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Temporal Concept Logic Objects, States, and Systems. Karl Erich Wolff University of Applied Sciences Darmstadt Ernst Schröder Center for Conceptual Knowledge Processing Research Group Concept Analysis at Darmstadt University of Technology. Old Unsolved Problems in Temporal Logic. - PowerPoint PPT Presentation
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Temporal Concept Logic Objects, States, and Systems
Karl Erich Wolff
University of Applied Sciences Darmstadt
Ernst Schröder Center
for Conceptual Knowledge Processing
Research Group Concept Analysis at
Darmstadt University of Technology
Old Unsolved Problems in Temporal Logic
Aristotle asked (Vol. 6, book III – VI):
What is the meaning of ...
• movement, change
• continuum, infinity
• place, empty place
• solid body, object
• time, point of time
• (state) ?
Aristotle
Famous Space-Time Scientists
• Zenon of Elea 495 – 430
• Aristotle 384 – 322
• Galileo Galilei 1564 – 1642
• Isaac Newton 1643 – 1727
• Immanuel Kant 1724 – 1804
• Jules Henri Poincaré 1854 – 1912
• Max Planck 1858 – 1947
• Albert Einstein 1879 – 1955
• Werner Heisenberg 1901 – 1976
Recent Approaches in Temporal Logic
• Arthur Norman Prior: Tense Logic (1957)
P "It has at some time been the case that … ,
F "It will at some time be the case that … ,
H "It has always been the case that … ,
G "It will always be the case that ... .“
• Hans Kamp (Linguistics, Stuttgart University)
Spq "q has been true since a time when p was true"
Upq "q will be true until a time when p is true"
• Johan van Benthem (ILLC Amsterdam, CSLI Stanford)
The Logic of Time (1983)
Logic and the Flow of Information (1991)
Precision and Granularity
Aristotle
(Physics, book VI, 239a, 23):
During the time when a system is moving, not only moving in some of its parts,
it is impossible that the moving system is precisely at a certain place.
We need a theoretical treatment of granularity!
Einstein’s Granularity Remark
Albert Einstein:
„Zur Elektrodynamik bewegter Körper“
Annalen der Physik 17 (1905): 891-921
Footnote on page 893:
„Die Ungenauigkeit, welche in dem Begriff der Gleichzeitigkeit zweier Ereignisse an (annähernd) demselben Orte steckt und gleichfalls durch eine Abstraktion überbrückt werden muß, soll hier nicht erörtert werden.“
Partial Solutions in Modern Time Theories
• Automata Theory: States, Transitions
generalized by
• Labeled Transition Systems with Attributes:
No explicit time representation
• Petri Nets: States, Transitions, Token, Time
States and Transitions are
not related to Time
• Mathematical System Theory
What is a state? (Zadeh 1964)
What is a system? (Lin 1999)
• Something which is stable for a moment?
• Stable: Constant as to some granularity
• Moment (instant, point of time): A time object described in some context
• For each moment the system should be in exactly one state!
What do we mean by a State of a System?
Conceptual Time Systems
Description of ...
• Temporal systems as scaled many-valued contexts
• Instants as formal objects (time granules)
• Granularity by Conceptual Scales
Then
• States are defined as object concepts
Example: Air Conditioning Plant
At any hour:
Temperatures:
OutdoorRoom 3Room 2Hot Water
In the following:Only for 3 days
Time Object Day Hour Outdoor-Temp Room3-Temp Room2-Temp Hot Water Temp100 1 0 3,3 19,6 19,7 32,6101 1 1 3,3 19,6 19,7 32,6102 1 2 3,3 19,5 19,7 32,6103 1 3 3,3 19,5 19,7 32,6104 1 4 3,3 19,5 19,7 32,6105 1 5 3,3 19,5 19,6 32,6106 1 6 3,3 19,5 19,8 33,8107 1 7 3,3 20,3 21,1 39,9108 1 8 3,3 21 21,4 40109 1 9 3,4 21,5 21,4 40110 1 10 3,3 21,8 21,5 40111 1 11 3,7 22,1 21,6 40112 1 12 4,1 22,4 21,6 40113 1 13 4,4 22,6 21,7 40114 1 14 4,2 22,8 21,7 40115 1 15 4,4 22,9 21,7 40116 1 16 4,7 22,9 21,7 40117 1 17 4,9 23 21,8 40118 1 18 5,3 23 21,8 40119 1 19 5,3 23,1 21,8 40120 1 20 5,4 23,1 21,8 39,2121 1 21 5,4 22,5 21,6 36,5122 1 22 5,1 21,9 21,2 34,8123 1 23 4,7 21,5 20,9 33,5
Temperatures during the first three days
0
5
10
15
20
25
30
35
40
45
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71
Outdoor Temperature
Room 3 Temperature
Room 2 Temperature
Hot Water Temperature
What is a state of this system?
0
5
10
15
20
25
30
35
40
45
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71
Outdoor Temperature
Room 3 Temperature
Room 2 Temperature
Hot Water Temperature
The same state in three situations!
Situations: state-time pairsfor the outdoor temperatureand the hours (not the days)
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
First Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
First Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
First Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
First Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
First Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
First Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
First Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Second Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
Stundeo5iaus6
aussen: <=3.0
aussen: <=4.0
aussen: <=5.0
aussen: <=6.0
aussen: <=7.0
aussen: <=7.4Stunde: 0-23
Stunde: >=12
Stunde: >=16
Stunde: >=20
Stunde: >=8
214215
213
212
118119216316
120121122321
200201202206
211311
112113114115312313314315 116
117217218317318319
123320322323
100101102103104105106107203<...>
108109110111208209210308309310
219
220
300301302303304305306
221221223
A movie in situation space:
Outdoor Temperature
Third Day
What is a transition?
In Temporal Concept Logic:For the definition of states we do not need an ordering on the set G time objects.For introducing transitions we use a time relation R on G.A basic R-transition is defined as an element (g,h) of R.For f: G X the induced transition on X is defined as( (g,h), f(g), f(h) ).Example: Transitions in the state space or in the situation space.
Life tracks and objects
birth
copy
cooperation
One object at the same point of time
in two different situations??
Objects as subsystems
forbidden
Each object is in each moment g
in exactly one situation.
The state space of a family (in the language of the therapeut)
Transitions in Automata
In Automata Theory: A transition is a labeled pair of states.The order of time is represented only by the names„initial states“ and „terminal states“.
Automata and even the more general structure of Labeled Transitions Systems with Attributes (LTSA)can be represented by conceptual time systems (with actual objects and a time relation) such that the automaton-transitions are represented by conceptual transitions.(Map Reconstruction Theorem)
The Map Reconstruction Theorem
