Logical and Mathematical Aspects of Dynamic Knowledge Engineering

1. Knowledge Engineering and Mathematical Geology 2015 Logical and Mathematical Aspectsof Dynamic Knowledge Engineering Cyril Pshenichny, ITMO University

2. Logical Inference Traditional definition (Gentzen, 1938): , where  is a list of formulae A1, A2, …, Am, and  is a list of formulae B1, B2, …, Bn: A1, A2, …, Am B1, B2, …, Bn or (A1&A2&…&An)(B1B2…Bm ). If all A1, A2, …, Am are true, then: at least one of B1, B2, …, Bn must be true either. Impossible: AiAi (& - conjunction,  - disjunction,  - negation,  - implication,  - logical entailment/inference).

3. Logical Inference Let us suggest a classification of inference based on the length of lists of formulae (Pshenichny and Mouromtsev, 2015): 1. A1, A2, …, Am B1, B2, …, Bn(zhivot), 2. A1, A2, …, Am B (shta), 3. A B1, B2, …, Bn(yon), 4. A B (izhen).

4. Logical Inference In terms of predicate logic and sets theory – Impossible: x(Pi (x)&Pi(x)) ( - universal quantifier, x – individual variable, Pi– i-th predicate of this variable). xk Pi(x) Pi(x)

5. Logical Inference BUT…

6. Volcanic edifice is intact  Volcanic edifice has a hole in its side Groundwater does not boil  Groundwater is boiling Ash is in the air  Ash is on the ground ETC. Photo by Boris Behncke and INGV staff, 2006

7. Logical Inference: Still the Case? Remedies suggested so far: • Introduction of time, temporal labels; • Nonmonotonous (“dynamic”) reasoning about incompleteness of knowledge; • Multi-valued logics; • Modal logic; • Probabilistic approach; • Possibilistic/belief-based approach.

8. Logical Inference: Extended Definition Extended definition: , where  is a list of formulae A1, A2, …, Am, and  is a list of formulae B1, B2, …, Bn: A1, A2, …, Am B1, B2, …, Bn if truth values of B1, B2, …, Bnare defined by truth values of A1, A2, …, Am by some law. Inference is a function F(A1, A2, …, Am, B1, B2, …, Bn) that returns values T or F to B1, B2, …, Bn, in which A1, A2, …, Am are independent variables. Of course, no such calculi as in the classical logic

9. Logical Inference: Extended Definition In terms of sets theory… an extension is needed too! AA: Assumption: elements may move from set to set. Pi(x) Pi(x) This idea can be elaborated in different directions. For example, as follows…

10. Proposed Extension to Sets Theory • Class is a set that has meaningful name. • Open class is a class with “passable walls” (i.e., a class that elements may enter and leave). • Crisp open class is an open class which elements leave all at once (not in portions). • System of crisp open classes is a finite set of crisp open classes that may exchange elements with each other. • Crisp system of crisp open classes implies that elements “jump” from class to class immediately (are never “stopped in between”).

11. Proposed Extension to Sets Theory • Crisp systems of crisp open classes may be open and closed. Open one allows elements come “from the space” and leave there, closed requires them to move only from class to class. We will be dealing with closed crisp systems of crisp open classes only. • Elements of one class may, if defined so, spawn or absorb elements of another class.

12. Proposed Extension to Sets Theory Bitie (Russian бытие, being)is a property of open class to contain elements. If a class is having elements, it possesses being. Empty class has no being. Class may acquire or lose its being. In the latter case, it gets nebitie. Sets with meaningless names may not include elements and therefore cannot be. Being of a class is assumed equivalent to the value True of a statement/formula that says this, and non-being, to the value False of the said formula.

13. Proposed Extension to Sets Theory Crisp open classes in a crisp system may be predmets (subjects, S) and primetas (properties, P). Predmets (S) may have elements regardless of their relationship with other classes, primetas (P) – only in the intersection with a subject. S P

14. Proposed Extension to Sets Theory Predmets (S) may never intersect with each other, primetas (P) intersect with each other and with subjects. S1 P1 S2 P2

15. Proposed Extension to Sets Theory In a system, if a predmet (S) has one intersection with primeta, it must be all covered by intersecting primetas (P). Thus, different subclasses of the predmet can be defined as follows: S1 – P1, P2, P3, P4; S1 – P1, P2, P3, P4; S1 – P1,P2, P3, P4; S1 – P1, P2, P3, P4; …; S1 – P1, P2, P3, P4, BUT NOT S1 – P1, P2, P3, P4 P3 S1 P1 P4 P2

16. Proposed Extension to Sets Theory The only type of subclass (i.e., open subclass) of predmets and primetas is intersection. Predmets, in addition, may be split into subsets called slices (Russian dol’ka). S1-1 S1-2 S1-3 S1 P

17. Proposed Extension to Sets Theory Event (Russian sobitie, co-being) is being of an intersection of predmet and primeta. S P Sobitie (event): S – P

18. Proposed Extension to Sets Theory Probitie is being of a predmet that has primeta of the same meaning. S PS Probitie

19. Proposed Extension to Sets Theory Soprobitie is an event in presence of another predmet that has the same meaning as the primeta involved in the event. S1 PS2 S2 Soprobitie

20. Proposed Extension to Sets Theory – Transitions (Processes) and Scenarios From event to event S – P1, P2 S – P1, P2 P1 AA P2 S The same with soprobitie

21. Proposed Extension to Sets Theory – Transitions (Processes) and Scenarios From event to event S1 – P S2 – P S1 S2 P

22. Proposed Extension to Sets Theory – Transitions (Processes) and Scenarios From event to event S1 P1 S1 – P1 S2 – P2 S2 P2 The same with soprobitie

23. Proposed Extension to Sets Theory – Transitions (Processes) and Scenarios From event to event P1 AA P3 P2 S S – P1, P2 , P3 {S – P1, P2 , P3 }, {S – P1, P2 , P3 } The same with soprobitie?

24. AND SO FORTH…

25. Link to Existing Methods Petri net

26. Link to Existing Methods Magma rises Magma rises to the surface Magma stops at depth Magma is not fragmented Magma is fragmented Fragmented magma erupts Lava erupts Tephra falls Fragments of magma erupt Gas emits Lava flows Lava piles up Pyroclastic flow forms UML activity diagram

27. Link to Existing Methods Customer Note sent to customer Order received Order processed Sales dept. employee Request received Offer formulated Offer sent to customer Offer rejected Offer closed Seller accepted rejected Offer considered Sales dept. manager Business process model and notation, or sequence diagram

28. Link to Existing Methods Atmosphere Wind blows Ash falls Ash drifts Volcano edifice Heavy rain falls Lahars wash slope out Ash plume rises Ash falls down No wind blows No eruption Volcano Eruption Magma rises Magma chamber Business process model and notation, or sequence diagram

29. Link to Existing Methods Bayesian Belief Network (Aspinall et al., 2003) Semantics of nodes not determined at all!

30. PRIMARY EXTERNAL EVENTS TERTIARY EVENTS SECONDARY EVENTS PRIMARY INTERNAL EVENTS Link to Existing Methods Basic syntax of the event bush (Pshenichny et al., 2009) Processes Products ENVIRONMENTAL INPUTS PROCESSES AND EVENTS END RESULTS PRIMARY INPUTS

31. Connectives of the event bush(Pshenichny and Kanzheleva, 2011) a) flux, b) influx, c) furcation, d) conflux

32. Connectives of the event bush(Pshenichny and Kanzheleva, 2011) Flux connective describes one event (Ei) producing another (Ej): Ei Flux Ej. Influx connective describes two events (Ei, Ej) producing another (Ek), but playing different roles: Ej, Ej Influx Ek. Furcation connective describes production of multiple events (Ei+1, Е i+2, …, Еn) by one (Ei): Ei Furcation Ei+1, Еi+2, …, Еn. Conflux connective describes production of one event (En) by multiple events (Ei, Ei+1, Еi+2, …, Еn-1): Ei, Ei+1, Еi+2, …, Еn-1 Conflux En.

33. Modi of Flux Connective 1. Change of subject from one secondary event to another with the same predicate: ii SiPk Flux Modus 1 ii SjPk 2. Change of subject from one secondary event to another with changing predicate without a semantic interrelationship between subjects and predicates: ii SiPk Flux Modus 2 ii SjPl 3. Change of subject from one secondary event to another with changing predicate and a semantic interrelationship between subjects and predicates: ii SiPj Flux Modus 3 ii SPjPSi 4. Change (negation vs. assertion) of predicate from one secondary event to another: ii Si~Pk Flux Modus 4 ii SiPk, ii SiPk Flux Modus 4 ii Si~Pk 5. Change of type of event: from primary internal to secondary, from secondary to tertiary, from primary internal to tertiary: ia SiPk Flux Modus 5 ii SiPk; ii SiPk Flux Modus 5 iii SiPk; ia SiPk Flux Modus 5 iii SiPk 6. Change of generality of subject – from general to particular: ia any SiPj Flux Modus 6 ii some SiPj; ii any SiPj Flux Modus 6 ii some SiPj.

34. Questions to Modus 1 of Flux Connective • Change of subject from one secondary event to another with the same predicate: ii SiPk Flux Modus 1 ii SjPk Does Si transform into Sj? Does Si produce (emit) Sj and continues to exist? Does Si just influence Sj and both exist regardless of each other?

35. Developing a bush: an example SEC stands for Etna’s South East Crater

36. Modeling and comparison of three eruptions by means of the event bush (for detail see Behncke and Pshenichny, 2009; Pshenichny et al., 2009) 16 Nov. 19 Nov. 24 Nov.

37. Link to Quantitative Models: Step One

38. Link to Quantitative Models: Step Two

39. Link to Quantitative Models:Step Three

40. Thank you very much! Sincerely yours, cpshenichny@yandex.ru