Interpretation of Automata in Temporal Concept Analysis

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Interpretation of Automata inTemporal Concept
Analysis
Karl Erich Wolff University of Applied Sciences
Darmstadt Ernst Schröder Center for Conceptual
Knowledge Processing Research Group Concept
Analysis at Darmstadt University of Technology
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Content

• Problems with Temporal Phenomena
• Automata and Labeled Transition Systems with
  Attributes
• Temporal Concept Analysis
• The Map Reconstruction Theorem

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Reality and Representation
Reality ?
Representations

• Signs
• Concepts
• Conceptual Structures

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Movements

• Place ?
• Space?
• Time ?
• Point of Time ?
• State ?
• Transition ?
• Object ?
• Granularity ?

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Precision and Granularity
Aristotle (Vol. 6, book VI, p. 163) 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!
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Einsteins 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.
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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)

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Automata
Initial state
Terminal states
No explicit time description!
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Labeled Transition Systemswith Attributes
Automaton
LTSA
No explicit time description! States not
attribute-specified
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An attribute-specified LTSA
Automaton
LTSA
attribute-specified
What is a state? What is a transition?
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What do we mean by a State of a System?

• 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!

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Conceptual Time Systems
(with a Time Relation)
Time part T
Event part C
g
v
w
(Time Relation)
h
Time scales
Event scales
Object concepts in
K(C)
K(T)
Time states
States
Situations
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Example Air Conditioning Plant

At any hour Temperatures Outdoor Room 3 Room
2 Hot Water
In the following Only for 3 days
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Temperatures during the first three days
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What is a state of this system?
The same state in three situations!
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A Situation SpaceOutdoor temperatureand hours
(not the days)
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A movie in situation space Outdoor
Temperature First Day
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A movie in situation space Outdoor
Temperature First Day
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A movie in situation space Outdoor
Temperature First Day
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A movie in situation space Outdoor
Temperature First Day
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A movie in situation space Outdoor
Temperature First Day
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A movie in situation space Outdoor
Temperature First Day
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A movie in situation space Outdoor
Temperature First Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Second Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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A movie in situation space Outdoor
Temperature Third Day
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Transitions and Life Tracks

Transition ( (g,h), (f(g), f(h)) ) Life Track f
(g,f(g)) g G
?
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Life tracks and objects
birth
cooperation
One object at the same point of time in two
different situations?
?
copy
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Objects as subsystems
Each object is in each moment g in exactly one
situation.
forbidden
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Conceptual Time Systems with Objects
and a Time Relation (CTSOT)
Time part T
Event part C
g
v
w
Object 1
h
i
Object 2
j
Time scales
Event scales
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The state space of a family (in the language of
the therapeut)

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The Map Reconstruction Theorem (1)
A state-transition covering
Linearly ordered CTSOT
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The derived context
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The Map Reconstruction Theorem (2)
LTSA isomorphic to State-LTSA (of a
CTSOT)
States
States Paths
Life Tracks
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Thank you!