Coalgebraic fixpoint logic Expressivity and completeness result
Fatemeh Seifan
Abstract:
This dissertation studies the expressivity and completeness of the coalgebraic µcalculus. This logic is a coalgebraic generalization of the standard µ-calculus, which creates a uniform framework to study different modal fixpoint logics. Our main objective is to show that several important results, such as uniform interpolation, expressive completeness and axiomatic completeness of the standard µ-calculus can be generalized to the level of coalgebras. To achieve this goal we develop automata and game-theoretic tools to study properties of coalgebraic µ-calculus.
In Chapter 3 we prove a uniform interpolation theorem for the coalgebraic µcalculus. This theorem generalizes a result by D’Agostino and Hollenberg [DH00] to a wider class of fixpoint logics, including the monotone µ-calculus, which is the extension of monotone modal logic with fixpoint operators. To this aim, first we consider a key property of automata, namely closure under projection. We prove this property, which is known to hold for weak pullback-preserving functors, for a wider class of functors, i.e., functors with a certain type of relation lifting called a quasi-functorial lax extension. Then we show that closure under projection implies definability of the bisimulation quantifier in the language of the coalgebraic µ-calculus. Finally, we use this result to prove a uniform interpolation theorem for the coalgebraic µ-calculus.
In Chapter 4 we generalize the Janin-Walukiewicz theorem [JW96], which states that the modal µ-calculus captures exactly the bisimulation invariant fragment of monadic second-order logic, to the level of coalgebras. For that, we first introduce a notion of coalgebraic monadic second-order logic MSOT for coalgebras of type T. Following the tradition of automata-theoretic approaches toward monadic second-order logic we define a class of parity automata that corresponds to MSOT. Similar to well-known results for monadic second-order logic over trees, we provide a translation from MSOT to these automata, which is truth-preserving on the class of tree-like T-coalgebras. We then proceed by identifying the class of functors T for which the coalgebraic µ-calculus µMLT corresponds to the fragment 219 220 Abstract of MSOT that is invariant under behavioural equivalence. We approach this at the level of one-step languages and show that to prove a coalgebraic characterization theorem, it suffices to find what we call an adequate uniform construction for the functor T. As applications of this result we obtain a partly new proof of the Janin-Walukiewicz theorem, and bisimulation invariance results for the bag functor (graded modal logic), and all exponential polynomial functors. In the last part of this chapter we consider in some detail the monotone neighborhood functor M, which provides coalgebraic semantics for monotone modal logic. It turns out that there is no adequate uniform construction for M. We resolve this problem by proving a second version of our general characterization theorem.
In Chapter 5 we prove an axiomatic completeness result for the coalgebraic µ-calculus. Here we follow the same track as in Chapter 4: a crucial role in our proofs is played by the notions of one-step logic and automata. Applying ideas from automata theory and coalgebra, our aim is to generalise Walukiewicz’ proof of completeness for the modal µ-calculus [Wal00] to the level of coalgebras. Our main contribution is to bring automata explicitly into the proof theory. This automata-theoretic approach lets us distinguish two key aspects of the coalgebraic µ-calculus (and the standard µ-calculus): the one-step dynamic encoded in the semantics of the modal operators, and the combinatorics involved in dealing with nested fixpoints. This distinction allows us to work with these two features in a largely independent manner. More in detail, the main tools that we employ in our automata-theoretic approach are two kinds of games for modal automata: the satisfiability game and the consequence game, and two special kinds of modal automata: disjunctive and semi-disjunctive automata. The consequence game between two automata is an original contribution of our approach. It is an infinite two-player game concentrating on establishing structural connections between the automata. In addition to the disjunctive automata that are known from the work of Janin and Walukiewicz [JW95], we define the class of semi-disjunctive automata and show that similar to the disjunctive ones, semi-disjunctive automata also have a fairly simple trace theory (combinatorics) with regards to the satisfiability and consequence games. As our main result we then provide a generalization of Walukiewicz’ main technical result, which states that every formula of the modal µ-calculus provably implies the translation of a disjunctive automaton, to the level of coalgebras. From this the completeness theorem is almost immediate.