Phase Semantics for Linear Logic with Least and Greatest Fixed Points

The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (?MALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to ?MALL with explicit (co)induction (i.e. ?MALL^{ind}). We introduce a Tait-style system for ?MALL called ?MALL_? where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property

Some model theory for the modal μ\mu-calculus: syntactic characterisations of semantic properties

This paper contributes to the theory of the modal $\mu$-calculus by proving some model-theoretic results. More in particular, we discuss a number of semantic properties pertaining to formulas of the modal $\mu$-calculus. For each of these properties we provide a corresponding syntactic fragment, in the sense that a $\mu$-formula $\xi$ has the given property iff it is equivalent to a formula $\xi'$ in the corresponding fragment. Since this formula $\xi'$ will always be effectively obtainable from $\xi$, as a corollary, for each of the properties under discussion, we prove that it is decidable in elementary time whether a given $\mu$-calculus formula has the property or not. The properties that we study all concern the way in which the meaning of a formula $\xi$ in a model depends on the meaning of a single, fixed proposition letter $p$. For example, consider a formula $\xi$ which is monotone in $p$; such a formula a formula $\xi$ is called continuous (respectively, fully additive), if in addition it satisfies the property that, if $\xi$ is true at a state $s$ then there is a finite set (respectively, a singleton set) $U$ such that $\xi$ remains true at $s$ if we restrict the interpretation of $p$ to the set $U$. Each of the properties that we consider is, in a similar way, associated with one of the following special kinds of subset of a tree model: singletons, finite sets, finitely branching subtrees, noetherian subtrees (i.e., without infinite paths), and branches. Our proofs for these characterization results will be automata-theoretic in nature; we will see that the effectively defined maps on formulas are in fact induced by rather simple transformations on modal automata. Thus our results can also be seen as a contribution to the model theory of modal automata

Aleph1 and the Modal mu-Calculus

For a regular cardinal kappa, a formula of the modal mu-calculus is kappa-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of kappa-directed sets. We define the fragment C1 (x) of the modal mu-calculus and prove that all the formulas in this fragment are aleph_1-continuous. For each formula phi(x) of the modal mu-calculus, we construct a formula psi(x) in C1 (x) such that phi(x) is kappa-continuous, for some kappa, if and only if psi(x) is equivalent to phi(x). Consequently, we prove that (i) the problem whether a formula is kappa-continuous for some kappa is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment C0(x) studied by Fontaine and the fragment C1 (x). We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal mu-calculus. An ordinal alpha is the closure ordinal of a formula phi(x) if its interpretation on every model converges to its least fixed-point in at most alpha steps and if there is a model where the convergence occurs exactly in alpha steps. We prove that omega_1, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, omega, omega_1 by using the binary operator symbol + gives rise to a closure ordinal