The exogenous approach to enriching any given base logic for probabilistic and quantum reasoning is brought into the realm of institutions. The theory of institutions helps in capturing the precise relationships between the logics that are obtained, and, furthermore, helps in analyzing some of the key design decisions and opens the way to make the approach more useful and, at the same time, more abstract. 1

The status of the Common Framework Initiative (CoFI) and the Common Algebraic Specification Language (Casl) are briefly presented.

Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. The first stored program computers were programmed in binary, which soon gave way to assembly languages that allowed symbolic codes for operations and addresses. fortran began the spread of "high level" programming languages, though at the time it was strongly opposed by many assembly programmers; important features that developed later include blocks, recursive procedures, flexible types, classes, inheritance, modules, and genericity. Without going into the philosophical problems raised by abstraction (which in view of the discussion of realism in Section 4 may be considerable), it seems clear that the mathematics used to describe programming concepts should in general get more abstract as the programming concepts get more abstract. Nevertheless, there has been great resistance to u...

Abstract: We investigate behavioral institutions and refinements in the context of the object oriented paradigm. The novelty of our approach is the application of generalized abstract algebraic logic theory of hidden heterogeneous deductive systems (called hidden k-logics) to the algebraic specification of object oriented programs. This is achieved through the Leibniz congruence relation and its combinatorial properties. We reformulate the notion of hidden k-logic as well as the behavioral logic of a hidden k-logic as institutions. We define refinements as hidden signature morphisms having the extra property of preserving logical consequence. A stricter class of refinements, the ones that preserve behavioral consequence, is studied. We establish sufficient conditions for an ordinary signature morphism to be a behavioral refinement.

Abstract: Verifying design instead of code can be an effective and practical approach to obtaining verified software. This paper argues that proof scores are an attractive method for verifying design, in that they achieve a balance in which the respective capabilities of humans and machines are utilized optimally. 1 Verifying Code or Design Although creation of a verifying compiler is a difficult challenge, recent developments suggest that there are ways to make it easier. Systems that generate lexical analyzers and parsers already have a long history (e.g. Lex and Yacc), and recent work of Sorin Lerner [23] shows that the same can be done for compiler backends; there is also work suggesting that code generation modules can be automatically generated (e.g. using intermediate languages). Unfortunately, a great number of different compilers are needed in today’s software world, and the underlying machine architectures are evolving, as are the languages, so it would be difficult to create verifying compilers for all useful combinations of language and platform, and code verification for such tools still remains very difficult. Major impediments include the unsolvability of discovering loop invariants, the potential unsolvability of loop once they are found, and the further difficulties raised by interactivity, nondeterminism, concurrency, distribution, active agents, and unreliable communication. A long term approach is to use high level, application specific source languages, in order to greatly simplify source program verification by eliminating many obscure features of current languages. In the meantime, a currently feasible approach is to verify the design of software, instead of its code; experience shows that design verification often leads to better design, and nearly always leads to greater conceptual clarity. An aditional motivation is that the main sources of errors in software are in areas other than code, namely, requirements, specification, and design. 2

Abstract. This paper explores reasoning about space and time, e.g., in metaphors of time as space; an important method is to find minimal assumptions needed to reach the same conclusions that humans reach. Some mathematical language, including the notion of triad, is introduced for this purpose, formalizing and generalizing the cognitive semantics approaches to conceptual spaces (in the senses of both Fauconnier & Turner and of Gärdenfors), blending, and metaphor; in particular, continuous mathematics is used to model space and time. A new explanation of emergent structure in blend spaces is also discussed, and proposed as a source of creativity. Four main examples illustrate the approach, and an appendix encapsulates the most difficult mathematics. 1

We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial algebraic specifications through means specific to ordinary algebraic reasoning, such as term rewriting.

The Semantic Web (SW) is viewed as the next generation of the Web that enables intelligent software agents to process and aggregate data autonomously. Ontology languages provide basic vocabularies to semantically markup data on the SW. We have witnessed an increase of numbers of SW languages in the last years. These languages, such as RDF, RDF Schema (RDFS), the OWL suite of languages, the OWL − suite, SWRL, are based on different semantics, such as the RDFS-based, description logic-based, Datalog-based semantics. The relationship among the various semantics poses a challenge for the SW community for making the languages interoperable. Institutions provide a means of reasoning about software specifications regardless of the logical system. This makes it an ideal candidate to represent and reason about the various languages in the Semantic Web. In this paper, we construct institutions for the SW languages and use institution morphisms to relate them. We show that RDF framework together with the RDF serializations of SW languages form an indexed institution. This allows the use of Grothendieck institutions to combine Web ontologies described in various languages.

We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the Curry-Howard-Tait paradigm. We then prove logic-independent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.

Abstract. In this paper, we show how the web ontology language OWL can be accommodated within the larger framework of the heterogeneous common algebraic specification language HETCASL. Through this change in perspective, OWL can benefit from various useful HETCASL features concerning structuring, modularity, and heterogeneity. This tackles a major problem area in ontology engineering: re-use of ontologies and re-combination of ontological modules. We discuss in particular: (1) the extension of the Manchester syntax for OWL with structuring mechanisms of CASL, allowing for explicit modularisation; (2) automatic translations between ontology languages to support ontology design across different ontology languages (heterogeneity); (3) heterogeneous ontology refinements, and corresponding automated reasoning support for different logics. 1