A system of relational syllogistic incorporating full Boolean reasoning

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: Some A are R-related to some B; Some A are R-related to all B; All A are R-related to some B; All A are R-related to all B. Such primitives formalize sentences from natural language like `All students read some textbooks'. Here A and B denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.Comment: Available at http://link.springer.com/article/10.1007/s10849-012-9165-

Graphs Identified by Logics with Counting

We classify graphs and, more generally, finite relational structures that are identified by C2, that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by C2. Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures. We provide constructions that solve the inversion problem for finite structures in linear time. This problem has previously been shown to be polynomial time solvable by Martin Otto. For graphs, we conclude that every C2-equivalence class contains a graph whose orbits are exactly the classes of the C2-partition of its vertex set and which has a single automorphism witnessing this fact. For general k, we show that such statements are not true by providing examples of graphs of size linear in k which are identified by C3 but for which the orbit partition is strictly finer than the Ck-partition. We also provide identified graphs which have vertex-colored versions that are not identified by Ck.Comment: 33 pages, 8 Figure

Prompt interval temporal logic

Interval temporal logics are expressive formalisms for temporal representation and reasoning, which use time intervals as primitive temporal entities. They have been extensively studied for the past two decades and successfully applied in AI and computer science. Unfortunately, they lack the ability of expressing promptness conditions, as it happens with the commonly-used temporal logics, e.g., LTL: whenever we deal with a liveness request, such as \u201csomething good eventually happens\u201d, there is no way to impose a bound on the delay with which it is fulfilled. In the last years, such an issue has been addressed in automata theory, game theory, and temporal logic. In this paper, we approach it in the interval temporal logic setting. First, we introduce PROMPT-PNL, a prompt extension of the well-studied interval temporal logic PNL, and we prove the undecidability of its satisfiability problem; then, we show how to recover decidability (NEXPTIME-completeness) by imposing a natural syntactic restriction on it

Modal Logics of Topological Relations

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity

Interval temporal logic model checking: The border between good and bad HS fragments

The model checking problem has thoroughly been explored in the context of standard point-based temporal logics, such as LTL, CTL, and CTL 17, whereas model checking for interval temporal logics has been brought to the attention only very recently. In this paper, we prove that the model checking problem for the logic of Allen\u2019s relations started-by and finished-by is highly intractable, as it can be proved to be EXPSPACE-hard. Such a lower bound immediately propagates to the full Halpern and Shoham\u2019s modal logic of time intervals (HS). In contrast, we show that other noteworthy HS fragments, namely, Propositional Neighbourhood Logic extended with modalities for the Allen relation starts (resp., finishes) and its inverse started-by (resp., finished-by), turn out to have\u2014maybe unexpectedly\u2014the same complexity as LTL (i.e., they are PSPACE-complete), thus joining the group of other already studied, well-behaved albeit less expressive, HS fragments

Controlled English for Reasoning on the Semantic Web

The existing Semantic Web languages have a very technical focus and fail to provide good usability for users with no background in formal methods. We argue that controlled natural languages like Attempto Controlled English (ACE) can solve this problem. ACE is a subset of English that can be translated into various logic based languages, among them the Semantic Web standards OWL and SWRL. ACE is accompanied by a set of tools, namely the parser APE, the Attempto Reasoner RACE, the ACE View ontology and rule editor, the semantic wiki AceWiki, and the Protune policy framework. The applications cover a wide range of Semantic Web scenarios, which shows how broadly ACE can be applied. We conclude that controlled natural languages can make the Semantic Web better understandable and more usable