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In this paper we provide the first (as far as we know) direct calculus deciding satisfiability of formulae in negation normal form in the fragment of FHL (full hybrid logic with the binder, including the global and converse modalities), where no occurrence of a universal operator is in the scope of a binder. By means of a satisfiability preserving translation of formulae, the calculus can be turned into a satisfiability decision procedure for the fragment FHL \ 2↓2, i.e. formulae in negation normal form where no occurrence of the binder is both in the scope of and contains in its scope a universal operator.The calculus is based on tableaux and termination is achieved by means of a form of anywhere blocking with indirect blocking. Direct blocking is a relation between nodes in a tableau branch, holding whenever the respective labels (formulae) are equal up to (a proper form of) nominal renaming. Indirect blocking is based on a partial order on the nodes of a tableau branch, which arranges them into a tree-like structure.

Abstract-Planning problems are usually expressed by specifying which actions can be performed to obtain a given goal. In temporal planning problems, actions come with a time duration and can overlap in time, which noticeably increase the complexity of the reasoning process. Action-based temporal planning has been thoroughly studied from the complexity-theoretic point of view, and it has been proved to be EXPSPACEcomplete in its general formulation. Conversely, timelinebased planning problems are represented as a collection of variables whose time-varying behavior is governed by a set of temporal constraints, called synchronization rules. Timelines provide a unified framework to reason about planning and execution under uncertainty. Timeline-based systems are being successfully employed in real-world complex tasks, but, in contrast to action-based planning, little is known on their computational complexity and expressiveness. In particular, a comparison of the expressiveness of the action-and timelinebased formalisms is still missing. This paper contributes a first step in this direction by proving that timelines are expressive enough to capture action-based temporal planning, showing as a byproduct the EXPSPACE-completeness of timeline-based planning with no temporal horizon and bounded temporal relations only.

Planning for real world problems with explicit temporal constraints is a challenging problem. Among several approaches, the use of flexible timelines in Planning and Scheduling (P&S) has demonstrated to be successful in a number of concrete applications, such as, for instance, autonomous space systems. A flexible timeline describes an envelope of possible solutions which can be exploited by an executive system for robust on-line execution. A remarkable research effort has been dedicated to design, build and deploy software environments, like EUROPA, ASPEN, and APSI-TRF, for the synthesis of timeline-based P&S applications. Several attempts have also been made to characterize the concept of timelines. Nevertheless, a formal characterization of flexible timelines and plans is still missing.\ud \ud This paper presents a formal account of flexible timelines aiming at providing a general semantics for related planning concepts such as domains, goals, problems, constraints and flexible plans. Some basic properties of the defined concepts are also stated and proved. A simple running example inspired by a real world planning domain is exploited to illustrate the proposed formal notions. Finally, a planning tool, called Extensible Planning and Scheduling Library (EPSL), is briefly presented, which is able to generate flexible plans that are compliant with the given semantics

In this work, the problem of performing abduction in modal logics is addressed, along the lines of 3], where a proof theoretical abduction method for full rst order classical logic is de ned, based on tableaux and Gentzen-type systems. This work applies the same methodology to face modal abduction. The non-classical context enforces the value of analytical proof systems as tools to face the meta-logical and proof-theoretical questions involved in abductive reasoning. The similarities and di erences between quanti ers and modal operators are investigated and proof theoretical abduction methods for the modal systems K, D, T and S4 are de ned, that are sound and complete. The construction of the abductive explanations is in strict relation with the expansion rules for the modal logics, in a modular manner that makes local modi cations possible. The method given in this paper is general, in the sense that it can be adapted to any propositional modal logic for which analytic tableaux are provided. Moreover, the way towards an extension to rst order modal logic is straightforward.

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