Logical Agents. Propositional Logic [Ch 6] Syntax, Semantics, Entailment, Derivation. Predicate Calculus Representation [Ch 7]

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1 Logical Agents Reasoning [Ch 6] Propositional Logic [Ch 6] Syntax Semantics Entailment Derivation Predicate Calculus Representation [Ch 7] Syntax Semantics Expressiveness... Situation Calculus Predicate Calculus Inference [Ch 9] Resolution Implemented Systems [Ch 10] Applications [Ch 8] Planning [Ch 11] Predicate-Calculus 1

2 Predicate-Calculus 2

3 Types of Logic Logics are characterized by what they commit to as primitives Ontological commitment: What exists: facts? objects? time? beliefs? Epistemological commitment: What states of knowledge? Language Ontological Epistemological Commitment Commitment Propositional logic facts true/false/unknown First-order logic facts objects true/false/unknown relations Temporal logic facts objects true/false/unknown relations times Probability theory facts degree of belief [0 1] Fuzzy logic degree of truth degree of truth [0 1] Predicate-Calculus 3

4 Prop. Logic vs Predicate Calculus Propositional Logic World consists of propositions... either true in world or not. W 12 W 11 W 12 W 44 (W 11 W 12 ) (W 11 W 13 ) (W 34 W 44 ) Note: W 12 (syntactically) unrelated to W 13 Predicate Calculus World consists of Objects Predicates (properties relations functions) (which participate in propositions that could be true in world) At(Wumpus [ 1 2 ]) l At( Wumpus l ) l 1 l 2 At(Wumpus l 1 ) At(Wumpus l 2 ) l 1 = l2 Predicate calculus is MORE POWERFUL than Propositional Logic Predicate-Calculus 4

5 Parts of Predicate Calculus Objects: things w/ individual identity (people houses numbers theories colors wars Ronald McDonald... ) Predicates: distinguish objects from one another properties (of single object) (red round bogus prime... ) relations amoung sets of objects (brother-of part-of has-color owns... ) functions: rel n w/ single value for each input (father-of one-more-than... ) Each predicate SET of object-tuples: Red = { rose#1 blood#7 ruby# } Brother = { Russ Miles Tom Mark... } Successor = { } NEUTRAL: can describe categories time... but nothing is built in... Predicate-Calculus 5

6 Predicate Calculus: Syntax Atomic Propositions basic statements about world At(Wumpus [ 3 4 ]) Adjacent([ 3 4 ] l 2 ) Smelly( [ 3 5 ] )... built from... Predicate Symbols: At Adjacent =... Constant Symbols: Agent Wumpus Pit S 1 [ 3 5 ]... Variable Symbols: s a l... Function Symbols: Turn Result Next... Well-formed atomic proposition is P ( t 1... t n ) where P is Predicate of arity n & Each t i is a term constant variable functional term: F ( t 1... t m ) if F is a Function of arity m; and each t i is a term. (representing object) Predicate-Calculus 6

7 Examples of Atomic Propositions Smelly( [ 3 5 ] ): Smelly has arity 1; [ 3 5 ] is term as it is constant At( Wumpus l 2 ): At has arity 2; Wumpus is term as it is constant l 2 is term as it is variable AgentAt( l 2 Next(S 0 ) ): AgentAt has arity 2; l 2 is term as it is variable Next(S 0 ) is term as Next is function of 1 arg and S 0 is term as constant Predicate-Calculus 7

8 Logical Connectives Build sentences from atomic prop s using: connectives: quantifiers: and or not implies equivalence if... then (biconditional) for all exists Examples Adjacent( [ 1 2 ] [ 1 3 ] ) At( Wumpus [ 1 3 ] ) At( Wumpus [ 2 2 ] ) l 1 l 2 At(Wumpus l 1 ) Adjacent(l 1 l 2 ) Smelly(l 2 ) l 1 l 2 Pit(l 1 ) Adjacent(l 1 l 2 ) Breezy(l 2 ) See p. 187 R&N. Predicate-Calculus 8

9 Predicate Calculus: Semantics Proposition Logic: Model m f based on f : Var {T F } { } A B C D f = m f = A m f = B C... Predicate Calculus Models... based on Extensions of Objects Relations Functions f : Symbol Extension f( RG ) = f( P rof ) = RG f( JS ) = RG JS JS DL TM f( P erson ) = RG JvR... Predicate-Calculus 9

10 f( Red ) = { Block1 FireEngine... } FireEngine JS f( T aller ) = RG JS FireEngine RG RG Block1...

11 Using Extension Q: Given extension f( ) is m f =? P rof(rg)? A: True as f( RG ) f( P rof ) RG RG JS DL TM... YES! m f =? P rof(block1)? False as f( Block1 ) f( P rof ) Block1 RG JS DL TM... Predicate-Calculus 10

12 Extension #2 m f = T aller(rg JS): f( RG ) f( JS ) f( T aller ) RG JS FireEngine RG JS FireEngine RG JS RG Block1... In general: m f = A(B C) iff f(b) f(c) f(a) Similar issues for 0-ary relations (aka constants ) n-ary relations functions Predicate-Calculus 11

13 Meaning of Quantifiers x Q(x) Q(v 1 ) Q(v 2 ) Q(v n ) over all objects v i ( number of them) Eg: All cats are mammals x Cat(x) Mammal(x) Cat( Spot ) Mammal( Spot ) Cat(Rebeca) Mammal(Rebeca) Cat( Felix ) Mammal( Felix ) Cat(Richard) Mammal(Richard) Cat( John ) Mammal( John ) x Q(x) Q(v 1 ) Q(v 2 ) Q(v n ) over all objects v i Eg: Spot has a sister who is a cat x Sister(x Spot) Cat(x) Sister( Spot Spot) Cat( Spot ) Sister(Rebeca Spot) Cat(Rebeca) Sister( Felix Spot) Cat( Felix ) Sister(Richard Spot) Cat(Richard) Sister( John Spot) Cat( John ) Predicate-Calculus 12

14 Notes on Quantifiers Common Mistakes: x Cat(x) Mammal(x) means everything is BOTH a cat and a mammal! x Sister(x Spot) Cat(x) means either something is not a sister of Spot or something is cat! Nesting: x y Parent(x y) Child(y x) y xparent(x y) Child(y x) Similarly x y... y x... But... x y Loves(x y) y x Loves(x y) (Everyone loves his mother... vs... GoodAngel loved by all) Duality: Everyone dislikes Parsnips there is no one who likes Parsnips x Likes(x Parsnips) xlikes(x Parsnips) De Morgan Rules: x P x P P Q (P Q) x P x P (P Q) P Q x P x P P Q ( P Q) x P x P ( P Q) P Q Predicate-Calculus 13

15 Sufficient Representation Kinship Domain Definition: A mother is a female parent m c Mother(m c) Female(m) Parent(m c) Disjointness: Males and females are disjoint x Male(x) Female(x) Inverse: Parent and child are inverse relations p c Parent(p c) Child(c p) Intervening Entity: A grandparent is a parent of one s (intervening) parent g c GrandParent(g c) p Parent(g p) Parent(p c) Exclusive: A sibling is another child of one s parents... x y Sibling(x y) x y p Parent(p x) Parent(p y) + Sets lists arithmetic... R/N p.197ff... Situations... Predicate-Calculus 14

16 Implementation of Agent Tell stores facts Tell(KB l 1 l 2 At(Wumpus l 1 ) Adjacent(l 1 l 2 ) Smelly(l 2 ) ) Tell(KB Adjacent([ 3 4 ] [ 2 4 ]) ) Tell(KB Smelly([ 3 1 ]) )... Ask answers queries (using inference rules) Ask(KB Adj([ 3 4 ] [ 2 4 ])) Yes Ask(KB Adj([ 3 4 ] l) ) Yes[l/[ 2 4 ]] Ask(KB Action(x 5) ) Yes[x/Grab] NOTE: Not just Yes but also a value required How does it get answers...? Predicate-Calculus 15

17 Substitution Given sentence S and substitution σ Sσ denotes result of plugging σ into S Eg S = Smarter(x y) σ = {x/hillary y/bill} Sσ = Smarter(HillaryBill) Ask(KB S) returns all σ s such that KB = Sσ More about substitutions... LATER! Predicate-Calculus 16

18 Formalization C1 1 2 X1 X2 1 3 A2 A1 O1 2 one-bit full adder : Inputs: two inputs and a carry Output: one output and carry Four gates types: AND OR XOR NOT Goal#1: see if design matches specification Consider: circuits terminals signals. Keep task in mind Eg for fault diagnosis: if wires can be broken may want to specify wires (eg Wire(xy)) Predicate-Calculus 17

19 Vocabulary Constants: X 1 X 2... XOR AND... On Off (signal values) Functions: Type(X 1 ) = XOR Q. Advantage of function vs Type( X 1 XOR)? Out(1 X 1 ) In(1 X 1 ) In(2 X 1 ) In(3 X 1 ) terms representing terminals Signal(x) signal value fn (eg Signal( In(1 X 1 ) )) Relations: Connected( Out(1 X 1 ) In(1 X 2 )) for connectivity: Note: don t have to name terminals explicity! Semantics of function will assign some unique object to it. Predicate-Calculus 18

20 General Rules Tell agent... How signals behave: R1 : t 1 t 2 Connected(t 1 t 2 ) Signal(t 1 ) = Signal(t 2 ) R2a : t Signal(t) = On Signal(t) = Off R2b : On Off R3 : t 1 t 2 Connected(t 1 t 2 ) Connected(t 2 t 1 ) How gates behave: R4 : g Type(g) = OR Signal(Out(1 g)) = On n Signal(In(n g)) = On R5 : g Type(g) = AND Signal(Out(1 g)) = On n Signal(In(n g)) = On R6 : g Type(g) = XOR Signal(Out(1 g)) = On (Signal(In(1 g)) Signal(In(2 g))) R7 : g Type(g) = NOT Signal(Out(1 g)) = On (Signal(In(1 g)) = Off Predicate-Calculus 19

21 Current Situation General Rules are Few (7) good ontology Clear good vocabulary Now what? Describe SPECIFIC circuit Ask questions about this specific circuit Predicate-Calculus 20

22 Describing Specific Circuit C1 1 2 X1 X2 1 3 A2 A1 O1 2 Tell agent: Types of gates: Type(X 1 ) = XOR Type(X 2 ) = XOR Type(A 1 ) = AND... Conectivity Connected( Out(1X 1 ) In(1X 2 ) ) Connected( In(1C 1 ) In(1X 1 ) ) Connected( Out(1X 1 ) In(1A 2 ) ) Connected( In(1C 1 ) In(1A 1 ) )... Predicate-Calculus 21

23 Queries Our KB captures full behavior. Can Ask different queries about behavior Q: i 1 i 2 i 3 Signal(In(1C 1 )) = i 1 Signal(In(2C 1 )) = i 2 Signal(In(3C 1 )) = i 3 Signal(Out(1C 1 )) = Off Signal(Out(2C 1 )) = On? A: (i 1 = On i 2 = On i 3 = Off) (i 1 = On i 2 = Off i 3 = On) (i 1 = Off i 2 = On i 3 = On) Q: What is the advantage over direct simulation? A: Allows agent to reason about overall behavior. Eg What inputs give a particular output? Predicate-Calculus 22

24 Pro/Con wrt Formalizing + Used in analysis of circuits/systems Contrast with Truth table method + Formalization is somewhat facilitated by closeness between logical formalisms and digital circuitry... allows for very powerful design methods (but did not prevent Pentium bug... ) Defining natural kinds: game or chair... difficulty w/ necessary and sufficient conditions. [R&N p.232] Problem with strict definition [Quine 1953] the Pope is a bachelor Predicate-Calculus 23

25 Summary First-order logic: objects relations are semantic primitives syntax: constants functions predicates equality quantifiers Increased expressive power: sufficient to define Wumpus World Situation calculus: conventions for describing actions and change in FOL can formulate planning as inference wrt a situation calculus KB Predicate-Calculus 24

26 Useful Equivalencies [Needs only & ] P P P Q [( P ) ( Q)] [P Q] ( P ) ( Q) P Q ( P ) Q (P Q) P Q [(P Q) (Q P )] (P Q) (Q P ) x. φ(x) [ x. φ(x)] [ x. φ(x)] x. φ(x) ϕ τ τ ϕ!x. φ(x) x. [φ(x) z. φ(z) z = x]... Exactly n values of ϕ... Predicate-Calculus 25

27 Example of Using U = Set of natural numbers N n. 6 n 2 n n. 6 n 2 n A = { n : 6 n } = { } B = { n : 2 n } = { } Notice A B = N (Hence each n N n satisfies either 6 n or 2 n) So n 6 n 2 n Predicate-Calculus 26

RN, Chapter 8 Predicate Calculus 1

RN, Chapter 8 Predicate Calculus 1 Predicate Calculus 1 RN, Chapter 8 Logical Agents Reasoning [Ch 6] Propositional Logic [Ch 7] Predicate Calculus Representation [Ch 8] Syntax, Semantics, Expressiveness Example: Circuits Inference [Ch