example of UML Activity Diagram (UML AD) and Event-B as a source and a target formalism.Before doing the transformation, a formal semantic is given to the source formalism.We use the institution theory to define the intended semantic.With this theory, we gain a algebraic specification for this formalism.Thus, the source formalism will be defined in its own natural semantic meaning without any intermediate semantic.Model transformation will be performed by a set of transformation schema which preserve the semantic expressed in the source model during the transformation process.The generated model expressed in Event-B language will be used for the formal verification of the source model.As a result, some model expressed in a precise formalism, the verification of this model can be seen as the verification of the Event-B model semantically equivalent to the source model.Then, in the present work we combine the institution theory, Event-Bmethod and graph grammar to develop an approach supporting the specification, the transformation and the verification of UML AD.

Formal verification efforts in the area of robotics are still comparatively scarce. In this paper we report on our experiences with one such effort, which was concerned with designing, implementing and certifying a safety function for autonomous vehicles and robots. We outline the algorithm which was specifically designed with safety through formal verification in mind, and present our verification methodology, which is based on formal proof and verification using the theorem prover Isabelle. The necessary normative measures that are covered are discussed. The algorithm and our methodology have been certified for use in applications up to SIL 3 of IEC61508-3 by a certification authority. Throughout, issues we recognised as being important for a successful application of formal methods in the domain at hand are highlighted. These pertain to the development process, the abstraction level at which specifications should be formulated, and the interplay between simulation and verification, among others.

Grid workflows are becoming a mainstream paradigm for implementing complex grid applications. In addition to existing grid enabling techniques, various grid ensuring techniques are emerging, e.g. workflow analysis and temporal reasoning, to probe potential pitfalls and errors and guarantee quality of services (QoS) at a design phase. A new state p calculus is proposed in this work, which not only enables flexible abstraction and management of historical grid system events, but also facilitates modeling and verification of grid workflows. Some typical patterns in grid workflows are captured and both static and dynamic formal verification issues are investigated, including structural correctness, specification satisfiability, logic satisfiability and consistency. A grid workflow modeling and verification environment, GridPiAnalyzer, is implemented using formal modeling and verification methods proposed in this work. Performance evaluation results are included using a grid workflow for gravitational wave data analysis.

Cognitive network is a new approach to solve self-management of complex heterogeneous network, and its model and analysis method are key problems to be solved. a formal modeling and analysis method for cognitive network is proposed based on stochastic process algebra. Firstly, basic concept of PEPA is introduced. then system behaviors are abstracted combining with cognitive essence, and formal models for cognitive network are constructed from the angles of macroscopic operating states and internal services with PEPA respectively. on the basis, quantitative analysis method is discussed. the proposed modeling and analysis method has the features logical accuracy, which can apply in any phases and direct the designation of cognitive network.

I think that the conjecture P + NP is not as widely taught in courses of mathematical logic as it should be, in view of its capital importance for the foundation of mathematics. Therefore I am writing this note in the hope that all logicians will always include it in the introductory courses of their subject although it does not appear yet in the appropriate books. The original paper of S. Cook [1], where the conjecture was formulated, was indeed written from the point of view of logic but it became the domain of computer scientists (see [2]), particularly because of a paper of Karp [3] where the combinatorial or computational aspects of the conjecture were developed in a very suggestive way. Let us assume that the teacher has already presented the concept of a first order theory and the concept of a Turing machine (neither the concepts of a decidable theory nor that of a recursive function are needed). Then he may proceed as follows: By a normal theory we shall mean a theory which is formalized with a finite alphabet in first order logic with equality and is axiomatizable by a finite set of axioms and axiom schemata in which one can prove 3xy [ x + y ]. (In [4] it is proved that every theory which is recursively axiomatizable and contains a minimal amount of arithmetic or set theory is normal.) By a proof in a normal theory we mean a Hilbert style proof from the axioms. Let E be a finite alphabet and E* the set of all words, i.e., finite sequences of elements of E. For any E*, I denotes the length of (. Now we introduce a more abstract concept of a theory which we will call a T7T-theory. A T7T-theory is a set of pairs T c E* x E* such that there exists a polynomial P(x, y) and a Turing machine M such that, for any (T, 1) E E* x E*, M can decide in time < P(ITI, I7TI) if (T, 7T) E T. If (i, 7T) E T, thenT is called a theorem of T and 7 is called a proof of i-in T. Every normal theory defines a Tg-theory since the time necessary to check the correctness of a Hilbert style proof in a normal theory can be estimated from above by a polynomial in the length of that proof. Now, a T7n-theory T will be called amenable (to automatization) iff there exists another polynomial Po(x, y) and another Turing machine Mo such that, given any word XE * and any positive integer n, MO can decide in time < Po(ITi-, n) if there exists a X E =* with Ig I < n and such that (i, s) E T. (Notice that if we replaced the condition < Po(iTI, n) by the condition < Po(ITI, Cn) where c = card E, then the concept would trivialize since every T7T-theory would be amenable. In fact, given a time Po(lTI, cn), the machine can form all sequences of symbols of length n and find out if any of them is a proof of i.) It is clear that, after Godel's 1931 discovery that all sufficiently strong theories are undecidable,

The Consistency of A Higher Order Predicate Calculus and Set Theory Based on Combinatory Logic

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Attribute Topology (AT) and Attribute Partial Ordered Structure Diagram (APOSD) are both important research methods of Formal Concept Analysis (FCA) and Knowledge Discovery. Each of them can transform into the other with solid mathematical foundation and specific algorithm. Basing on set theory, this paper mainly researches and identifies the properties between related attributes, between attribute and AT, between attribute and APOSD. All of above is for the purpose of clarifying the relationships and properties in the transformation of AT and APOSD.

Bidimensionality Theory is a meta-algorithmic theory whose main ingredients are the Grid Exclusion Theorem of the Graph Minors series of Robertson and Seymour and the celebrated Courcelle’s Theorem. The grid exclusion theorem states that if a graph excludes a bidimensional grid as a minor (a graph H is a minor of a graph G if H can be obtained by some subgraph of G by con- tracting edges) then its structure topologically resembles the structure of a tree (in technical terms, it has small treewidth). Intuitively, this result says that the absence of the bidimensional structure of a grid implies that a graph has the “mono-dimensional” structure of a tree. On the other side, Courcelle’s theorem states that if a problem on graphs is expressible in Monadic Second Order Logic (MSOL), then it is possible to solve it in linear time when the treewidth of their input graphs is fixed. Intuitively, this theorem expresses the fact that the mono- dimensional structure of a tree (i.e., small treewidth) makes it possible to treat a graph as the input string of a finite-state tree automaton, where the finiteness of its states is guaranteed by the MSOL-expressibility of the problem. Combin- ing these two results together, we derive that the absence of the bidimensional structure of a grid, enables the applicability of the “divide-and-conquer” tech- nique for problems of certain descriptive complexity. It appears that for many graph theoretic problems the existence of a grid-minor (or other bidimensional strucures) on the input graph provides a certificate for an immediate negative (or positive) answer and, for the remaining instances, a dynamic programming approach on graphs of bounded treewidth may give an answer to the problem. This phenomenon reveals fruitful interleave between graph structure and logic in graph algorithms. Bidimensionality Theory aims at systematizing this idea and extending its applicability in diverse paradigms of algorithm design.

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).