A Model of Mizar Concepts - Unification

The aim of this paper is to develop a formal theory of Mizar linguistic concepts following the ideas from [6] and [7]. The theory presented is an abstraction from the existing implementation of the Mizar system and is devoted to the formalization of Mizar expressions. The concepts formalized here are: standarized constructor signature, arity-rich signatures, and the unification of Mizar expressions.The University of Finance and Management, Białystok-Ełk, PolandGrzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547-552, 1991.Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.Grzegorz Bancerek. Subtrees. Formalized Mathematics, 5(2):185-190, 1996.Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra. Formalized Mathematics, 6(2):279-287, 1997.Grzegorz Bancerek. On the structure of Mizar types. In Herman Geuvers and Fairouz Kamareddine, editors, Electronic Notes in Theoretical Computer Science, volume 85. Elsevier, 2003.Grzegorz Bancerek. Towards the construction of a model of Mizar concepts. Formalized Mathematics, 16(2):207-230, 2008, doi:10.2478/v10037-008-0027-x.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Artur Korniłowicz. Yet another construction of free algebra. Formalized Mathematics, 9(4):779-785, 2001.Grzegorz Bancerek and Yatsuka Nakamura. Full adder circuit. Part I. Formalized Mathematics, 5(3):367-380, 1996.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67-74, 1996.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

More on Descriptive Complexity of Second-Order HORN Logics

This paper concerns Gradel's question asked in 1992: whether all problems which are in PTIME and closed under substructures are definable in second-order HORN logic SO-HORN. We introduce revisions of SO-HORN and DATALOG by adding first-order universal quantifiers over the second-order atoms in the bodies of HORN clauses and DATALOG rules. We show that both logics are as expressive as FO(LFP), the least fixed point logic. We also prove that FO(LFP) can not define all of the problems that are in PTIME and closed under substructures. As a corollary, we answer Gradel's question negatively

An insertion operator preserving infinite reduction sequences

International audienceA common way to show the termination of the union of two abstract reduction systems, provided both systems terminate, is to prove that they enjoy a specific property (some sort of 'commutation' for instance). This specific property is actually used to show that, for the union not to terminate, one of the systems must itself be non-terminating, which leads to a contradiction. Unfortunately, the property may be impossible to prove because some of the objects that are reduced do not enjoy an adequate form. Hence the purpose of this paper is threefold: - First, it introduces an operator enabling us to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property. Of course, some new properties will need to be verified. - Second, as an instance of our technique, the operator is applied to relax a well-known lemma stating the termination of the union of two termination abstract reduction systems. - Finally, this lemma is applied in a peculiar and then in a more general way to show the termination of some lambda calculi with inductive types augmented with specific reductions dealing with: (i) copies of inductive types; (ii) the representation of symmetric groups

Proof-carrying Bytecode

AbstractIn the Mobile Resource Guarantees project's Proof Carrying Code implementation, .class files are associated with Isabelle [Tobias Nipkow, Lawrence C. Paulson, Markus Wenzel, Isabelle/HOL: A Proof Assistant for Higher-Order Logic, volume 2283 of LNCS. Springer-Verlag, 2002] proof scripts containing proofs of bounds on their resource consumption. By using the tools gf and isabelle on the consumer-side, it is possible to verify after download, that a piece of code conforms to a particular resource policy specified by the consumer, and prevent execution in the event that it does not. We present here a prototype implementation using certain features of the J2SE 5.0 Platform [Sun Microsystems, Inc. Java 2 Platform, Standard Edition 1.5.0, http://java.sun.com/j2se/1.5.0/, May 27, 2004]. The (unmodified) bytecode and its proof are packaged as a JAR file for convenient distribution. The codebase uses Java agents providing the Instrumentation interface, and implements a custom permission class and Security Manager. The external tools are invoked from within Java. Two system commands makeMRGjar and MRGjava provide a convenient way of using this implementation

Context Aware Service Oriented Computing in Mobile Ad Hoc Networks

These days we witness a major shift towards small, mobile devices, capable of wireless communication. Their communication capabilities enable them to form mobile ad hoc networks and share resources and capabilities. Service Oriented Computing (SOC) is a new emerging paradigm for distributed computing that has evolved from object-oriented and component-oriented computing to enable applications distributed within and across organizational boundaries. Services are autonomous computational elements that can be described, published, discovered, and orchestrated for the purpose of developing applications. The application of the SOC model to mobile devices provides a loosely coupled model for distributed processing in a resource-poor and highly dynamic environment. Cooperation in a mobile ad hoc environment depends on the fundamental capability of hosts to communicate with each other. Peer-to-peer interactions among hosts within communication range allow such interactions but limit the scope of interactions to a local region. Routing algorithms for mobile ad hoc networks extend the scope of interactions to cover all hosts transitively connected over multi-hop routes. Additional contextual information, e.g., knowledge about the movement of hosts in physical space, can help extend the boundaries of interactions beyond the limits of an island of connectivity. To help separate concerns specific to different layers, a coordination model between the routing layer and the SOC layer provides abstractions that mask the details characteristic to the network layer from the distributed computing semantics above. This thesis explores some of the opportunities and challenges raised by applying the SOC paradigm to mobile computing in ad hoc networks. It investigates the implications of disconnections on service advertising and discovery mechanisms. It addresses issues related to code migration in addition to physical host movement. It also investigates some of the security concerns in ad hoc networking service provision. It presents a novel routing algorithm for mobile ad hoc networks and a novel coordination model that addresses space and time explicitly

Linda[m] and Tiamat: Providing generative communications in a changing world

When generative communications, as exemplified by Linda [Gel85], were originally proposed, they were intended as a mechanism for coordination of parallel processes. Since that time, they have been adapted to a variety of distributed environments with great success, as can be seen in commercial systems such as T Spaces [WMLF98]. The time, space and identity decoupling afforded to coordinating entities by generative communications also seems to be ideally suited to mobile environments where devices can come and go frequently and often without warning. Such a rapidly changing environment, however, presents a new set of challenges and attempts to introduce the generative communications paradigm into these environments have, so far, met with limited success. Indeed evaluation of research platforms, such as LIME (Linda In a Mobile Environment) [PMR99.MPR01] and L[2]imbo [DFWB98] have led some to conclude that the generative communication paradigm is not well suited to mobile environments. It is my belief, however, that it is the research platforms in question, rather than the paradigm, which do not fit well with mobile environments. These platforms either attempt to impose tight constraints on an inherently loosely constrained environment, or require significant alterations to the semantics of generative communications. I believe that these systems do not work well as they are not designed around the environment, rather they are forced onto the environment. I will begin by examining why these systems do not suit their environment. This done, I will then show that the conclusions drawn from these systems, namely that generative communications are unsuitable for mobile environments, are incorrect. Further, through construction and examination of a proof of concept system built around an environment-centric design, I will show that generative communications can be provided in a mobile environment with few (minor) semantic alterations. An evaluation of some of the mechanisms used will also be presented along with characterisation of the operation of the system. A comparison with existing mobile solutions will be used to highlight how the environment-driven design results in a system which better suits the nature of the target environment