Probabilistic logics based on Riesz spaces

We introduce a novel real-valued endogenous logic for expressing properties of probabilistic transition systems called Riesz modal logic. The design of the syntax and semantics of this logic is directly inspired by the theory of Riesz spaces, a mature field of mathematics at the intersection of universal algebra and functional analysis. By using powerful results from this theory, we develop the duality theory of the Riesz modal logic in the form of an algebra-to-coalgebra correspondence. This has a number of consequences including: a sound and complete axiomatization, the proof that the logic characterizes probabilistic bisimulation and other convenient results such as completion theorems. This work is intended to be the basis for subsequent research on extensions of Riesz modal logic with fixed-point operators

Quantitative Graded Semantics and Spectra of Behavioural Metrics

Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the classical linear-time/branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics come in various degrees of granularity, depending on the observer's ability to interact with the system. Graded monads have been shown to provide a unifying framework for spectra of behavioural equivalences. Here, we transfer this principle to spectra of behavioural metrics, working at a coalgebraic level of generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we discuss presentations of graded monads on the category of metric spaces in terms of graded quantitative equational theories. Moreover, we obtain a canonical generic notion of invariant real-valued modal logic, and provide criteria for such logics to be expressive in the sense that logical distance coincides with the respective behavioural distance. We thus recover recent expressiveness results for coalgebraic branching-time metrics and for trace distance in metric transition systems; moreover, we obtain a new expressiveness result for trace semantics of fuzzy transition systems. We also provide a number of salient negative results. In particular, we show that trace distance on probabilistic metric transition systems does not admit a characteristic real-valued modal logic at all

One-variable fragments of intermediate logics over linear frames

A correspondence is established between one-variable fragments of (first-order) intermediate logics defined over a fixed countable linear frame and Gödel modal logics defined over many-valued equivalence relations with values in a closed subset of the real unit interval. It is also shown that each of these logics can be interpreted in the one-variable fragment of the corresponding constant domain intermediate logic, which is equivalent to a Gödel modal logic defined over (crisp) equivalence relations. Although the latter modal logics in general lack the finite model property with respect to their frame semantics, an alternative semantics is defined that has this property and used to establish co-NP-completeness results for the one-variable fragments of the corresponding intermediate logics both with and without constant domains

(Metric) Bisimulation Games and Real-Valued Modal Logics for Coalgebras

Behavioural equivalences can be characterized via bisimulations, modal logics and spoiler-defender games. In this paper we review these three perspectives in a coalgebraic setting, which allows us to generalize from the particular branching type of a transition system. We are interested in qualitative notions (classical bisimulation) as well as quantitative notions (bisimulation metrics). Our first contribution is to introduce a spoiler-defender bisimulation game for coalgebras in the classical case. Second, we introduce such games for the metric case and furthermore define a real-valued modal coalgebraic logic, from which we can derive the strategy of the spoiler. For this logic we show a quantitative version of the Hennessy-Milner theorem

A behavioural pseudometric for probabilistic transition systems

AbstractDiscrete notions of behavioural equivalence sit uneasily with semantic models featuring quantitative data, like probabilistic transition systems. In this paper, we present a pseudometric on a class of probabilistic transition systems yielding a quantitative notion of behavioural equivalence. The pseudometric is defined via the terminal coalgebra of a functor based on a metric on the space of Borel probability measures on a metric space. States of a probabilistic transition system have distance 0 if and only if they are probabilistic bisimilar. We also characterize our distance function in terms of a real-valued modal logic

Approximate reasoning for real-time probabilistic processes

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -- such as probabilities and time -- in the model. We give a fixed point characterization of the pseudo-metric. This makes available coinductive reasoning principles for reasoning about distances. We demonstrate that our approach is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity. We provide a logical characterization of this uniformity using a real-valued modal logic. We show that several quantitative properties of interest are continuous with respect to the pseudo-metric. Thus, if two processes are metrically close, then observable quantitative properties of interest are indeed close.Comment: Preliminary version appeared in QEST 0

The Logic of Internal Rational Agent

In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev's Logic of Rational Agent \textbf{LRA} \cite{LRA}. We call our logic $\bf LIRA$ (Logic of Internal Rational Agency). In contrast to \textbf{LRA}, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko's bi-facial truth logic \cite{ZS}. This logic may be useful in a situation when according to an agent's point of view (i.e. internal point of view) her/his reasoning is rational, while from the external one it might be not the case. One may use \textbf{LIRA}, if one wants to reconstruct an agent's way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev's necessity and possibility operators for \textbf{LRA} definable by means of four-valued Kripke-style semantics and show that, due to two negations (as well as their combination) of \textbf{LRA}, two more possibility operators for \textbf{LRA} can be defined. Then we slightly modify all these modalities to be appropriate for $\bf LIRA$. Finally, we formalize all the truth-functional $n$-ary extensions of the negation fragment of $\bf LIRA$ (including $\bf LIRA$ itself) as well as their basic modal extension via linear-type natural deduction systems

The Logic of Internal Rational Agent

In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev's Logic of Rational Agent \textbf{LRA} \cite{LRA}. We call our logic $\bf LIRA$ (Logic of Internal Rational Agency). In contrast to \textbf{LRA}, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko's bi-facial truth logic \cite{ZS}. This logic may be useful in a situation when according to an agent's point of view (i.e. internal point of view) her/his reasoning is rational, while from the external one it might be not the case. One may use \textbf{LIRA}, if one wants to reconstruct an agent's way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev's necessity and possibility operators for \textbf{LRA} definable by means of four-valued Kripke-style semantics and show that, due to two negations (as well as their combination) of \textbf{LRA}, two more possibility operators for \textbf{LRA} can be defined. Then we slightly modify all these modalities to be appropriate for $\bf LIRA$. Finally, we formalize all the truth-functional $n$-ary extensions of the negation fragment of $\bf LIRA$ (including $\bf LIRA$ itself) as well as their basic modal extension via linear-type natural deduction systems