Interpretation of Automata in Temporal Concept Analysis

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

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

Reality and Representation Reality ? Representations: • Signs • Concepts • Conceptual Structures

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

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!

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)

Automata Initial state Terminal states No explicit time description!

Labeled Transition Systemswith Attributes Automaton LTSA No explicit time description! States: not attribute-specified

An attribute-specified LTSA Automaton LTSA attribute-specified What is a state? What is a transition?

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!

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(T) K(C) Time states States Situations

Example: Air Conditioning Plant At any hour: Temperatures: Outdoor Room 3 Room 2 Hot Water In the following: Only for 3 days

Temperatures during the first three days

What is a state of this system? The same state in three situations!

A Situation Space:Outdoor temperatureand hours (not the days)

A movie in situation space: Outdoor Temperature First Day

A movie in situation space: Outdoor Temperature Second Day

A movie in situation space: Outdoor Temperature Third Day

Transitions and Life Tracks Transition: ( (g,h), (f(g), f(h)) ) Life Track f := {(g,f(g)) | g G } 

Life tracks and objects birth cooperation One object at the same point of time in two different situations? ? copy

Interpretation of Automata in Temporal Concept Analysis