Algebraic Properties of Qualitative Spatio-Temporal Calculi
Introduction
Requirements
Algebraic properties
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Algebraic Properties of Qualitative Spatio-Temporal Calculi1 Frank Dyllaa Till Mossakowskia,b Thomas Schneider c Diedrich Wolter a a Collaborative
Research Center on Spatial Cognition (SFB/TR 8), Univ. Bremen Research Center for Artificial Intelligence (DFKI), Bremen c Dept of Mathematics and Computer Science, University of Bremen b German
2 August 2013
1
Thanks for discussions at the “Spatial Reasoning Teatime”.
Dylla, Mossakowski, Schneider, Wolter
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And now . . .
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Requirements to qualitative calculi
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Algebraic properties
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Information-preservation properties
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Conclusion and outlook
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Qualitative spatio-temporal representation and reasoning
= Symbolic way to represent spatio-temporal knowledge and draw inferences from it
Common approach: define set R of relations to describe spatial relationships use R as primitives for representation employ techniques from constraint and qualitative reasoning to reason about the primitives various domains, typically infinite
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Applications by domain Time interval relations I Medical diagnostics I Law texts
Simplified Allen Allen-13
I Business and manufacturing: diagnostics
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Allen-13
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Applications by domain Positions, regions (topology) I Planning, robotics, navigation I Natural language processing I Image understanding
9-int, CarDir, RCC 9-intersection
I CAD and manufacturing
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Rectangle Alg., . . . 9-int, CarDir, RCC, Allen
I GIS, spatial query answering I Traffic tracking
RCC, Block Algebra, ROC
LR, RCC
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Applications by domain Moving point objects, directional information
OPRA, Dipole Flipflop, StarVars
I Robotics, navigation, motion planning I GIS, spatial query answering
QTC
I Traffic tracking
Dipole
I Ambient intelligence, smart environments (scene analysis, task modelling)
OPRA, RCC
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Representation and reasoning tasks required I Knowledge representation
I Data interpretation I Inference
I Constraint-based reasoning
(CSP-SAT, -ENT, -MOD, -MIN)
I Neighbourhood-based reasoning
(Relaxing constraints, continuity constraints, dominance space)
I Logical reasoning
(Deduction, abduction)
I Learning
(Inductive logic programming)
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What is a qualitative calculus? Some answers from the literature A weak representation of a non-associative relation algebra
[Egenhofer & Rodríguez 1999; Ligozat et al. 2003; Ligozat & Renz 2004]
A system of relations forming a constraint algebra [Nebel & Scivos 2002]
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What is a qualitative calculus? Some answers from the literature A weak representation of a non-associative relation algebra
[Egenhofer & Rodríguez 1999; Ligozat et al. 2003; Ligozat & Renz 2004]
A system of relations forming a constraint algebra [Nebel & Scivos 2002]
Commonly agreed ingredients Set R of relations Operations ∪, ∩, ¯, ◦, ˘ with certain properties (closure, algebraic properties) Mapping to a domain with certain properties (e.g., JEPD) Dylla, Mossakowski, Schneider, Wolter
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And in reality? Zoo of qualitative calculi 36 implemented in SparQ; many more in the literature “classical” calculi, usually with strong algebraic properties (e.g., Allen-13, RCC-8)
more recent calculi, often with weaker algebraic properties
(e.g., Cardinal Direction Relations, Rectangular Cardinal Relations)
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And in reality? Zoo of qualitative calculi 36 implemented in SparQ; many more in the literature “classical” calculi, usually with strong algebraic properties (e.g., Allen-13, RCC-8)
more recent calculi, often with weaker algebraic properties
(e.g., Cardinal Direction Relations, Rectangular Cardinal Relations)
;
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And in reality? Zoo of qualitative calculi 36 implemented in SparQ; many more in the literature “classical” calculi, usually with strong algebraic properties (e.g., Allen-13, RCC-8)
more recent calculi, often with weaker algebraic properties
(e.g., Cardinal Direction Relations, Rectangular Cardinal Relations)
; Research question To what extent do existing calculi meet the imposed requirements? ; If not, what are minimal requirements? ; How can we classify calculi according to their properties? Dylla, Mossakowski, Schneider, Wolter
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On the agenda today
Revisit and generalise the definition of a qualitative calculus Identify notions of algebras that cover existing calculi Discuss relevance of algebraic properties for spatial reasoning Evaluate the algebraic properties of existing calculi ; derive improved generic reasoning procedure Examine information-preservation properties of calculi during reasoning
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And now . . .
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Introduction
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Requirements to qualitative calculi
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Algebraic properties
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Information-preservation properties
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Conclusion and outlook
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Calculi à la Ligozat & Renz (1) Basic notions Universe (domain) U: spatio-temporal entities Set R of base relations over U Uncertain information ; union of base relations Restriction to binary relations in this work
R is JEPD: jointly exhaustive and pairwise disjoint Jointly exhaustive: R covers U × U
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Calculi à la Ligozat & Renz (1) Basic notions Universe (domain) U: spatio-temporal entities Set R of base relations over U Uncertain information ; union of base relations Restriction to binary relations in this work
R is JEPD: jointly exhaustive and pairwise disjoint Jointly exhaustive: R covers U × U
Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Dylla, Mossakowski, Schneider, Wolter
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition)
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition) Converse: ϕ(r˘) = ϕ(r )˘
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition) Converse: ϕ(r˘) = ϕ(r )˘ Weak composition: r s = smallest set T of base rel.s with ϕ(T ) ⊇ ϕ(r ) ◦ ϕ(s)
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition) Converse: ϕ(r˘) = ϕ(r )˘ Weak composition: r s = smallest set T of base rel.s with ϕ(T ) ⊇ ϕ(r ) ◦ ϕ(s) ˘, usually given by tables Dylla, Mossakowski, Schneider, Wolter
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These requirements are strong
Some calculi violate them e.g.: Cardinal Direction Relations Rectangular Cardinal Relations Their converse only satisfies ϕ(r˘) ⊇ ϕ(r )˘
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These requirements are strong
Some calculi violate them e.g.: Cardinal Direction Relations Rectangular Cardinal Relations Their converse only satisfies ϕ(r˘) ⊇ ϕ(r )˘ ; Weaken requirements to partition schemes and calculi!
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation = mapping to an abstract part. scheme Plus symbolic operations ˘, (abstract converse & compos.)
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation = mapping to an abstract part. scheme Plus symbolic operations ˘, (abstract converse & compos.) Abstract converse: ϕ(r˘) ⊇ ϕ(r )˘
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation = mapping to an abstract part. scheme Plus symbolic operations ˘, (abstract converse & compos.) Abstract converse: ϕ(r˘) ⊇ ϕ(r )˘ Abstract composition: ϕ(r s) ⊇ ϕ(r ) ◦ ϕ(s)
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Qualitative spatio-temporal reasoning
Qualitative constraint Formula xRy with x , y variables, R relation from a calculus C
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Qualitative spatio-temporal reasoning
Qualitative constraint Formula xRy with x , y variables, R relation from a calculus C Qualitative constraint satisfaction problem (QCSP) Input: set of constraints Question: Is there a mapping from variables to C’s domain that satisfies all constraints?
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Qualitative spatio-temporal reasoning
Qualitative constraint Formula xRy with x , y variables, R relation from a calculus C Qualitative constraint satisfaction problem (QCSP) Input: set of constraints Question: Is there a mapping from variables to C’s domain that satisfies all constraints? (Analogous definition for other reasoning problems)
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Qualitative spatio-temporal reasoning Common techniques for solving QCSPs Some taken over from finite-domain CSPs (constraint propagation, k-consistency)
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Qualitative spatio-temporal reasoning Common techniques for solving QCSPs Some taken over from finite-domain CSPs (constraint propagation, k-consistency) Algebraic closure (a-closure) sufficient condition for consistency guaranteed by “⊇” of abstract composition For some calculi, a-closure known to be necessary too
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Qualitative spatio-temporal reasoning Common techniques for solving QCSPs Some taken over from finite-domain CSPs (constraint propagation, k-consistency) Algebraic closure (a-closure) sufficient condition for consistency guaranteed by “⊇” of abstract composition For some calculi, a-closure known to be necessary too
Composition of arbitrary relations R, S is uniquely determined by the composition results of the base relations in R, S
(composition table)
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Existing qualitative spatio-temporal calculi Algebraic Algebraic Properties Properties of Qualitative of Qualitative Spatio-Temporal Spatio-Temporal CalculiCalculi 7
decides
Name Name Ref. Ref. Domain Domain #BR #BR RM RM 9-Intersection 9-Intersection [9] [9] simple simple 2D regions 2D regions 8 I [12,16] 8 I [12, • Allen’s Allen’s intervalinterval relations relations [1] [1] intervals intervals (order)(order) 13 A [42] 13 A [42] n n • Algebra Block Block Algebra [2] [2] n-dimensional n-dimensional blocks blocks 13 A13 [2] A [2] • Cardinal Cardinal Dir. Calculus Dir. Calculus CDC CDC [10,17] [10,17] directions directions (point (point abstr.)abstr.) 9 A [17] 9 A [17] Cardinal Cardinal Dir. Relations Dir. Relations CDR CDR [38] [38] regionsregions 218 P 218 P CycOrd, CycOrd, binary binary CYCb CYCb [14] [14] oriented oriented lines lines 4 U 4 U Dependency Calculus Calculus [33] [33] points points (partial(partial order) order) 5 A [33] 5 A [33] • Dependency a a Dipole Dipole Calculus Calculus DRAf DRAf [25,24] [25,24] directions directions from line from segm. line segm. 72 I [46] 72 I [46] DRAfpDRAfp [24] [24] directions directions from line from segm. line segm. 80 I 80 I DRA-connectivity DRA-connectivity [45] [45] connectivity connectivity of line of segm. line segm.7 U 7 U Geometric Geometric Orientation Orientation [7] [7] relativerelative orientation orientation 4 U 4 U INDU INDU [32] [32] intervals intervals (order,(order, rel. dur.n) rel. dur.n) 25 P 25 P OPRAOPRA ,m . . .=, 81, . . . , 8 [23,28] [23,28] oriented oriented points points 4m · (4m 4m+· (4m 1) + 1) m, m = m1, (Oriented (Oriented Point Rel. Point Algebra) Rel. Algebra) I [46] I [46] Point Point Calculus [42] [42] points points (total order) (total order) 3 A [42] 3 A [42] • Calculus Qualitat. Qualitat. Traject. Traject. Calc. QTC Calc.B11 QTC [40,41] movingmoving point obj.s pointinobj.s 1D in 1D9 U 9 U B11 [40,41] QTCB12 QTCB12 ” ”” ” 17 U 17 U QTCB21 QTCB21 ” ”movingmoving point obj.s pointinobj.s 2D in 2D9 U 9 U QTCB22 QTCB22 ” ”” ” 27 U 27 U QTCC12 QTCC12 ” ”” ” 81 U 81 U QTCC22 QTCC22 ” ”” ” 305 U305 U Region Connection Connection Calc. RCC-5 Calc. RCC-5 [34] [34] regionsregions 5 A [15] 5 A [15] • Region RCC-8 [34] [34] regionsregions 8 A [35] 8 A [35] • RCC-8 Cardinal Cardinal Rel.s RDR Rel.s[30] RDR [30] regionsregions 36 A [30] 36 A [30] a-closure Rectangular —• Rectangular Star Algebra STAR4STAR4 [36] [36] directions directions from a from pointa point 9 P 9 P consistencyaStar Algebra a Variant Variant DRAc DRA is notc based is not on based a weak on apartition weak partition schemescheme – JEPD – JEPD is violated is violated [24]. [24].
#BR: #BR: numbernumber of baseofrelations base relations Dylla, Mossakowski, Schneider, Wolter Algebraic Properties of Qualitative Spatio-Temporal Calculi
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And now . . .
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Requirements to qualitative calculi
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Algebraic properties
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Information-preservation properties
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Conclusion and outlook
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Why relation algebras?
If a calculus is a relation algebra (RA), then . . . certain optimisations in reasoners are permitted e.g., associativity of ensures fast processing of paths: if a QCSP contains xRy , ySz, zTu, then compute relation between x , u “from left to right” without associativity, compute (R S) T and R (S T )
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Why relation algebras?
If a calculus is a relation algebra (RA), then . . . certain optimisations in reasoners are permitted e.g., associativity of ensures fast processing of paths: if a QCSP contains xRy , ySz, zTu, then compute relation between x , u “from left to right” without associativity, compute (R S) T and R (S T )
RAs have been considered for spatio-temporal calculi before. [Ligozat & Renz 2004, Düntsch 2005, F. Mossakowski 2007]
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Frank is Dylla, Till Mossakowski, What a relation algebra? Thomas Schneider, and Diedrich Wolter R1 r∪s = R2 r ∪ (s ∪ t) = R3 r¯ ∪ s¯ ∪ r¯ ∪ s = R4 r � (s � t) = R5 (r ∪ s) � t = R6 r � id = R7 (r˘)˘ = R8 (r ∪ s)˘ = R9 (r � s)˘ = R10 r˘ � r � s ∪ s¯ = WA ((r ∩ id) � 1) � 1 = SA (r � 1) � 1 = R6l id � r = PL (r � s) ∩ t˘ = ∅ ⇔
s∪r ∪-commutativity (r ∪ s) ∪ t ∪-associativity r Huntington’s axiom (r � s) � t �-associativity (r � t) ∪ (s � t) �-distributivity r identity law r ˘-involution r˘ ∪ s˘ ˘-distributivity s˘ � r˘ ˘-involutive distributivity s¯ Tarski/de Morgan axiom (r ∩ id) � 1 weak �-associativity r�1 � semi-associativity r left-identity law (s � t) ∩ r˘ = ∅ Peircean law
Table 2. Axioms for relation algebras and weaker variants [22].
given TableSchneider, 2. AllWolter axioms except PL can be weakened to only one 21of Dylla,in Mossakowski, Algebraic Properties of Qualitative Spatio-Temporal Calculi
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Frank is Dylla, Till Mossakowski, What a relation algebra? Thomas Schneider, and Diedrich Wolter R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 WA SA R6l PL
r∪s = s∪r ∪-commutativity r ∪ (s ∪ t) = (r ∪ s) ∪ t ∪-associativity r¯ ∪ s¯ ∪ r¯ ∪ s = r Huntington’s axiom r � (s � t) = (r � s) � t �-associativity (r ∪ s) � t = (r � t) ∪ (s � t) �-distributivity r � id = r identity law (r˘)˘ = r ˘-involution (r ∪ s)˘ = r˘ ∪ s˘ ˘-distributivity (r � s)˘ = s˘ � r˘ ˘-involutive distributivity r˘ � r � s ∪ s¯ = s¯ Tarski/de Morgan axiom ((r ∩ id) � 1) � 1 = (r ∩ id) � 1 weak �-associativity algebra R1 , . . . , R10 RA (r �relation 1) � 1 = r�1 � semi-associativity id � r = r left-identity R1 , . . . , Rlaw NA non-associative RA 10 minus R4 (r � s) ∩ t˘ = ∅ ⇔ (s � t) ∩ r˘ = ∅ Peircean law WA, SA weakly/semi-associative RA weakenings of R4 Table 2. Axioms for relation algebras and weaker variants [22].
given TableSchneider, 2. AllWolter axioms except PL can be weakened to only one 21of Dylla,in Mossakowski, Algebraic Properties of Qualitative Spatio-Temporal Calculi
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Frank is Dylla, Till Mossakowski, What a relation algebra? Thomas Schneider, and Diedrich Wolter R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 WA SA R6l PL
r∪s = s∪r ∪-commutativity r ∪ (s ∪ t) = (r ∪ s) ∪ t ∪-associativity r¯ ∪ s¯ ∪ r¯ ∪ s = r Huntington’s axiom r � (s � t) = (r � s) � t �-associativity (r ∪ s) � t = (r � t) ∪ (s � t) �-distributivity r � id = r identity law (r˘)˘ = r ˘-involution (r ∪ s)˘ = r˘ ∪ s˘ ˘-distributivity (r � s)˘ = s˘ � r˘ ˘-involutive distributivity r˘ � r � s ∪ s¯ = s¯ Tarski/de Morgan axiom ((r ∩ id) � 1) � 1 = (r ∩ id) � 1 weak �-associativity algebra R1 , . . . , R10 RA (r �relation 1) � 1 = r�1 � semi-associativity id � r = r left-identity R1 , . . . , Rlaw NA non-associative RA 10 minus R4 (r � s) ∩ t˘ = ∅ ⇔ (s � t) ∩ r˘ = ∅ Peircean law WA, SA weakly/semi-associative RA weakenings of R4 Table 2. Axioms for relation algebras and weaker variants [22].
Calculi à la Ligozat & Renz: based on NA’s (by definition)
given TableSchneider, 2. AllWolter axioms except PL can be weakened to only one 21of Dylla,in Mossakowski, Algebraic Properties of Qualitative Spatio-Temporal Calculi
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Testing algebraic properties of calculi Research questions 1
Which calculi correspond to RAs (NAs, WAs, SAs)?
2
Which weaker algebra notions correspond to other calculi?
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Testing algebraic properties of calculi Research questions 1
Which calculi correspond to RAs (NAs, WAs, SAs)?
2
Which weaker algebra notions correspond to other calculi?
Experimental setup Corpus: 31 calculi listed before Used HETS (Heterogeneous Tool Set) to test the SparQ implementation of each calculus against CASL specifications of the RA axioms (+ weakenings)
Some axioms trivially hold ; no need to test them
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Testing algebraic properties of calculi Research questions 1
Which calculi correspond to RAs (NAs, WAs, SAs)?
2
Which weaker algebra notions correspond to other calculi?
Experimental setup Corpus: 31 calculi listed before Used HETS (Heterogeneous Tool Set) to test the SparQ implementation of each calculus against CASL specifications of the RA axioms (+ weakenings)
Some axioms trivially hold ; no need to test them Parallel tests via SparQ’s built-in function analyze-calculus Dylla, Mossakowski, Schneider, Wolter
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Test12Frank results calculus Dylla, Frankper Till Dylla, Mossakowski, Till Mossakowski, ThomasThomas Schneider, Schneider, and Diedrich and Diedrich Wolter Wolter
CalculusCalculus Testsa Tests R4 a SA R4 WASA R6WAR6l R6R7 R6lR9 R7PL R9R10PL R10 Allen Allen MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Block Algebra Block Algebra HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ CardinalCardinal Direction Direction CalculusCalculus MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ CYCb , Geometric CYCb , Geometric Orientation Orientation HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ DRAfp , DRA DRA-conn. HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ fp , DRA-conn. Point Calculus Point Calculus HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ RCC-5, RCC-5, Dependency Dependency Calc. Calc. MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ RCC-8, RCC-8, 9-Intersection 9-Intersection MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ STAR4 STAR4 HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ DRAf DRAf MHS MHS 19 ✓ 19 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ INDU INDU MHS MHS 12 ✓ 12 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ OPRAnOPRA , n � 8n , n � 8 MHS 21–91 MHSb 21–91 ✓ b✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ QTCBxxQTCBxx MHS MHS ✓ ✓✓ ✓ ✓ 89–100 ✓ 89–100 ✓ ✓ ✓✓ ✓ ✓ ✓ ✓ QTCC21QTCC21 HS HS 55 ✓ 55 ✓ ✓ 99✓ 99 99✓ 99 2 ✓
Requirements
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Algebraic Properties of Qualitative Spatio-Temporal Calculi1 Frank Dyllaa Till Mossakowskia,b Thomas Schneider c Diedrich Wolter a a Collaborative
Research Center on Spatial Cognition (SFB/TR 8), Univ. Bremen Research Center for Artificial Intelligence (DFKI), Bremen c Dept of Mathematics and Computer Science, University of Bremen b German
2 August 2013
1
Thanks for discussions at the “Spatial Reasoning Teatime”.
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And now . . .
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Qualitative spatio-temporal representation and reasoning
= Symbolic way to represent spatio-temporal knowledge and draw inferences from it
Common approach: define set R of relations to describe spatial relationships use R as primitives for representation employ techniques from constraint and qualitative reasoning to reason about the primitives various domains, typically infinite
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Applications by domain Time interval relations I Medical diagnostics I Law texts
Simplified Allen Allen-13
I Business and manufacturing: diagnostics
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Allen-13
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Applications by domain Positions, regions (topology) I Planning, robotics, navigation I Natural language processing I Image understanding
9-int, CarDir, RCC 9-intersection
I CAD and manufacturing
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Rectangle Alg., . . . 9-int, CarDir, RCC, Allen
I GIS, spatial query answering I Traffic tracking
RCC, Block Algebra, ROC
LR, RCC
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Applications by domain Moving point objects, directional information
OPRA, Dipole Flipflop, StarVars
I Robotics, navigation, motion planning I GIS, spatial query answering
QTC
I Traffic tracking
Dipole
I Ambient intelligence, smart environments (scene analysis, task modelling)
OPRA, RCC
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Representation and reasoning tasks required I Knowledge representation
I Data interpretation I Inference
I Constraint-based reasoning
(CSP-SAT, -ENT, -MOD, -MIN)
I Neighbourhood-based reasoning
(Relaxing constraints, continuity constraints, dominance space)
I Logical reasoning
(Deduction, abduction)
I Learning
(Inductive logic programming)
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What is a qualitative calculus? Some answers from the literature A weak representation of a non-associative relation algebra
[Egenhofer & Rodríguez 1999; Ligozat et al. 2003; Ligozat & Renz 2004]
A system of relations forming a constraint algebra [Nebel & Scivos 2002]
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What is a qualitative calculus? Some answers from the literature A weak representation of a non-associative relation algebra
[Egenhofer & Rodríguez 1999; Ligozat et al. 2003; Ligozat & Renz 2004]
A system of relations forming a constraint algebra [Nebel & Scivos 2002]
Commonly agreed ingredients Set R of relations Operations ∪, ∩, ¯, ◦, ˘ with certain properties (closure, algebraic properties) Mapping to a domain with certain properties (e.g., JEPD) Dylla, Mossakowski, Schneider, Wolter
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And in reality? Zoo of qualitative calculi 36 implemented in SparQ; many more in the literature “classical” calculi, usually with strong algebraic properties (e.g., Allen-13, RCC-8)
more recent calculi, often with weaker algebraic properties
(e.g., Cardinal Direction Relations, Rectangular Cardinal Relations)
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And in reality? Zoo of qualitative calculi 36 implemented in SparQ; many more in the literature “classical” calculi, usually with strong algebraic properties (e.g., Allen-13, RCC-8)
more recent calculi, often with weaker algebraic properties
(e.g., Cardinal Direction Relations, Rectangular Cardinal Relations)
;
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And in reality? Zoo of qualitative calculi 36 implemented in SparQ; many more in the literature “classical” calculi, usually with strong algebraic properties (e.g., Allen-13, RCC-8)
more recent calculi, often with weaker algebraic properties
(e.g., Cardinal Direction Relations, Rectangular Cardinal Relations)
; Research question To what extent do existing calculi meet the imposed requirements? ; If not, what are minimal requirements? ; How can we classify calculi according to their properties? Dylla, Mossakowski, Schneider, Wolter
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On the agenda today
Revisit and generalise the definition of a qualitative calculus Identify notions of algebras that cover existing calculi Discuss relevance of algebraic properties for spatial reasoning Evaluate the algebraic properties of existing calculi ; derive improved generic reasoning procedure Examine information-preservation properties of calculi during reasoning
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And now . . .
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Calculi à la Ligozat & Renz (1) Basic notions Universe (domain) U: spatio-temporal entities Set R of base relations over U Uncertain information ; union of base relations Restriction to binary relations in this work
R is JEPD: jointly exhaustive and pairwise disjoint Jointly exhaustive: R covers U × U
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Calculi à la Ligozat & Renz (1) Basic notions Universe (domain) U: spatio-temporal entities Set R of base relations over U Uncertain information ; union of base relations Restriction to binary relations in this work
R is JEPD: jointly exhaustive and pairwise disjoint Jointly exhaustive: R covers U × U
Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Dylla, Mossakowski, Schneider, Wolter
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition)
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition) Converse: ϕ(r˘) = ϕ(r )˘
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition) Converse: ϕ(r˘) = ϕ(r )˘ Weak composition: r s = smallest set T of base rel.s with ϕ(T ) ⊇ ϕ(r ) ◦ ϕ(s)
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Calculi à la Ligozat & Renz (2) Partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation ϕ = mapping to a partition scheme Plus symbolic operations ˘, (converse, weak composition) Converse: ϕ(r˘) = ϕ(r )˘ Weak composition: r s = smallest set T of base rel.s with ϕ(T ) ⊇ ϕ(r ) ◦ ϕ(s) ˘, usually given by tables Dylla, Mossakowski, Schneider, Wolter
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These requirements are strong
Some calculi violate them e.g.: Cardinal Direction Relations Rectangular Cardinal Relations Their converse only satisfies ϕ(r˘) ⊇ ϕ(r )˘
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These requirements are strong
Some calculi violate them e.g.: Cardinal Direction Relations Rectangular Cardinal Relations Their converse only satisfies ϕ(r˘) ⊇ ϕ(r )˘ ; Weaken requirements to partition schemes and calculi!
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation = mapping to an abstract part. scheme Plus symbolic operations ˘, (abstract converse & compos.)
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation = mapping to an abstract part. scheme Plus symbolic operations ˘, (abstract converse & compos.) Abstract converse: ϕ(r˘) ⊇ ϕ(r )˘
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Our notion of a calculus Abstract partition scheme Pair (U, R) with R being JEPD R contains the identity relation id and is closed under converse ˘ Qualitative calculus Set of symbolic relations Plus interpretation = mapping to an abstract part. scheme Plus symbolic operations ˘, (abstract converse & compos.) Abstract converse: ϕ(r˘) ⊇ ϕ(r )˘ Abstract composition: ϕ(r s) ⊇ ϕ(r ) ◦ ϕ(s)
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Qualitative spatio-temporal reasoning
Qualitative constraint Formula xRy with x , y variables, R relation from a calculus C
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Qualitative spatio-temporal reasoning
Qualitative constraint Formula xRy with x , y variables, R relation from a calculus C Qualitative constraint satisfaction problem (QCSP) Input: set of constraints Question: Is there a mapping from variables to C’s domain that satisfies all constraints?
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Qualitative spatio-temporal reasoning
Qualitative constraint Formula xRy with x , y variables, R relation from a calculus C Qualitative constraint satisfaction problem (QCSP) Input: set of constraints Question: Is there a mapping from variables to C’s domain that satisfies all constraints? (Analogous definition for other reasoning problems)
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Qualitative spatio-temporal reasoning Common techniques for solving QCSPs Some taken over from finite-domain CSPs (constraint propagation, k-consistency)
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Qualitative spatio-temporal reasoning Common techniques for solving QCSPs Some taken over from finite-domain CSPs (constraint propagation, k-consistency) Algebraic closure (a-closure) sufficient condition for consistency guaranteed by “⊇” of abstract composition For some calculi, a-closure known to be necessary too
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Qualitative spatio-temporal reasoning Common techniques for solving QCSPs Some taken over from finite-domain CSPs (constraint propagation, k-consistency) Algebraic closure (a-closure) sufficient condition for consistency guaranteed by “⊇” of abstract composition For some calculi, a-closure known to be necessary too
Composition of arbitrary relations R, S is uniquely determined by the composition results of the base relations in R, S
(composition table)
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Existing qualitative spatio-temporal calculi Algebraic Algebraic Properties Properties of Qualitative of Qualitative Spatio-Temporal Spatio-Temporal CalculiCalculi 7
decides
Name Name Ref. Ref. Domain Domain #BR #BR RM RM 9-Intersection 9-Intersection [9] [9] simple simple 2D regions 2D regions 8 I [12,16] 8 I [12, • Allen’s Allen’s intervalinterval relations relations [1] [1] intervals intervals (order)(order) 13 A [42] 13 A [42] n n • Algebra Block Block Algebra [2] [2] n-dimensional n-dimensional blocks blocks 13 A13 [2] A [2] • Cardinal Cardinal Dir. Calculus Dir. Calculus CDC CDC [10,17] [10,17] directions directions (point (point abstr.)abstr.) 9 A [17] 9 A [17] Cardinal Cardinal Dir. Relations Dir. Relations CDR CDR [38] [38] regionsregions 218 P 218 P CycOrd, CycOrd, binary binary CYCb CYCb [14] [14] oriented oriented lines lines 4 U 4 U Dependency Calculus Calculus [33] [33] points points (partial(partial order) order) 5 A [33] 5 A [33] • Dependency a a Dipole Dipole Calculus Calculus DRAf DRAf [25,24] [25,24] directions directions from line from segm. line segm. 72 I [46] 72 I [46] DRAfpDRAfp [24] [24] directions directions from line from segm. line segm. 80 I 80 I DRA-connectivity DRA-connectivity [45] [45] connectivity connectivity of line of segm. line segm.7 U 7 U Geometric Geometric Orientation Orientation [7] [7] relativerelative orientation orientation 4 U 4 U INDU INDU [32] [32] intervals intervals (order,(order, rel. dur.n) rel. dur.n) 25 P 25 P OPRAOPRA ,m . . .=, 81, . . . , 8 [23,28] [23,28] oriented oriented points points 4m · (4m 4m+· (4m 1) + 1) m, m = m1, (Oriented (Oriented Point Rel. Point Algebra) Rel. Algebra) I [46] I [46] Point Point Calculus [42] [42] points points (total order) (total order) 3 A [42] 3 A [42] • Calculus Qualitat. Qualitat. Traject. Traject. Calc. QTC Calc.B11 QTC [40,41] movingmoving point obj.s pointinobj.s 1D in 1D9 U 9 U B11 [40,41] QTCB12 QTCB12 ” ”” ” 17 U 17 U QTCB21 QTCB21 ” ”movingmoving point obj.s pointinobj.s 2D in 2D9 U 9 U QTCB22 QTCB22 ” ”” ” 27 U 27 U QTCC12 QTCC12 ” ”” ” 81 U 81 U QTCC22 QTCC22 ” ”” ” 305 U305 U Region Connection Connection Calc. RCC-5 Calc. RCC-5 [34] [34] regionsregions 5 A [15] 5 A [15] • Region RCC-8 [34] [34] regionsregions 8 A [35] 8 A [35] • RCC-8 Cardinal Cardinal Rel.s RDR Rel.s[30] RDR [30] regionsregions 36 A [30] 36 A [30] a-closure Rectangular —• Rectangular Star Algebra STAR4STAR4 [36] [36] directions directions from a from pointa point 9 P 9 P consistencyaStar Algebra a Variant Variant DRAc DRA is notc based is not on based a weak on apartition weak partition schemescheme – JEPD – JEPD is violated is violated [24]. [24].
#BR: #BR: numbernumber of baseofrelations base relations Dylla, Mossakowski, Schneider, Wolter Algebraic Properties of Qualitative Spatio-Temporal Calculi
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And now . . .
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Why relation algebras?
If a calculus is a relation algebra (RA), then . . . certain optimisations in reasoners are permitted e.g., associativity of ensures fast processing of paths: if a QCSP contains xRy , ySz, zTu, then compute relation between x , u “from left to right” without associativity, compute (R S) T and R (S T )
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Why relation algebras?
If a calculus is a relation algebra (RA), then . . . certain optimisations in reasoners are permitted e.g., associativity of ensures fast processing of paths: if a QCSP contains xRy , ySz, zTu, then compute relation between x , u “from left to right” without associativity, compute (R S) T and R (S T )
RAs have been considered for spatio-temporal calculi before. [Ligozat & Renz 2004, Düntsch 2005, F. Mossakowski 2007]
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Frank is Dylla, Till Mossakowski, What a relation algebra? Thomas Schneider, and Diedrich Wolter R1 r∪s = R2 r ∪ (s ∪ t) = R3 r¯ ∪ s¯ ∪ r¯ ∪ s = R4 r � (s � t) = R5 (r ∪ s) � t = R6 r � id = R7 (r˘)˘ = R8 (r ∪ s)˘ = R9 (r � s)˘ = R10 r˘ � r � s ∪ s¯ = WA ((r ∩ id) � 1) � 1 = SA (r � 1) � 1 = R6l id � r = PL (r � s) ∩ t˘ = ∅ ⇔
s∪r ∪-commutativity (r ∪ s) ∪ t ∪-associativity r Huntington’s axiom (r � s) � t �-associativity (r � t) ∪ (s � t) �-distributivity r identity law r ˘-involution r˘ ∪ s˘ ˘-distributivity s˘ � r˘ ˘-involutive distributivity s¯ Tarski/de Morgan axiom (r ∩ id) � 1 weak �-associativity r�1 � semi-associativity r left-identity law (s � t) ∩ r˘ = ∅ Peircean law
Table 2. Axioms for relation algebras and weaker variants [22].
given TableSchneider, 2. AllWolter axioms except PL can be weakened to only one 21of Dylla,in Mossakowski, Algebraic Properties of Qualitative Spatio-Temporal Calculi
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Frank is Dylla, Till Mossakowski, What a relation algebra? Thomas Schneider, and Diedrich Wolter R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 WA SA R6l PL
r∪s = s∪r ∪-commutativity r ∪ (s ∪ t) = (r ∪ s) ∪ t ∪-associativity r¯ ∪ s¯ ∪ r¯ ∪ s = r Huntington’s axiom r � (s � t) = (r � s) � t �-associativity (r ∪ s) � t = (r � t) ∪ (s � t) �-distributivity r � id = r identity law (r˘)˘ = r ˘-involution (r ∪ s)˘ = r˘ ∪ s˘ ˘-distributivity (r � s)˘ = s˘ � r˘ ˘-involutive distributivity r˘ � r � s ∪ s¯ = s¯ Tarski/de Morgan axiom ((r ∩ id) � 1) � 1 = (r ∩ id) � 1 weak �-associativity algebra R1 , . . . , R10 RA (r �relation 1) � 1 = r�1 � semi-associativity id � r = r left-identity R1 , . . . , Rlaw NA non-associative RA 10 minus R4 (r � s) ∩ t˘ = ∅ ⇔ (s � t) ∩ r˘ = ∅ Peircean law WA, SA weakly/semi-associative RA weakenings of R4 Table 2. Axioms for relation algebras and weaker variants [22].
given TableSchneider, 2. AllWolter axioms except PL can be weakened to only one 21of Dylla,in Mossakowski, Algebraic Properties of Qualitative Spatio-Temporal Calculi
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Frank is Dylla, Till Mossakowski, What a relation algebra? Thomas Schneider, and Diedrich Wolter R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 WA SA R6l PL
r∪s = s∪r ∪-commutativity r ∪ (s ∪ t) = (r ∪ s) ∪ t ∪-associativity r¯ ∪ s¯ ∪ r¯ ∪ s = r Huntington’s axiom r � (s � t) = (r � s) � t �-associativity (r ∪ s) � t = (r � t) ∪ (s � t) �-distributivity r � id = r identity law (r˘)˘ = r ˘-involution (r ∪ s)˘ = r˘ ∪ s˘ ˘-distributivity (r � s)˘ = s˘ � r˘ ˘-involutive distributivity r˘ � r � s ∪ s¯ = s¯ Tarski/de Morgan axiom ((r ∩ id) � 1) � 1 = (r ∩ id) � 1 weak �-associativity algebra R1 , . . . , R10 RA (r �relation 1) � 1 = r�1 � semi-associativity id � r = r left-identity R1 , . . . , Rlaw NA non-associative RA 10 minus R4 (r � s) ∩ t˘ = ∅ ⇔ (s � t) ∩ r˘ = ∅ Peircean law WA, SA weakly/semi-associative RA weakenings of R4 Table 2. Axioms for relation algebras and weaker variants [22].
Calculi à la Ligozat & Renz: based on NA’s (by definition)
given TableSchneider, 2. AllWolter axioms except PL can be weakened to only one 21of Dylla,in Mossakowski, Algebraic Properties of Qualitative Spatio-Temporal Calculi
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Testing algebraic properties of calculi Research questions 1
Which calculi correspond to RAs (NAs, WAs, SAs)?
2
Which weaker algebra notions correspond to other calculi?
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Testing algebraic properties of calculi Research questions 1
Which calculi correspond to RAs (NAs, WAs, SAs)?
2
Which weaker algebra notions correspond to other calculi?
Experimental setup Corpus: 31 calculi listed before Used HETS (Heterogeneous Tool Set) to test the SparQ implementation of each calculus against CASL specifications of the RA axioms (+ weakenings)
Some axioms trivially hold ; no need to test them
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Testing algebraic properties of calculi Research questions 1
Which calculi correspond to RAs (NAs, WAs, SAs)?
2
Which weaker algebra notions correspond to other calculi?
Experimental setup Corpus: 31 calculi listed before Used HETS (Heterogeneous Tool Set) to test the SparQ implementation of each calculus against CASL specifications of the RA axioms (+ weakenings)
Some axioms trivially hold ; no need to test them Parallel tests via SparQ’s built-in function analyze-calculus Dylla, Mossakowski, Schneider, Wolter
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Information preservation
Conclusion
Test12Frank results calculus Dylla, Frankper Till Dylla, Mossakowski, Till Mossakowski, ThomasThomas Schneider, Schneider, and Diedrich and Diedrich Wolter Wolter
CalculusCalculus Testsa Tests R4 a SA R4 WASA R6WAR6l R6R7 R6lR9 R7PL R9R10PL R10 Allen Allen MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Block Algebra Block Algebra HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ CardinalCardinal Direction Direction CalculusCalculus MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ CYCb , Geometric CYCb , Geometric Orientation Orientation HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ DRAfp , DRA DRA-conn. HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ fp , DRA-conn. Point Calculus Point Calculus HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ RCC-5, RCC-5, Dependency Dependency Calc. Calc. MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ RCC-8, RCC-8, 9-Intersection 9-Intersection MHS MHS ✓ ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ STAR4 STAR4 HS ✓HS ✓✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ DRAf DRAf MHS MHS 19 ✓ 19 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ INDU INDU MHS MHS 12 ✓ 12 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ OPRAnOPRA , n � 8n , n � 8 MHS 21–91 MHSb 21–91 ✓ b✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ QTCBxxQTCBxx MHS MHS ✓ ✓✓ ✓ ✓ 89–100 ✓ 89–100 ✓ ✓ ✓✓ ✓ ✓ ✓ ✓ QTCC21QTCC21 HS HS 55 ✓ 55 ✓ ✓ 99✓ 99 99✓ 99 2 ✓
