Abstract State Machines 1988-1998: Commented ASM Bibliography

An annotated bibliography of papers which deal with or use Abstract State Machines (ASMs), as of January 1998.Comment: Also maintained as a BibTeX file at http://www.eecs.umich.edu/gasm

Two Decades of Maude

This paper is a tribute to José Meseguer, from the rest of us in the Maude team, reviewing the past, the present, and the future of the language and system with which we have been working for around two decades under his leadership. After reviewing the origins and the language's main features, we present the latest additions to the language and some features currently under development. This paper is not an introduction to Maude, and some familiarity with it and with rewriting logic are indeed assumed.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories

In flowchart languages, predicates play an interesting double role. In the textual representation, they are often presented as conditions, i.e., expressions which are easily combined with other conditions (often via Boolean combinators) to form new conditions, though they only play a supporting role in aiding branching statements choose a branch to follow. On the other hand, in the graphical representation they are typically presented as decisions, intrinsically capable of directing control flow yet mostly oblivious to Boolean combination. While categorical treatments of flowchart languages are abundant, none of them provide a treatment of this dual nature of predicates. In the present paper, we argue that extensive restriction categories are precisely categories that capture such a condition/decision duality, by means of morphisms which, coincidentally, are also called decisions. Further, we show that having these categorical decisions amounts to having an internal logic: Analogous to how subobjects of an object in a topos form a Heyting algebra, we show that decisions on an object in an extensive restriction category form a De Morgan quasilattice, the algebraic structure associated with the (three-valued) weak Kleene logic $\mathbf{K}^w_3$. Full classical propositional logic can be recovered by restricting to total decisions, yielding extensive categories in the usual sense, and confirming (from a different direction) a result from effectus theory that predicates on objects in extensive categories form Boolean algebras. As an application, since (categorical) decisions are partial isomorphisms, this approach provides naturally reversible models of classical propositional logic and weak Kleene logic.Comment: 19 pages, including 6 page appendix of proofs. Accepted for MFPS XXX

Matching Logic

This paper presents matching logic, a first-order logic (FOL) variant for specifying and reasoning about structure by means of patterns and pattern matching. Its sentences, the patterns, are constructed using variables, symbols, connectives and quantifiers, but no difference is made between function and predicate symbols. In models, a pattern evaluates into a power-set domain (the set of values that match it), in contrast to FOL where functions and predicates map into a regular domain. Matching logic uniformly generalizes several logical frameworks important for program analysis, such as: propositional logic, algebraic specification, FOL with equality, modal logic, and separation logic. Patterns can specify separation requirements at any level in any program configuration, not only in the heaps or stores, without any special logical constructs for that: the very nature of pattern matching is that if two structures are matched as part of a pattern, then they can only be spatially separated. Like FOL, matching logic can also be translated into pure predicate logic with equality, at the same time admitting its own sound and complete proof system. A practical aspect of matching logic is that FOL reasoning with equality remains sound, so off-the-shelf provers and SMT solvers can be used for matching logic reasoning. Matching logic is particularly well-suited for reasoning about programs in programming languages that have an operational semantics, but it is not limited to this

An alternative Gospel of structure: order, composition, processes

We survey some basic mathematical structures, which arguably are more primitive than the structures taught at school. These structures are orders, with or without composition, and (symmetric) monoidal categories. We list several `real life' incarnations of each of these. This paper also serves as an introduction to these structures and their current and potentially future uses in linguistics, physics and knowledge representation.Comment: Introductory chapter to C. Heunen, M. Sadrzadeh, and E. Grefenstette. Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse. Oxford University Press, 201

Formalized proof, computation, and the construction problem in algebraic geometry

An informal discussion of how the construction problem in algebraic geometry motivates the search for formal proof methods. Also includes a brief discussion of my own progress up to now, which concerns the formalization of category theory within a ZFC-like environment