The intuitionistic temporal logic of dynamical systems

A dynamical system is a pair $(X,f)$, where $X$ is a topological space and $f\colon X\to X$ is continuous. Kremer observed that the language of propositional linear temporal logic can be interpreted over the class of dynamical systems, giving rise to a natural intuitionistic temporal logic. We introduce a variant of Kremer's logic, which we denote ${\sf ITL^c}$, and show that it is decidable. We also show that minimality and Poincar\'e recurrence are both expressible in the language of ${\sf ITL^c}$, thus providing a decidable logic expressive enough to reason about non-trivial asymptotic behavior in dynamical systems

Axiomatic systems and topological semantics for intuitionistic temporal logic

We propose four axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. Our topological semantics features a new interpretation for the `henceforth' modality that is a natural intuitionistic variant of the classical one. Using the soundness results, we show that the four logics obtained from the axiomatic systems are distinct. Finally, we show that when the language is restricted to the `henceforth'-free fragment, the set of valid formulas for the relational and topological semantics coincide

Modal Logics for Mobile Processes Revisited

We revisit the logical characterisations of various bisimilarity relations for the finite fragment of the ?-calculus. Our starting point is the early and the late bisimilarity, first defined in the seminal work of Milner, Parrow and Walker, who also proved their characterisations in fragments of a modal logic (which we refer to as the MPW logic). Two important refinements of early and late bisimilarity, called open and quasi-open bisimilarity, respectively, were subsequently proposed by Sangiorgi and Walker. Horne, et. al., showed that open and quasi-bisimilarity are characterised by intuitionistic modal logics: OM (for open bisimilarity) and FM (for quasi-open bisimilarity). In this work, we attempt to unify the logical characterisations of these bisimilarity relations, showing that they can be characterised by different sublogics of a unifying logic. A key insight to this unification derives from a reformulation of the four bisimilarity relations (early, late, open and quasi-open) that uses an explicit name context, and an observation that these relations can be distinguished by the relative scoping of names and their instantiations in the name context. This name context and name substitution then give rise to an accessibility relation in the underlying Kripke semantics of our logic, that is captured logically by an S4-like modal operator. We then show that the MPW, the OM and the FM logics can be embedded into fragments of our unifying classical modal logic. In the case of OM and FM, the embedding uses the fact that intuitionistic implication can be encoded in modal logic S4

Ruitenburg's Theorem via Duality and Bounded Bisimulations

For a given intuitionistic propositional formula A and a propositional variable x occurring in it, define the infinite sequence of formulae { A \_i | i$\ge$1} by letting A\_1 be A and A\_{i+1} be A(A\_i/x). Ruitenburg's Theorem [8] says that the sequence { A \_i } (modulo logical equivalence) is ultimately periodic with period 2, i.e. there is N $\ge$ 0 such that A N+2 $\leftrightarrow$ A N is provable in intuitionistic propositional calculus. We give a semantic proof of this theorem, using duality techniques and bounded bisimulations ranks

Towards a Generic Model Theory: Automatic Bisimulations for Atomic, Molecular and First-order Logics

After observing that the truth conditions of connectives of non-classical logics are generally defined in terms of formulas of first-order logic (FOL), we introduce protologics, a class of logics whose connectives are defined by arbitrary first-order formulas. Then, we identify two subclasses of protologics which are particularly well-behaved. We call them atomic and molecular logics. Notions of invariance for atomic and molecular logics can be automatically defined from the truth conditions of their connectives, bisimulations do not need to be defined by hand on a case by case basis for each logic. Moreover, molecular logics behave as 'paradigmatic logics': every first-order logic and every protologic is as expressive as a molecular logic. Then, we prove a series of model-theoretical results for molecular logics which characterize them as fragments of FOL and which provide criteria for axiomatizability and definability of a class of models in these logics. In particular, we rediscover van Benthem's theorem for modal logic as a specific instance of our generic theorems and other results for modal intuitionistic logic and temporal logic. We also discover a wide range of novel results, such as for the Lambek calculus. Then, we apply our method and generic results to FOL and find out novel invariance notions for FOL, that we call predicate bisimulation and first-order bisimulation. They refine the usual notions of isomorphism and partial isomorphism. We prove generalizations as well as new versions of the Keisler theorems for countable languages in which isomorphisms are replaced by predicate bisimulations and first-order bisimulations