Logics of Space with Connectedness Predicates Tinko Tinchev and - - PowerPoint PPT Presentation

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Logics of Space with Connectedness Predicates

Tinko Tinchev and Dimiter Vakarelov

Faculty of Mathematics and Computer Science Department of Mathematical logic Sofia University

Advances in Modal Logic 2010 (AiML-2010), 24-27 August, Moscow, Russia

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Outline

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Whiteheadean theory of space

The paper is in the field of a new theory of space originated by Whitehead:

• A. N. Whitehead, The Organization of Though, London,

William and Norgate, 1917, page 195 “...It follows from the relative theory that a point should be definable in terms of the relations between material things. So far as I am aware, this outcome of the theory has escaped the notice of mathematicians, who have invariably assumed the point as the ultimate starting ground of their reasoning.Many years ago I explained some types of ways in which we might achieve such a definition, and more recently have added some

• thers...” The main contribution of Whitehead is in the book
• A. N. Whitehead, Process and Reality, New York, MacMillan,

1929.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Whiteheadean theory of space

The paper is in the field of a new theory of space originated by Whitehead:

• A. N. Whitehead, The Organization of Though, London,

William and Norgate, 1917, page 195 “...It follows from the relative theory that a point should be definable in terms of the relations between material things. So far as I am aware, this outcome of the theory has escaped the notice of mathematicians, who have invariably assumed the point as the ultimate starting ground of their reasoning.Many years ago I explained some types of ways in which we might achieve such a definition, and more recently have added some

• thers...” The main contribution of Whitehead is in the book
• A. N. Whitehead, Process and Reality, New York, MacMillan,

1929.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Whiteheadean theory of space

The paper is in the field of a new theory of space originated by Whitehead:

• A. N. Whitehead, The Organization of Though, London,

William and Norgate, 1917, page 195 “...It follows from the relative theory that a point should be definable in terms of the relations between material things. So far as I am aware, this outcome of the theory has escaped the notice of mathematicians, who have invariably assumed the point as the ultimate starting ground of their reasoning.Many years ago I explained some types of ways in which we might achieve such a definition, and more recently have added some

• thers...” The main contribution of Whitehead is in the book
• A. N. Whitehead, Process and Reality, New York, MacMillan,

1929.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Whiteheadean theory of space

The theory of space based on the Whiteheadean approach is known now as Region Based Theory of Space (RBTS). Unlike the classical Euclidean approach, based on the primitive notions point, line and plane, which are abstract features having no existence in reality as separate things, RBTS is based on a more realistic primitive notions like region and some spatial relations between regions like part-of and contact.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Contact algebras and RBTS It is now commonly accepted that the simplest version of RBTS, capturing only some topological aspects of regions is the notion

• f contact algebra. Contact algebras are in fact Boolean

algebras extended with one binary relation called contact, satisfying some obvious axioms. The elements of the Boolean algebra are called regions and the Boolean operations can be considered as operations of how to define new regions from given ones. The Boolean part of contact algebra can be considered as a formalization of the mereological part of RBTS, considered in mereology as the theory of parts. For instance Boolean ordering corresponds to part-of relation.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Contact algebras and RBTS It is now commonly accepted that the simplest version of RBTS, capturing only some topological aspects of regions is the notion

• f contact algebra. Contact algebras are in fact Boolean

algebras extended with one binary relation called contact, satisfying some obvious axioms. The elements of the Boolean algebra are called regions and the Boolean operations can be considered as operations of how to define new regions from given ones. The Boolean part of contact algebra can be considered as a formalization of the mereological part of RBTS, considered in mereology as the theory of parts. For instance Boolean ordering corresponds to part-of relation.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Contact algebras and RBTS It is now commonly accepted that the simplest version of RBTS, capturing only some topological aspects of regions is the notion

• f contact algebra. Contact algebras are in fact Boolean

algebras extended with one binary relation called contact, satisfying some obvious axioms. The elements of the Boolean algebra are called regions and the Boolean operations can be considered as operations of how to define new regions from given ones. The Boolean part of contact algebra can be considered as a formalization of the mereological part of RBTS, considered in mereology as the theory of parts. For instance Boolean ordering corresponds to part-of relation.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

MEREOTOPOLOGY The contact relation captures some properties of Whitehead’s connection relation. Standard point models of contact algebra are of topological nature, namely the Boolean algebra of regular closed subsets of a given topological space and the main representation theorem for contact algebras in fact realizes the Whitehead’s idea to define points by means of regions and some relations between regions. Topological models of contact algebras show that contact has some topological nature and that in certain sense CONTACT ALGEBRAS = MEREOLOGY + TOPOLOGY This suggests also to use another terminology and equation: CONTACT ALGEBRAS = MEREOTOPOLOGY

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

MEREOTOPOLOGY The contact relation captures some properties of Whitehead’s connection relation. Standard point models of contact algebra are of topological nature, namely the Boolean algebra of regular closed subsets of a given topological space and the main representation theorem for contact algebras in fact realizes the Whitehead’s idea to define points by means of regions and some relations between regions. Topological models of contact algebras show that contact has some topological nature and that in certain sense CONTACT ALGEBRAS = MEREOLOGY + TOPOLOGY This suggests also to use another terminology and equation: CONTACT ALGEBRAS = MEREOTOPOLOGY

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

RBTS and computer science

It is important to mention that Whitehead’s ideas of region-based theory of space flourished, and in a sense have been reinvented, and found applications in some areas of computer science: Qualitative Spatial Reasoning (QSR), Knowledge representation (KR), Geographical information systems (GIS), Formal ontologies in information systems (FOIS), Image processing, natural language semantics etc. The reason is that the language of region-based theory of space is more simple to describe some qualitative spatial features and properties of space bodies.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Spatial logics in QSR: RCC and RCC-8

One of the most popular system in QSR is the system RCC (Regular Connection Calculus) introduced by Randell, Cui and Cohn. Similar system, denoted by RCC-8, considered by many authors, is based on the following 8 spatial relations between regular closed sets in a topological space:

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Spatial logics in QSR: RCC and RCC-8

One of the most popular system in QSR is the system RCC (Regular Connection Calculus) introduced by Randell, Cui and Cohn. Similar system, denoted by RCC-8, considered by many authors, is based on the following 8 spatial relations between regular closed sets in a topological space:

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

RCC-8 spatial relations

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Extensions of RCC-8

The language of RCC-8 is too weak to distinguish general topological models from the models over connected and Euclidean spaces. Wolter and Zakharyaschev extended RCC-8 with Boolean terms (BRCC-8). This extension can separate general topological models from the models over connected spaces. However, the language of BRCC-8 is not enough to define the property of connectedness ( c(a) ) and the number of connected components of a region (c≤n(a), n = 1, 2 . . . ). Extensions of BRCC-8 with predicates c(a) and c≤n(a) have been studied recently with respect to expressivity and complexity by Kontchakov, Pratt-Hartmann, Wolter and Zakharyaschev.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Extensions of RCC-8

The language of RCC-8 is too weak to distinguish general topological models from the models over connected and Euclidean spaces. Wolter and Zakharyaschev extended RCC-8 with Boolean terms (BRCC-8). This extension can separate general topological models from the models over connected spaces. However, the language of BRCC-8 is not enough to define the property of connectedness ( c(a) ) and the number of connected components of a region (c≤n(a), n = 1, 2 . . . ). Extensions of BRCC-8 with predicates c(a) and c≤n(a) have been studied recently with respect to expressivity and complexity by Kontchakov, Pratt-Hartmann, Wolter and Zakharyaschev.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Extensions of RCC-8

The language of RCC-8 is too weak to distinguish general topological models from the models over connected and Euclidean spaces. Wolter and Zakharyaschev extended RCC-8 with Boolean terms (BRCC-8). This extension can separate general topological models from the models over connected spaces. However, the language of BRCC-8 is not enough to define the property of connectedness ( c(a) ) and the number of connected components of a region (c≤n(a), n = 1, 2 . . . ). Extensions of BRCC-8 with predicates c(a) and c≤n(a) have been studied recently with respect to expressivity and complexity by Kontchakov, Pratt-Hartmann, Wolter and Zakharyaschev.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Extensions of RCC-8

The language of RCC-8 is too weak to distinguish general topological models from the models over connected and Euclidean spaces. Wolter and Zakharyaschev extended RCC-8 with Boolean terms (BRCC-8). This extension can separate general topological models from the models over connected spaces. However, the language of BRCC-8 is not enough to define the property of connectedness ( c(a) ) and the number of connected components of a region (c≤n(a), n = 1, 2 . . . ). Extensions of BRCC-8 with predicates c(a) and c≤n(a) have been studied recently with respect to expressivity and complexity by Kontchakov, Pratt-Hartmann, Wolter and Zakharyaschev.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

THE AIM OF THE PAPER is to study various extensions of BRCC-8 with predicates c(a) and c≤n(a) with respect to complete axiomatizations. The language of the logics is based on the language of contact algebras extended with predicates c(a) and c≤n(a). The completeness theorems are with respect to three different semantics: algebraic semantics - based on contact algebras. topological semantics - based on contact algebras over topological spaces, relational semantics - over relational structures (W, R) with reflexive and symmetric relation R called by Galton adjacency spaces.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Contact algebras

Definition By a Contact Algebra(CA) we mean any system B = (B, C) = (B, 0, 1, ., +, ∗, C), where (B, 0, 1, ., +, ∗) is a non-degenerate Boolean algebra with Boolean complement denoted by “∗” and C – a binary relation in B, called contact and satisfying the following axioms: (C1) If xCy, then x, y = 0, (C2) xC(y + z) iff xCy or xCz, (C3) If xCy, then yCx, (C4) If x.y = 0, then xCy. The elements of B are called regions.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Contact algebras satisfying additional axioms

We consider contact algebras satisfying some additional axioms: (Con) if a = 0 and a = 1, then aCa∗ – connectedness. (Ext) if a = 1, then ∃b = 0 such that aCb – extensionality (Nor) if aCb, then ∃c such that aCc and c∗Cb – normality Note that (Ext) is equivalent to the following axiom (Ext’) considered by Whitehead (Ext’) a = b iff (∀c)(aCc ↔ bCc)

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Examples of contact algebras

Examples ( (1). Topological example: the CA of regular closed sets.) Let X be an arbitrary topological space. A subset a of X is regular closed if a = Cl(Int(a)), where Cl and Int are the standard topological closure and interior operations in X. The set of all regular closed subsets of X will be denoted by RC(X). Facts:

1

Regular closed sets with the operations a + b = a ∪ b, a.b = Cl(Int(a ∩ b)), a∗ = Cl(X \ a), 0 = ∅ and 1 = X form a Boolean algebra.

2

If we define the contact by a CX b iff a ∩ b = ∅, then RC(X) with the above contact is a contact algebra.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Examples of contact algebras

Examples ( (1). Topological example: the CA of regular closed sets.) Let X be an arbitrary topological space. A subset a of X is regular closed if a = Cl(Int(a)), where Cl and Int are the standard topological closure and interior operations in X. The set of all regular closed subsets of X will be denoted by RC(X). Facts:

1

Regular closed sets with the operations a + b = a ∪ b, a.b = Cl(Int(a ∩ b)), a∗ = Cl(X \ a), 0 = ∅ and 1 = X form a Boolean algebra.

2

If we define the contact by a CX b iff a ∩ b = ∅, then RC(X) with the above contact is a contact algebra.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Examples of contact algebras

Examples ( (1). Topological example: the CA of regular closed sets.) Let X be an arbitrary topological space. A subset a of X is regular closed if a = Cl(Int(a)), where Cl and Int are the standard topological closure and interior operations in X. The set of all regular closed subsets of X will be denoted by RC(X). Facts:

1

Regular closed sets with the operations a + b = a ∪ b, a.b = Cl(Int(a ∩ b)), a∗ = Cl(X \ a), 0 = ∅ and 1 = X form a Boolean algebra.

2

If we define the contact by a CX b iff a ∩ b = ∅, then RC(X) with the above contact is a contact algebra.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Examples of contact algebras

Examples ( (2) Non-topological example,related to Kripke frames from modal logic.) Let (X, R) be a reflexive and symmetric modal frame and let B(X) be the Boolean algebra of all subsets of X. Define a contact CR for a, b ∈ B(X) by aCRb iff (∃x ∈ a)(∃y ∈ b)(xRy) Then: B(X) = CA(W, R) equipped with the contact CR is a contact algebra. Moreover:

1

CA(W, R) satisfies (Con) iff (W, R) is a connected graph.

2

CA(W, R) satisfies (Nor) iff R is an equivalence relation. The above example is related to Galton’s adjacency spaces and discrete mereotopology.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Relational example

Non-topological contact

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Topological representation of CA

Theorem (Topological representation of contact algebras[Dimov,Vakarelov].) For every contact algebra (B, C) there exists a semi-regular and compact T0 space X and an embedding h into the contact algebra RC(X). Similar theorems are proved for contact algebras satisfying some of the axioms (Con), (Ext) and (Nor). Theorem (Discrete representation of contact algebras [Duntsch-Vakarelov].) For every contact algebra (B, C) there exists an adjacency space (W, R) and an embedding h into the contact algebra CA(W, R) over (W, R). Moreover, (B, C) satisfies (Nor) iff R is an equivalence relation on W.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Topological representation of CA

Theorem (Topological representation of contact algebras[Dimov,Vakarelov].) For every contact algebra (B, C) there exists a semi-regular and compact T0 space X and an embedding h into the contact algebra RC(X). Similar theorems are proved for contact algebras satisfying some of the axioms (Con), (Ext) and (Nor). Theorem (Discrete representation of contact algebras [Duntsch-Vakarelov].) For every contact algebra (B, C) there exists an adjacency space (W, R) and an embedding h into the contact algebra CA(W, R) over (W, R). Moreover, (B, C) satisfies (Nor) iff R is an equivalence relation on W.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Topological representation of CA

Theorem (Topological representation of contact algebras[Dimov,Vakarelov].) For every contact algebra (B, C) there exists a semi-regular and compact T0 space X and an embedding h into the contact algebra RC(X). Similar theorems are proved for contact algebras satisfying some of the axioms (Con), (Ext) and (Nor). Theorem (Discrete representation of contact algebras [Duntsch-Vakarelov].) For every contact algebra (B, C) there exists an adjacency space (W, R) and an embedding h into the contact algebra CA(W, R) over (W, R). Moreover, (B, C) satisfies (Nor) iff R is an equivalence relation on W.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Connectedness predicates c(a) and c≤n(a).

In topological spaces c(a) means that the region a is connected in a topological sense, and c≤n(a) – that a is a sum of at most n (n ≥ 1) maximal connected components. Obviously c(a) iff c≤1(a). The following facts are well known: c(a) iff (∀b0 = ∅, b1 = ∅ ∈ RC(X))(a = b0 ∪ b1 → b0 ∩ b1 = ∅), c≤n(a) iff (∀b0 = ∅ . . . bn = ∅ ∈ RC(X))(a = b0 ∪ . . . ∪ bn → ∃i = j, ij = 0 . . . n, bi ∩ bj = ∅).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Connectedness predicates c(a) and c≤n(a).

In topological spaces c(a) means that the region a is connected in a topological sense, and c≤n(a) – that a is a sum of at most n (n ≥ 1) maximal connected components. Obviously c(a) iff c≤1(a). The following facts are well known: c(a) iff (∀b0 = ∅, b1 = ∅ ∈ RC(X))(a = b0 ∪ b1 → b0 ∩ b1 = ∅), c≤n(a) iff (∀b0 = ∅ . . . bn = ∅ ∈ RC(X))(a = b0 ∪ . . . ∪ bn → ∃i = j, ij = 0 . . . n, bi ∩ bj = ∅).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Connectedness predicates c(a) and c≤n(a).

In topological spaces c(a) means that the region a is connected in a topological sense, and c≤n(a) – that a is a sum of at most n (n ≥ 1) maximal connected components. Obviously c(a) iff c≤1(a). The following facts are well known: c(a) iff (∀b0 = ∅, b1 = ∅ ∈ RC(X))(a = b0 ∪ b1 → b0 ∩ b1 = ∅), c≤n(a) iff (∀b0 = ∅ . . . bn = ∅ ∈ RC(X))(a = b0 ∪ . . . ∪ bn → ∃i = j, ij = 0 . . . n, bi ∩ bj = ∅).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Definitions of connectedness predicates in contact algebras

The above facts suggest the following Definition (Connectedness predicates in contact algebras) Let B be a contact algebra. We define c(a) and c≤n(a) for a ∈ B, n = 1, 2, . . . as follows:

1

c(a) ↔def (∀b, d ∈ B)(b = 0 ∧ d = 0 ∧ a = b + d → bCd),

2

c≤n(a) ↔def (∀b0 . . . bn ∈ B)(b0 = 0 ∧ . . . ∧ bn = 0 ∧ a = b0 + · · · + bn → ∃i = j, 0 ≤ ij ≤ n : biCbj).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Definitions of connectedness predicates in contact algebras

The above facts suggest the following Definition (Connectedness predicates in contact algebras) Let B be a contact algebra. We define c(a) and c≤n(a) for a ∈ B, n = 1, 2, . . . as follows:

1

c(a) ↔def (∀b, d ∈ B)(b = 0 ∧ d = 0 ∧ a = b + d → bCd),

2

c≤n(a) ↔def (∀b0 . . . bn ∈ B)(b0 = 0 ∧ . . . ∧ bn = 0 ∧ a = b0 + · · · + bn → ∃i = j, 0 ≤ ij ≤ n : biCbj).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Connectedness predicates in contact algebras over adjacency spaces

Lemma Let (W, R) be adjacency space, a ⊆ W and Ra be the restriction of R over a. Then:

1

c(a) iff (∀x, y ∈ a)(xR∗

ay).

2

c≤n(a) iff (∀x0 . . . xn ∈ a)(∃i = j, 0 ≤ ij ≤ n)(xiR∗

axj).

The lemma explains that connectedness predicates over (W, R) have the standard graph-theoretic sense.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Connectedness predicates in contact algebras over adjacency spaces

Lemma Let (W, R) be adjacency space, a ⊆ W and Ra be the restriction of R over a. Then:

1

c(a) iff (∀x, y ∈ a)(xR∗

ay).

2

c≤n(a) iff (∀x0 . . . xn ∈ a)(∃i = j, 0 ≤ ij ≤ n)(xiR∗

axj).

The lemma explains that connectedness predicates over (W, R) have the standard graph-theoretic sense.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Topological representation of contact algebras with connectedness predicates

Theorem (Topological representation of connectedness) Let B = (B, C) be a contact algebra, X be a compact topological space and h be an embedding of (B, C) into the contact algebra RC(X) such that the set {h(b) : b ∈ B} forms a base for the closed sets in X. Then for any a ∈ B: c≤n(a) holds in B iff c≤n(h(a)) holds in RC(X). Corollary The embedding h from the Topological representation theorem for contact algebras preserves the predicates c≤n. Similar theorems hold for contact algebras satisfying some of the additional axioms (Con), (Ext), (Nor).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness

Topological representation of contact algebras with connectedness predicates

Theorem (Topological representation of connectedness) Let B = (B, C) be a contact algebra, X be a compact topological space and h be an embedding of (B, C) into the contact algebra RC(X) such that the set {h(b) : b ∈ B} forms a base for the closed sets in X. Then for any a ∈ B: c≤n(a) holds in B iff c≤n(h(a)) holds in RC(X). Corollary The embedding h from the Topological representation theorem for contact algebras preserves the predicates c≤n. Similar theorems hold for contact algebras satisfying some of the additional axioms (Con), (Ext), (Nor).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Representation of connectedness predicates over adjacency spaces

Theorem (Discrette representation of connectedness) Let B = (B, C) be a finite contact algebra. Then there exists a finite adjacency space (W, R) and an isomorphism h between B and the contact algebra CA(W, R) over (W, R) such that for every a ∈ B the following equivalence is true: c≤n(a) holds in B iff c≤n(h(a)) holds in CA(W, R). Moreover,

1

B satisfies (Con) iff the relation (W, R) is graph-connected;

2

B satisfies (Nor) iff the relation R is an equivalence relation.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Syntax

We consider quantifier-free spatial logics based on the language of contact algebras and connectedness predicates. Formally the language, denoted by L(≤, C, c≤n), consists of: a denumerable set Var of Boolean variables, Boolean operations: +, ., ∗ and 0, 1 (Boolean constants). Relational symbols: ≤ (part-of), C (contact), c≤n, n ≥ 1. Standard propositional connectives: ¬, ∧, ∨, ⇒, ⇔ and the propositional constants ⊥, ⊤.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Syntax

The set of Boolean terms is defined in a standard way from Boolean variables and constants by means of Boolean

• perations.

Atomic formulas are of the type: a ≤ b, aCb, c≤n(a), ⊥, ⊤, where a, b are Boolean terms. Formulas are defined from atomic formulas by using propositional connectives in a standard way. Abbreviations: c(a) =def c1(a), a = b =def (a ≤ b) ∧ (b ≤ a), a = b =def ¬(a = b), aCb =def ¬(aCb), a ≪ b =def aCb∗.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Semantics

The semantics of the language L(≤, C, c≤n) is the ordinary semantics of quantifier-free predicate languages based on the notion of interpretation. We consider three kinds of interpretations: algebraic – in various classes of contact algebras, topological – in contact algebras of regular closed sets of various classes of topological spaces, relational – in contact algebras over various classes of adjacency spaces.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

The system PWRCC and some extensions

The system PWRCC (Propositional Weak RCC-system) is the axiomatic system based on the language of contact algebras, studied in Balbiani, Tinchev and Vakarelov, Modal Logics for Region-based theory of space, Fundamenta Informaticae, vol 81,(2007), 29-82 Since the axioms of contact algebras are universal formulas PWRCC is based on the axioms of propositional logic, axioms

• f Boolean algebra, and axioms of contact C. The only rule is

Modus Ponens. This system has various extensions related to the axioms (Con), (Ext) and (Nor).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Difficulties with non-universal axioms

The extension of PWRCC with axiom of connectedness (Con) presents no problems - just add the corresponding universal formula p = 0 ∧ p = 1 ⇒ pCp∗ to the set of

• axioms. The obtained system is denoted by PWRCCCon.

The contact axioms (Ext) (∀p)(p = 1 ⇒ (∃q)(q = 0 ∧ pCq)) and (Nor) (∀p, q)(pCq ⇒ (∃r)(pCr ∧ r ∗Cq)) are not universal sentences and can not be added to the

• axiomatics. But they can be simulated by additional rules
• f inference, denoted correspondingly by (EXT) and

(NOR), which guarantee that the contact algebra in the canonical model construction satisfies the corresponding axiom.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Difficulties with non-universal axioms

The extension of PWRCC with axiom of connectedness (Con) presents no problems - just add the corresponding universal formula p = 0 ∧ p = 1 ⇒ pCp∗ to the set of

• axioms. The obtained system is denoted by PWRCCCon.

The contact axioms (Ext) (∀p)(p = 1 ⇒ (∃q)(q = 0 ∧ pCq)) and (Nor) (∀p, q)(pCq ⇒ (∃r)(pCr ∧ r ∗Cq)) are not universal sentences and can not be added to the

• axiomatics. But they can be simulated by additional rules
• f inference, denoted correspondingly by (EXT) and

(NOR), which guarantee that the contact algebra in the canonical model construction satisfies the corresponding axiom.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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The rules (EXT) and (NOR)

For an analog of the first-order axiom (Ext) we introduce the rule of extensionality (EXT) α ⇒ (p = 0 ∨ aCp) α ⇒ (a = 1) , where p is a Boolean variable that does not occur in a and α. For an analog of axiom (Nor) we introduce the rule of normality: (NOR) α ⇒ (aCp ∨ p∗Cb) α ⇒ aCb , where p is a Boolean variable that does not occur in a, b, and α.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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The rules (EXT) and (NOR)

For an analog of the first-order axiom (Ext) we introduce the rule of extensionality (EXT) α ⇒ (p = 0 ∨ aCp) α ⇒ (a = 1) , where p is a Boolean variable that does not occur in a and α. For an analog of axiom (Nor) we introduce the rule of normality: (NOR) α ⇒ (aCp ∨ p∗Cb) α ⇒ aCb , where p is a Boolean variable that does not occur in a, b, and α.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Propositional RCC type logics (Extensions of PWRCC)

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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The axiom and rule for c≤n

The definition of c≤n can be divided into one axiom and one rule of inference (Ax c≤n) c≤n(a) ∧ n

i=0 pi = 0 ∧ a = n i=0 pi ⇒ i=j,ij∈{0,...,n} piCpj

(Rule c≤n) c≤n) α ∧ n

i=0 pi = 0 ∧ a = n i=0 pi ⇒ i=j,ij∈{0,...,n} piCpj

α ⇒ c≤n(a) , where α is a formula and p0, . . . , pn are different Boolean variables not occurring in the term a and the formula α . We extend all RCC-type logics by the axioms (Ax c≤n) and the rules (Rule c≤n), n = 1, 2 . . .. For each of the logics we associate the corresponding class of contact algebras in an

• bvious way.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

The axiom and rule for c≤n

The definition of c≤n can be divided into one axiom and one rule of inference (Ax c≤n) c≤n(a) ∧ n

i=0 pi = 0 ∧ a = n i=0 pi ⇒ i=j,ij∈{0,...,n} piCpj

(Rule c≤n) c≤n) α ∧ n

i=0 pi = 0 ∧ a = n i=0 pi ⇒ i=j,ij∈{0,...,n} piCpj

α ⇒ c≤n(a) , where α is a formula and p0, . . . , pn are different Boolean variables not occurring in the term a and the formula α . We extend all RCC-type logics by the axioms (Ax c≤n) and the rules (Rule c≤n), n = 1, 2 . . .. For each of the logics we associate the corresponding class of contact algebras in an

• bvious way.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

The minimal logics with connectedness predicates

Note that the minimal logic Lmin with connectedness predicates coincides with the logic PWRCC+(Ax c≤n)+ (Rule c≤n), the minimal logic LCon with connectedness predicates containing the axiom (Con) coincides with Lmin+ (Con). These two logics, under other names, were studied by Kontchakov, Pratt-Harmann, Wolter and Zakharyschev with respect to complexity. They just coincide with the logics BRCC-8, extended with the predicates c≤n interpreted in the class of all topological spaces and in the class of all connected topological spaces respectively.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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The minimal logics with connectedness predicates

Note that the minimal logic Lmin with connectedness predicates coincides with the logic PWRCC+(Ax c≤n)+ (Rule c≤n), the minimal logic LCon with connectedness predicates containing the axiom (Con) coincides with Lmin+ (Con). These two logics, under other names, were studied by Kontchakov, Pratt-Harmann, Wolter and Zakharyschev with respect to complexity. They just coincide with the logics BRCC-8, extended with the predicates c≤n interpreted in the class of all topological spaces and in the class of all connected topological spaces respectively.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Outline

1

Hystorical introduction Whiteheadean approach RBTS and computer science THE AIM OF THE PAPER

2

Contact algebras with connectedness predicates Contact algebras Contact algebras with connectedness predicates Representation theory of contact algebras with connectedness predicates

3

Spatial RCC-like logics with connectedness predicates Syntax and semantics Axiomatizations Completeness theorems

4

Concluding remarks

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Algebraic and topological completeness theorems

Theorem (Algebraic and topological completeness.) Let L be any of the above defined logics with connectedness

• predicate. Then L is strongly complete both in the

corresponding class of contact algebras and in the corresponding class of topological contact algebras. Idea of the proof. Completeness with respect to algebraic models is obtained by an appropriate canonical-model construction, generalizing the Henkin construction from the predicate logic. Completeness for the topological models goes through the topological representation theorem for the corresponding class of contact algebras with connectedness predicates.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Algebraic and topological completeness theorems

Theorem (Algebraic and topological completeness.) Let L be any of the above defined logics with connectedness

• predicate. Then L is strongly complete both in the

corresponding class of contact algebras and in the corresponding class of topological contact algebras. Idea of the proof. Completeness with respect to algebraic models is obtained by an appropriate canonical-model construction, generalizing the Henkin construction from the predicate logic. Completeness for the topological models goes through the topological representation theorem for the corresponding class of contact algebras with connectedness predicates.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Algebraic and topological completeness theorems

Theorem (Algebraic and topological completeness.) Let L be any of the above defined logics with connectedness

• predicate. Then L is strongly complete both in the

corresponding class of contact algebras and in the corresponding class of topological contact algebras. Idea of the proof. Completeness with respect to algebraic models is obtained by an appropriate canonical-model construction, generalizing the Henkin construction from the predicate logic. Completeness for the topological models goes through the topological representation theorem for the corresponding class of contact algebras with connectedness predicates.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks Syntax and semantics Axiomatizations Completeness theorems

Theorem (Weak completeness with respect to relational semantics and finite model property.) Let L be any of the above defined logics with connectedness

• predicate. Then:

1

L is weakly complete with respect to the corresponding finite relational models.

2

Consequently L has fmp and hence is decidable.

3

All logics not containing the axiom (Con) collapse to the minimal logic Lmin, and all logics containing the axiom (Con) collapse to the logic LCon. Idea of the proof. (1) and (2) go by an appropriate filtration and (3) - by elimination of the rules (EXT) and (NOR).

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

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Other logics related to connectedness

We simplify the language L(≤, C, c≤n), replacing each predicate c≤n with a denumerable set C≤n of a new kind of Boolean variables called connectedness nominals. In the interpretation in contact algebras each such nominal a is interpreted by a region satisfying the predicate c≤n. We proved that the new language has the same expressivity power. The advantage of the new language is that the corresponding logics do require only axioms concerning nominals and not additional rules of inference. Axiomatizations and completeness theorems of all mentioned kinds are also obtained.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks

Concluding remarks

We introduced quantifier-free axiomatic systems for several spatial logics based on contact relation and connectedness predicates. The axiomatizations contain special rules of inference, replacing non-universal axioms from contact algebras. We proved for them strong completeness theorems with respect to algebraic and topological semantics, based on the topological representation theory for certain classes of contact algebras with connectedness predicates. We proved weak completeness theorem with respect to their intended finite algebraic, topological and relational models, which imply fmp and decidability of all logics. Another results is that with respect to their theorems, all logics collapse to two minimal systems.

Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates

tu-logo ur-logo Hystorical introduction Contact algebras with connectedness predicates Spatial RCC-like logics with connectedness predicates Concluding remarks

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Tinko Tinchev and Dimiter Vakarelov Logics of Space with Connectedness Predicates