On Modal Logics of Partial Recursive Functions

The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established

Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity

Step-Indexed Normalization for a Language with General Recursion

The Trellys project has produced several designs for practical dependently typed languages. These languages are broken into two fragments-a_logical_fragment where every term normalizes and which is consistent when interpreted as a logic, and a_programmatic_fragment with general recursion and other convenient but unsound features. In this paper, we present a small example language in this style. Our design allows the programmer to explicitly mention and pass information between the two fragments. We show that this feature substantially complicates the metatheory and present a new technique, combining the traditional Girard-Tait method with step-indexed logical relations, which we use to show normalization for the logical fragment.Comment: In Proceedings MSFP 2012, arXiv:1202.240

Generic Trace Semantics via Coinduction

Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category. This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets. Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity. Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

The Temporal Logic of two dimensional Minkowski spacetime is decidable

We consider Minkowski spacetime, the set of all point-events of spacetime under the relation of causal accessibility. That is, ${\sf x}$ can access ${\sf y}$ if an electromagnetic or (slower than light) mechanical signal could be sent from ${\sf x}$ to ${\sf y}$. We use Prior's tense language of ${\bf F}$ and ${\bf P}$ representing causal accessibility and its converse relation. We consider two versions, one where the accessibility relation is reflexive and one where it is irreflexive. In either case it has been an open problem, for decades, whether the logic is decidable or axiomatisable. We make a small step forward by proving, for the case where the accessibility relation is irreflexive, that the set of valid formulas over two-dimensional Minkowski spacetime is decidable, decidability for the reflexive case follows from this. The complexity of either problem is PSPACE-complete. A consequence is that the temporal logic of intervals with real endpoints under either the containment relation or the strict containment relation is PSPACE-complete, the same is true if the interval accessibility relation is "each endpoint is not earlier", or its irreflexive restriction. We provide a temporal formula that distinguishes between three-dimensional and two-dimensional Minkowski spacetime and another temporal formula that distinguishes the two-dimensional case where the underlying field is the real numbers from the case where instead we use the rational numbers.Comment: 30 page