Incremental complexity of a bi-objective hypergraph transversal problem

The hypergraph transversal problem has been intensively studied, from both a theoretical and a practical point of view. In particular , its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalization of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs (A, B), and the aim is to find minimal sets which hit all the hyperedges of A while intersecting a minimal set of hyperedges of B. In this paper, we formalize this problem, link it to a problem on monotone boolean $\land$ -- $\lor$ formulae of depth 3 and study its incremental complexity

Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them

Quantitative mu-calculus and CTL Based on Constraint Semirings

Model checking and temporal logics are boolean. The answer to the model checking question does a system satisfy a property? is either true or false, and properties expressed in temporal logics are defined over boolean propositions. While this classic approach is enough to specify and verify boolean temporal properties, it does not allow to reason about quantitative aspects of systems. Some quantitative extensions of temporal logics has been already proposed, especially in the context of probabilistic systems. They allow to answer questions like with which probability does a system satisfy a property? We present a generalization of two well-known temporal logics: CTL and the [mu]-calculus. Both extensions are defined over c-semirings, an algebraic structure that captures many problems and that has been proposed as a general framework for soft constraint satisfaction problems (CSP). Basically, a c-semiring consists of a domain, an additive operation and a multiplicative operation, which satisfy some properties. We present the semantics of the extended logics over transition systems, where a formula is interpreted as a mapping from the set of states to the domain of the c-semiring, and show that the usual connection between CTL and [mu]-calculus does not hold in general. In addition, we reason about the feasibility of computing the logics and illustrate some applications of our framework, including boolean model checking