praktikum ki teil v: logik. praktikum ki sose 2005 Überblick organisation des praktikums...
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Praktikum KI
Teil V: Logik
Praktikum KI SoSe 2005
Überblick
Organisation des Praktikums Einführung in die Künstliche Intelligenz Suche nach einer „intelligenten“ Lösung Problemlösung mit Heuristiken Logik Planung & Robotik Expertensysteme Lernen
Praktikum KI SoSe 2005
Wumpus World PEAS description Performance measure
gold +1000, death -1000 -1 per step, -10 for using the arrow
Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square
Sensors: Stench, Breeze, Glitter, Bump, Scream Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot
Praktikum KI SoSe 2005
Wumpus World characterization Fully Observable No – only local perception Deterministic Yes – outcomes exactly specified Episodic No – sequential at the level of actions Static Yes – Wumpus and Pits do not move Discrete Yes Single-agent? Yes – Wumpus is essentially a natural feature
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Exploring a wumpus world
Praktikum KI SoSe 2005
Logik – Grundlagen Aussagenlogik
Aussagenlogik kennt nur Fakten (und die Verknüpfungen , , v, , ) B21 (breeze in Feld (2,1)) B41 B42 P31 B43 B34 P33 v P44
Implikationen () lassen sich umschreiben: (A B) ist äqivalent zu (A v B) Beweis: Wahrheitstabelle:
A B A B A v B
True True True True
True False False False
False True True True
False False True True
Praktikum KI SoSe 2005
Logik – Knowledge Base
Fakten werden in Knowledge Base (KB) gespeichert Einträge der KB sind mit UND verknüpft Aus bestehenden Einträgen können neue generiert
werden KB: {A, A B} Daraus generiert (“geschlussfolgert”) B Neue KB: {A, A B, B}
Oder: KB: {A, A B C } Daraus generiert (“geschlussfolgert”) C Neue KB: {A, A B C, C}
Praktikum KI SoSe 2005
Logik - Interferenz
An die Knowledge Base können Anfragen gestellt werden KB: {A, A B C } Mögliche Anfragen (α): A, C, C, D, …
Eine Anfrage ist true, wenn gilt (KB α) Zur positiven Beantwortung einer Anfrage zeige also,
dass für jede Belegung der Symbole mit true / false (=Modell), die für KB true ergibt, auch α true ist
Alternativ zeige, dass (KB α) false ist Die Suche nach einem solchen Modell wird als
Interferenz bezeichnet
Praktikum KI SoSe 2005
Logik - Inference by enumeration One sentence entails another (A B) if B is true in all worlds where A is true An Inference Method is
sound if it derives only sentences which are entailed in the KB complete if it derives all sentences which are entailed in the KB
Depth-first enumeration of all models is sound and complete
For n symbols, time complexity is O(2n), space complexity is O(n)
Praktikum KI SoSe 2005
Logik – Kojunktive Normalform Konjunktive Normalform (CNF): Konjunktion () von
Disjunktionen (v) von Literalen, z.B.(A B) (B C D)
Jede Aussage kann in CNF gebracht werden A B ist äquivalent (≡) zu (A B) (B A) (A B) ≡ (A v B) de Morgan ((AvB) ≡ A B, …) Distributivität von und v
Zur Interferenz werden stets zwei Disjunktionen herangezogen, die das gleiche Literal einmal mit und einmal ohne Negation () enthalten (A B) (B C D)
Praktikum KI SoSe 2005
Resolution Resolution inference rule (for CNF):
Let li and mj be complementary literals (i.e. li ≡ mj). Then it holds
l1 … lk, m1 … mn
l1 … li-1 li+1 … lk m1 … mj-1 mj+1 ... mn
E.g.,
P1,3 P2,2, P2,2
P1,3
Resolution is sound and complete for propositional logic
Praktikum KI SoSe 2005
Logik – Beweis durch Resolution
Führt die Interferenz durch Resolution auf die leere Aussage {}, so ist die ursprüngliche CNF falsch {} entsteht ausschließlich durch
A, A
{} Wenn A und A in Konjunktion () auftreten, so ist die Aussage
stets False
Um KB ╞ α zu zeigen, versuche daher (KB α) via Resolution zum Widerspruch, d.h. zu {}, zu bringen.
Praktikum KI SoSe 2005
Resolution algorithm Proof by contradiction, i.e., show KBα unsatisfiable
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Resolution example KB = (B1,1 (P1,2 P2,1)) B1,1
α = P1,2
Praktikum KI SoSe 2005
Logik – Horn Clauses
Horn Clauses sind spezielle Form der CNF Jede Disjunktion enthält genau ein positives Literal Bsp: (A B) (B C D) Es gilt: (A B C … N ) ≡ (B C … N A)
Damit kann die “Modus Ponens”-Regel angewandt werden:
A B C, A B C D
D
Bei einer KB aus Horn Clauses können Forward- und Backward-Chaining angewandt werden
Praktikum KI SoSe 2005
Forward chaining Idea: fire any rule whose premises are satisfied in the KB,
add its conclusion to the KB, until query is found
Praktikum KI SoSe 2005
Forward chaining example
Praktikum KI SoSe 2005
Forward chaining example
Praktikum KI SoSe 2005
Forward chaining example
Praktikum KI SoSe 2005
Forward chaining example
Praktikum KI SoSe 2005
Forward chaining example
Praktikum KI SoSe 2005
Forward chaining example
Praktikum KI SoSe 2005
Forward chaining example
Praktikum KI SoSe 2005
Backward chainingIdea: work backwards from the query q:
to prove q by BC,check if q is known already, orprove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal stack
Avoid repeated work: check if new subgoal1. has already been proved true, or2. has already failed
3.
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Backward chaining example
Praktikum KI SoSe 2005
Forward vs. backward chaining FC is data-driven, automatic, unconscious processing,
e.g., object recognition, routine decisions
May do lots of work that is irrelevant to the goal
BC is goal-driven, appropriate for problem-solving, e.g., Where are my keys? How do I get into a PhD program?
Complexity of BC can be much less than linear in size of KB
Praktikum KI SoSe 2005
Efficient propositional inferenceTwo families of efficient algorithms for propositional inference:
Complete backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms
WalkSAT algorithm
Praktikum KI SoSe 2005
The DPLL algorithmDetermine if an input propositional logic sentence (in CNF) is satisfiable.
Improvements over truth table enumeration:1. Early termination
A clause is true if any literal is true.A sentence is false if any clause is false.
2. Pure symbol heuristicPure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), (B C), (C A), A and B are pure, C is impure. Make a pure symbol literal true.
3. Unit clause heuristicUnit clause: only one literal in the clauseThe only literal in a unit clause must be true.
Praktikum KI SoSe 2005
The DPLL algorithm
Praktikum KI SoSe 2005
The WalkSAT algorithm Incomplete, local search algorithm Evaluation function: The min-conflict heuristic of minimizing the number of
unsatisfied clauses Balance between greediness and randomness
Praktikum KI SoSe 2005
The WalkSAT algorithm
Praktikum KI SoSe 2005
Inference-based agents in the wumpus world
A wumpus-world agent using propositional logic:
P1,1
W1,1
Bx,y (Px,y+1 Px,y-1 Px+1,y Px-1,y)
Sx,y (Wx,y+1 Wx,y-1 Wx+1,y Wx-1,y)
W1,1 W1,2 … W4,4
W1,1 W1,2
W1,1 W1,3 …
64 distinct proposition symbols, 155 sentences
Praktikum KI SoSe 2005
Inference-based agents in the wumpus world
Praktikum KI SoSe 2005
KB contains "physics" sentences for every single square
For every time t and every location [x,y],
Lx,y FacingRightt Forwardt Lx+1,y
Rapid proliferation of clauses
Expressiveness limitation of propositional logic
t+1t
Praktikum KI SoSe 2005
Summary Logical agents apply inference to a knowledge base to derive new information and
make decisions Basic concepts of logic:
syntax: formal structure of sentences semantics: truth of sentences wrt models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences
Wumpus world requires the ability to represent partial and negated information, reason by cases, etc.
Resolution is complete for propositional logicForward, backward chaining are linear-time, complete for Horn clauses
Propositional logic lacks expressive power