Knowledge-based Systems for Industrial Applications - Propositional Logic

Transcription

1 Knowledge-based Systems for Industrial Applications - Propositional Logic P. Struss WS 16/17 WS 16/17 KBSIA 2A - 1

2 Logic 2A Logic und Knowledge Representation 2A.1 Propositional logic WS 16/17 KBSIA 2A - 2

3 Example: Problem with the Car ( [Reinfrank 91]) The engine of my car doesn t start What happens if you turn on the light? The light is on What do you hear if you try to start the engine? It sounds normal, but the engine doesn t start Then you probably forgot to buy gas Oh... (blushes) How did you know that? If the starter works fine, the battery is ok and there is enough gas, the engine would start. The noise shows, that the starter works fine and the battery has power. Hence, gas is missing. Can we use logic, to represent such knowledge and to draw proper conclusions? WS 16/17 KBSIA 2A - 3

4 Situations - Structures Specific situation described by aspects (features, objects properties, ) e.g. car problem: engine battery gasoline starter light sound starts OK OK on normal Situation: represented by a structure (vector): Presence or absence of properties: e.g. a car situation engine battery gasoline starter light sound starts OK OK on normal (F, F, T, T, F, F) WS 16/17 KBSIA 2A - 4

5 Situation Description Information about a specific situation: Constrains the set of vectors e.g. the problem situation engine battery gasoline starter light sound starts OK OK on normal (F, *, *, *, T, T) (* means no constraints) Not all possible vectors characterize real situations e.g. all situations engine battery gasoline starter light sound starts OK OK on normal (T, *, F, *, *, *) are impossible WS 16/17 KBSIA 2A - 5

6 Propositional Logic Atomic Formulas Knowledge: Constraint on the vector space Enumeration of all possible vectors impossible or unnecessary Description of sets of vectors with formulas of a symbolic language Symbolic names of elementary aspects, e.g. EngineStarts, BatteryOK Atomic Formulas e.g., the formula SoundNormal means: The starter sounds normal WS 16/17 KBSIA 2A - 6

7 Propositional Logic - Composite Formulas Composite formulas: consist of atomic formulas and/or composite formulas with logical relations (connectors) For instance: conjunction (AND) BatteryOK LightOn ( The battery is OK and the light is on ) For instance: Negation(NOT) ( BatteryOK LightOn) ( The battery is empty, and the light is on ) WS 16/17 KBSIA 2A - 7

8 Truth Value For each formula: true or false for a certain structure. Atomic formulas, p: Truth value directly given in structure S e.g. Val(LightON, S) = S (LightON) Composite formulas: Value (recursively) determined from the values of the parts dependent on the connectors e.g. Val( (BatteryOK LightOn), S) = T if Val(BatteryOK LightOn), S) = F = F if Val(BatteryOK LightOn), S) = T and Val (BatteryOK LightOn), S) = T if Val(BatteryOK, S) = T and Val(LightOn, S) = T = F else WS 16/17 KBSIA 2A - 8

9 Truth Tables Truth Values of different relations: p q q p q p q p q F F T F F T F T F F T T T F F T F T T T T T WS 16/17 KBSIA 2A - 9

10 Models Model of a formula p: A structure, S, for which p is true For instance, every structure engine battery gasoline starter light sound starts OK OK on normal (*, F, *, *, T, *) is a model of the formula LightOn but no model of the formula ( BatteryOK LightOn) A structure, S, is a model of a set of formulas, if it is a model of every formula in the set. Tautology: formula, for which every structure is a model. Inconsistent: formula (or set of formulas) that has no model. WS 16/17 KBSIA 2A - 10

11 Models (Cont d) Models of a set of formulas: Intersection of the formulas model sets. For instance, a structure engine battery gasoline starter light sound starts OK OK on normal (*, T, *, *, T, *) is a model of the set of formulas {LightOn, ( BatteryOK LightOn)} Basic Idea: Represent general knowledge about a domain and information about a particular situation by means of formulas and determine the models of this set of formulas. WS 16/17 KBSIA 2A - 11

12 Back to the Car Example Formulas and Models Information about the situation: ( 1 ) SoundNormal engine battery gasoline starter light sound starts OK OK on normal (*, *, *, *, *, T) ( 2 ) MotorStarts (F, *, *, *, *, *) Knowledge about the domain : (K 1 ) SoundNormal ( StarterOK BatteryOK ) (*, T, *, T, *, T) (*, *, *, *, *, F) (K 2 ) (StarterOK BatteryOK Gasoline ) MotorStarts (T, T, T, T, *, *) (*, *, *, F, *, *) (*, *, F, *, *, *) (*, F, *, *, *, *) Intersection (F, T, F, T, *, T) that means: no gas! WS 16/17 KBSIA 2A - 12

13 Entailment A formula, p, which is true for all models of a set of formulas, {p 1,..., p k }, is entailed by {p 1,..., p k } : {p 1,..., p k } p Car example: gasoline is entailed by general knowledge and information about the situation: { 1, 2, K 1, K 2 } gasoline Task: Replace ( 1 ) and (K 1 ) with ( 1) LightOn (K 1) ( Batterie OK LightOn) Then { 1, 2, K 1, K 2 } ( Gasoline StarterOK) WS 16/17 KBSIA 2A - 13

14 Another Car Example Information about the situation: ( 1) LightOn engine battery gasoline starter light sound starts OK OK on normal (*, *, *, *, T, *) ( 2 ) MotorStarts (F, *, *, *, *, *) Knowledge about the domain : (K 1) ( BatteryOK LightOn) (*, T, *, *, *, *) (*, *, *, *, F, *) (K 2 ) (StarterOK BatteryOK Gasoline) MotorStarts (T, T, T, T, *, *) (*, *, *, F, *, *) (*, *, F, *, *, *) (*, F, *, *, *, *) Intersection: Diagnosis? WS 16/17 KBSIA 2A - 14

15 Inference Rules Determine all models to find the logical implications? impossible Basic Idea: transform formula sets directly and prove the logical consequences symbolically. Inference rules generate formulas from sets of formulas WS 16/17 KBSIA 2A - 15

16 Inference Rules - Examples Modus Ponens p, p q / q If p holds, and p q holds, then also q holds Because: p q p q F F T F T T T F F T T T Other inference rules: q, p q / p p, (p q) / q p, p q / q WS 16/17 KBSIA 2A - 16

17 Symbolic Proofs A proof of a formula, q, from a set of formulas, {p 1,..., p k }, is a sequence of execution of n inference rules q m1, q m2,..., q mj / q m, 1 m n, where for each q mi holds q mi {p 1,..., p k } or q mi = q j for j m 1 is valid and q m = q If such a proof exists, q is provable from {p 1,..., p k }: p 1,..., p k q WS 16/17 KBSIA 2A - 17

18 A Proof for Gasoline ( 1) SoundNormal (K1) SoundNormal ( StarterOK BatteryOK) p, p q / q (K2) (StarterOK BatteryOK Gasoline) MotorStarts ( 2) MotorStarts q, p q / p (StarterOK BatteryOK) (StarterOK BatteryOK Gasoline ) p, (p q) / q Gasoline WS 16/17 KBSIA 2A - 18

19 Symbolic Proofs - Properties Interesting are sets of inference rules, that are correct, i.e. if {p 1,..., p k } q, then {p 1,..., p k } q or even (correct and) complete, i.e. if {p 1,..., p k } q, then {p 1,..., p k } q WS 16/17 KBSIA 2A - 19

20 Another Example: Problem with the Car The engine of my car doesn t start What happens if you turn on the light? The light is on What do you hear if you try to start the engine? It sounds normal, but the engine doesn t start Then you probably forgot to buy gas My car is an electric car Oh (Blushes) Why does the reasoning fail this time? WS 16/17 KBSIA 2A - 20

21 Proof for Gasoline: Based on Assumptions about the Car ( 1) SoundNormal (K1) SoundNormal ( StarterOK BatteryOK) p, p q / q (K2) (StarterOK BatteryOK Gasoline) MotorStarts ( 2) MotorStarts q, p q / p (StarterOK BatteryOK) (StarterOK BatteryOK Gasoline ) p, (p q) / q Gasoline WS 16/17 KBSIA 2A - 21

22 A Different Approach to Diagnosis Required Rule base not re-usable Applicability implicit Needed: Explicit representation of the structure of the car Organize (diagnostic) knowledge around (types of) components Model-based diagnosis WS 16/17 KBSIA 2A - 22