Logical Theories for Fragments of Elementary Geometry

Balbiani, Philippe; Goranko, Valentin; Kellerman, Ruaan; Vakarelov, Dimiter

doi:10.1007/978-1-4020-5587-4_7

Cited by 17 publications

(30 citation statements)

References 65 publications

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“…3. Then we express that d is in the triangle (a , b , c ) by the following formula: Note that the modal language for plane geometry with betweenness is quite expressive (see [2]), but BBL covers only a small fragment of it because it cannot refer directly to lines, points and incidence. Nevertheless, if cameras can change their positions in the plane the expressiveness of BBL increases substantially and its full power is still subject to further investigation.…”

Section: Defining Betweenness and Collinearitymentioning

confidence: 99%

Big Brother Logic: visual-epistemic reasoning in stationary multi-agent systems

Gasquet

Goranko

Schwarzentruber

2015

Auton Agent Multi-Agent Syst

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We consider multi-agent scenarios where each agent controls a surveillance camera in the plane, with fixed position and angle of vision, but rotating freely. The agents can thus observe the surroundings and each other. They can also reason about each other's observation abilities and knowledge derived from these observations. We introduce suitable logical languages for reasoning about such scenarios which involve atomic formulae stating what agents can see, multi-agent epistemic operators for individual, distributed and common knowledge, as well as dynamic operators reflecting the ability of cameras to turn around in order to reach positions satisfying formulae in the language. We also consider effects of public announcements. We introduce several different but equivalent versions of the semantics for these languages, discuss their expressiveness and provide translations in PDL style. Using these translations we develop algorithms and obtain complexity results for model checking and satisfiability testing for the basic logic BBL that we introduce here and for some of its extensions. Notably, we show that even for the extension with common knowledge, model checking and satisfiability testing remain in PSPACE. We also discuss the sensitivity of the set of validities to the admissible angles of vision of the agents' cameras. Finally, we discuss some further extensions: adding obstacles, positioning the cameras in 3D or enabling them to change positions. Our work has potential applications to automated reasoning, forThis is a revised and substantially extended version of [10]. B François Schwarzentruber 123Auton Agent Multi-Agent Syst mal specification and verification of observational abilities and knowledge of multi-robot systems.

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Section: Defining Betweenness and Collinearitymentioning

confidence: 99%

Big Brother Logic: visual-epistemic reasoning in stationary multi-agent systems

Gasquet

Goranko

Schwarzentruber

2015

Auton Agent Multi-Agent Syst

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“…If we assume that none of the conditions (i) to (iii) hold, then we have a) there exists a formula ♦ 4 . By modal logic and Proposition 5.4 (1)…”

Section: Proposition 54mentioning

confidence: 99%

Completeness of a functional system for surjective functions

Burrieza

Ruiz

Guzmán

2017

Mathematical Logic Qtrly

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Combining modalities has proven to have interesting applications and many approaches that combine time with other types of modalities have been developed. One of these approaches uses accessibility functions between flows of time to study the basic properties of the functions, such as being total or partial, injective, surjective, etc. The completeness of certain systems expressing many of these properties, with the exception of surjectivity, has been proven. In this paper we propose a language with nominals to denote the initial and final sets of each function in a model in order to obtain a complete system for reasoning about surjective partial functions.

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“…Other methods for automated reasoning in geometry include the Characteristic Set Method of Ritt and Wu (Chou (1988)) and the Gröbner Basis Method of Buchberger et al (1988). For an overview of automated reasoning in geometry see Chou and Gao (2001) and for a detailed discussion on logical theories of fragments of elementary geometry see Balbiani et al (2007).…”

Section: Axiomatic Logical and Model-theoretic Treatments Of Geometrymentioning

confidence: 99%

“…Moreover, unlike the general purpose first-order languages, modal logics are more suited for specific applications, and most importantly have better computational behavior: very often they are decidable, in PSPACE or at most EXPTIME, as opposed to the usually undecidable first-order counterparts. In particular, modal logics for parallelism, orthogonality, incidence, affine, and projective geometries have been introduced and studied in Balbiani et al (1997); Balbiani (1998);Venema (1999); Balbiani and Goranko (2002); for further details see Balbiani et al (2007). Other applications of modal logic to the study of space will be discussed in the next sections.…”

Section: Axiomatic Logical and Model-theoretic Treatments Of Geometrymentioning

confidence: 99%

Logic for physical space

et al. 2011

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Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the major milestones in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces.

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Logical Theories for Fragments of Elementary Geometry

Cited by 17 publications

References 65 publications

Big Brother Logic: visual-epistemic reasoning in stationary multi-agent systems

Big Brother Logic: visual-epistemic reasoning in stationary multi-agent systems

Completeness of a functional system for surjective functions

Logic for physical space

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