Temporal Concept Logic Objects, States, and Systems

Temporal Concept LogicObjects, 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)

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 • 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: 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!

Situations: state-time pairsfor the outdoor temperatureand the 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

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 Ron 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 cooperation One object at the same point of time in two different situations? ? copy

Objects as subsystems Each object is in each moment g in exactly one situation. forbidden

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)

Temporal Concept Logic Objects, States, and Systems

Temporal Concept Logic Objects, States, and Systems

Presentation Transcript

XCTL (Explicit Clock Temporal Logic)

Expressive Power of Temporal Logic

Ordering Systems: Temporal Sequential and Spatial

Identifying Objects Using Cluster and Concept Analysis

Temporal Parts of Four-Dimensional Objects

Axiomatizations of Temporal Logic

LINEAR TEMPORAL LOGIC

Temporal Logic

Axiomatizations of Temporal Logic

A Logic of Arbitrary and Indefinite Objects

Temporal Logic

Digital Systems and Logic Design

Spatial-temporal Database and moving objects management

Temporal Logic and Model Checking

Logic Gate Objects

Alternating Temporal Logic and Game-Based Properties

The Temporal Logic Sugar

Querying Mobile Objects in Spatio-Temporal Databases

XCTL (Explicit Clock Temporal Logic)

Interval Hybrid Temporal Logic

Temporal logic

A Logic of Arbitrary and Indefinite Objects