First-order logic. AI Slides (5e) c Lin

Transcription

1 First-order logic 6 AI Slides (5e) c Lin Zuoquan@PKU

2 6 First-Order Logic 2.1 First-order logic Syntax Semantics 2.2 Representation in FOL 2.3 Agents in first-order case AI Slides (5e) c Lin Zuoquan@PKU

3 Why FOL Pros and cons of propositional logic (PL) PL is declarative: pieces of syntax correspond to facts PL allows partial/disjunctive/negated information (unlike most data structures and databases) PL is compositional: meaning of B 1,1 P 1,2 is derived from meaning of B 1,1 and of P 1,2 Meaning in PL is context-independent (unlike natural language, where meaning depends on context) PL has very limited expressive power (unlike natural language) E.g., cannot say pits cause breezes in adjacent squares except by writing one sentence for each square AI Slides (5e) c Lin Zuoquan@PKU

4 FOL theses Mathematics: Hilbert s Thesis There is no logic beyond first-order logic that when one is forced to make all one s mathematical (extralogical) assumption explicit, these axioms can always be expressed in FOL, and that the informal notion of provable used in mathematics is made precise by the formal notion of provable in FOL AI: McCarthy s Thesis There is no declarative knowledge representation beyond firstorder logic FOL is very powerful can be used as a full programming language there is no way to detect in general when a program is looping AI Slides (5e) c Lin Zuoquan@PKU

5 First order logic Whereas PL assumes world contains facts, FOL (like natural language) assumes the world contains Objects: people, houses, numbers, theories, Ronald McDonald, colors, baseball games, wars, centuries... Relations: red, round, bogus, prime, multistoried..., brother of, bigger than, inside, part of, has color, occurred after, owns, comes between,... Functions: father of, best friend, third inning of, one more than, end of... AI Slides (5e) c Lin Zuoquan@PKU

6 Syntax Let L be a first-order language Vocabulary: Constants kingjohn, 2, pku,... Predicates Brother, >,... Functions sqrt, lef tlegof,... Variables x, y, a, b,... Connectives Equality = Quantifiers AI Slides (5e) c Lin Zuoquan@PKU

7 Syntax arity: number of arguments arity 0 predicates: propositional symbols arity 0 functions: constant symbols The predicates and functions are non-logical symbols predicate: mixed case capitalized, e.g., OlderThan functions: mixed case uncapitalized, e.g., brotherof Sometimes no distinction if no confusion Notation: occasionally add or omit (,) use [,] and {,} also AI Slides (5e) c Lin Zuoquan@PKU

8 Atomic sentences Atomic sentence (formula) = predicate(term 1,...,term n ) or term 1 = term 2 Term = function(term 1,...,term n ) or constant or variable E.g., Brother(kingJohn, richardt helionheart) > (Length(lef tlegof(richard)), Length(lef tlegof(kingjohn))) AI Slides (5e) c Lin Zuoquan@PKU

9 Complex sentences Complex sentences(well-formed formulas, wffs) are made from atomic sentences using connectives 1. Every atomic sentence is a wff 2. If S 1 and S 1 are wffs, and x is a variable, then S, S 1 S 2, S 1 S 2, S 1 S 2, S 1 S 2, xs 1 (x), xs 1 (x) are wffs E.g. Sibling(kingJohn, richard) Sibling(richard, kingjohn) >(1,2) (1,2) >(1,2) >(1,2) PL as FOL subset: no terms, no quantifiers AI Slides (5e) c Lin Zuoquan@PKU

10 variables sentence Everyone at Beida is smart: x At(x, beida) Smart(x) Universal quantification x P is true in a model m iff P is true with x being each possible object in the model Roughly speaking, equivalent to the conjunction of instantiations of P (At(kingJohn,beida) Smart(kingJohn)) (At(richard,beida) Smart(richard)) (At(lin,Beida) Smart(lin))... AI Slides (5e) c Lin Zuoquan@PKU

11 variables sentence Someone at Qinghua is smart: x At(x,qinghua) Smart(x) Existential quantification x P is true in a model m iff P is true with x being some possible object in the model Roughly speaking, equivalent to the disjunction of instantiations of P (At(kingJohn, Qinghua) Smart(kingJohn)) (At(richard, Qinghua) Smart(richard)) (At(wang, qinghua) Smart(wang))... AI Slides (5e) c Lin Zuoquan@PKU

12 Mistakes to avoid Typically, is the main connective with Mistake: using as the main connective with : x At(x,beida) Smart(x) means Everyone is at Beida and everyone is smart Typically, is the main connective with Mistake: using as the main connective with : x At(x, qinghua) Smart(x) is true if there is anyone who is not at Qinghua AI Slides (5e) c Lin Zuoquan@PKU

13 x y is the same as y x x y is the same as y x Properties of quantifiers x y is not the same as y x x y Loves(x,y) There is a person who loves everyone in the world y x Loves(x,y) Everyone in the world is loved by at least one person Quantifier duality: each can be expressed using the other x Likes(x, icecream) x Likes(x, broccoli) x Likes(x, icecream) x Likes(x, broccoli) AI Slides (5e) c Lin Zuoquan@PKU

14 Variable scope The variables have a scope determined by the quantifiers P(x) xp(x) Q(x) free bound occurrences of variables - sentences: wffs with no free variables (i.e., closed wffs) - usually, free variables assumed to be universally quantified -usedot. forthescope, e.g., x.p(x) Q(x)for x(p(x) Q(x)) Substitution: α[x/t] means α with all free occurrences of the x replaced by term t also, α[t 1,...,t n ] means α[x 1 /t 1,,x n /t n ], or simple α[x/t] x = (x 1,,x n ) stands for a tuple of variables (also terms) AI Slides (5e) c Lin Zuoquan@PKU

15 Semantics Consider how to interpret sentences what do sentences claim about the world? or, what does believing one amount to? Without meaning, sentences cannot be used to represent knowledge Comparing with PL, cannot fully specify interpretation of sentences because non-logical symbols reach outside the language Logical interpretation specification of how to understand predicate and function symbols Problem: cannot realistically expect to specify once and for all what a sentence means the non-logical symbols are used in an application dependent way E.g., Happy(lin), who slin, evenifweweretoagreeonwhat Happy means AI Slides (5e) c Lin Zuoquan@PKU

16 Semantics Abstract structure to specify interpretation 1. There are objects (in the world) 2. ForanypredicateP (ofarity1), someoftheobjectswill satisfy P and some will not each interpretation settles extension of P aach interpretation assigns to function f a mapping from objects to objects functions always well-defined and single-valued 3. No other aspects of the world matter The FOL assumption this is all you need to know about the non-logical symbols to understand which sentences of FOL are true or false AI Slides (5e) c Lin Zuoquan@PKU

17 Models Sentences are true w.r.t. a model and an interpretation Model contains 1 objects (domain elements) and relations among them Interpretation specifies referents for constant symbols objects predicate symbols relations function symbols functional relations An atomic sentence predicate(term 1,...,term n ) is true iff the objects referred to by term 1,...,term n are in the relation referred to by predicate AI Slides (5e) c Lin Zuoquan@PKU

18 Models: Example crown person brother brother on head person king R $ J left leg left leg AI Slides (5e) c Lin Zuoquan@PKU

19 Models: Truth Consider the interpretation in which richard Richard the Lionheart john the evil King John Brother the brotherhood relation Under this interpretation, Brother(richard, john) is true just in case Richard the Lionheart and the evil King John are in the brotherhood relation in the model AI Slides (5e) c Lin Zuoquan@PKU

20 Models: Entailment Entailment in PL can be computed by enumerating models We can enumerate the FOL models for a given KB vocabulary: For each number of domain elements n from 1 to For each k-ary predicate P k in the vocabulary For each possible k-ary relation on n objects For each constant symbol C in the vocabulary For each choice of referent for C from n objects... Computing entailment by enumerating FOL models is not easy AI Slides (5e) c Lin Zuoquan@PKU

21 Interpretation Interpretation I: the domain I 1.If σ is an object constant, then I(σ) I 2.If π is an n-ary function constant, then I(π) : I n I 3.If ρ is an n-ary relation constant, then I(ρ) I n AI Slides (5e) c Lin Zuoquan@PKU

22 Assignment Variable assignment U: a function from the variables of L to objects of I Term assignment T IU : given I and U 1.If τ is an object constant, then T IU (τ) = I(τ) 2.If τ is a variable, then T IU (τ) = U(τ) 3.If τ is a term of the form π(τ 1,,τ n ) and I(π) = g and T IU (τ i ) = x i, then T IU (τ) = g(x 1,,x n ) AI Slides (5e) c Lin Zuoquan@PKU

23 Satisfaction Satisfaction = I φ[u] (simply =): a sentence φ is satisfied by an interpretation I and a variable assignment U 1. = (σ = τ) iff T IU (σ) = T IU (τ) 2. = ρ(τ 1,,τ n ) iff < T IU (τ 1 ),,T IU (τ n ) > I(ρ) 3. = φ iff = φ 4. = φ ψ iff = φ and = ψ 5. = φ ψ iff = φ or = ψ 6. = φ ψ iff = φ or = ψ 7. = xφ(x) iff for all d I it is the case that = φ[v], where V(x) = d and V(y) = U(y) for x y 8. = xφ(x) iff for some d I it is the case that = φ[v], where V(x) = d and V(y) = U(y) for x y AI Slides (5e) c Lin Zuoquan@PKU

24 Models and entailment Model I: If an interpretation I satisfies a sentence φ for all variable assignments, then I is said to be a model of φ, written = I φ or I = φ Similarly (in PL), a sentence is valid if it is true in all models E.g., A A, A A, (A (A B)) B A sentence is unsatisfiable if it is true in no models E.g., A A Entailment =: Let KB be a set of sentences and φ a sentence, KB = φ iff φ is true in all models of KB AI Slides (5e) c Lin Zuoquan@PKU

25 Brothers are siblings Representation in FOL AI Slides (5e) c Lin Zuoquan@PKU

26 Representation in sentences Brothers are siblings x, y Brother(x, y) Sibling(x, y). Sibling is symmetric AI Slides (5e) c Lin Zuoquan@PKU

27 Representation in sentences Brothers are siblings x, y Brother(x, y) Sibling(x, y). Sibling is symmetric x,y Sibling(x,y) Sibling(y,x). One s mother is one s female parent AI Slides (5e) c Lin Zuoquan@PKU

28 Representation in sentences Brothers are siblings x, y Brother(x, y) Sibling(x, y). Sibling is symmetric x,y Sibling(x,y) Sibling(y,x). One s mother is one s female parent x,y Mother(x,y) (Female(x) Parent(x,y)). A first cousin is a child of a parent s sibling AI Slides (5e) c Lin Zuoquan@PKU

29 Brothers are siblings Representation in sentences x, y Brother(x, y) Sibling(x, y). Sibling is symmetric x,y Sibling(x,y) Sibling(y,x). One s mother is one s female parent x,y Mother(x,y) (Female(x) Parent(x,y)). A first cousin is a child of a parent s sibling x,y FirstCousin(x,y) p,ps Parent(p,x) Sibling(ps,p) P arent(ps, y) AI Slides (5e) c Lin Zuoquan@PKU

30 Equality term 1 = term 2 is true under a given interpretation if and only if term 1 and term 2 refer to the same object E.g., 1 = 2 and x (Sqrt(x),Sqrt(x)) = x are satisfiable 2 = 2 is valid E.g., definition of (full) Sibling in terms of Parent: x,y Sibling(x,y) [ (x=y) m,f (m=f) P arent(m, x) P arent(f, x) P arent(m, y) P arent(f, y)] AI Slides (5e) c Lin Zuoquan@PKU

31 Knowledge engineering KR is first and foremost about knowledge meaning and entailment find individuals and properties, then encode facts sufficient for entailments Before implementing, need to understand clearly what is to be computed? why and where inference is necessary? Task: KB with appropriate entailments what vocabulary? what facts to represent? AI Slides (5e) c Lin Zuoquan@PKU

32 Knowledge base KB is set of sentences explicit statement of sentences believed (including any assumed connections among non-logical symbols) KB = φ φ is a further consequence of what is believed explicit knowledge: KB implicit knowledge: {φ KB = φ} AI Slides (5e) c Lin Zuoquan@PKU

33 Knowledge-based systems Building (larger) KB to represent what is explicitly known e.g. what the system has been told or has learned Want to influence behaviour based on what is implicit in the KB Requires reasoning deductive inference process of calculating entailments of KB i.e., KB = φ AI Slides (5e) c Lin Zuoquan@PKU

34 Agents in first-order case Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t = 5: T ell(kb, P ercept([smell, Breeze, N one], 5)) Ask(KB, a Action(a, 5)) I.e., does KB entail any particular actions at t = 5? Answer: Y es, {a/shoot} substitution (binding list) Given a sentence S and a substitution σ, Sσ denotes the result of plugging σ into S; e.g., S = Smarter(x,y) σ = {x/hillary, y/bill} Sσ = Smarter(Hillary, Bill) Ask(KB,S) returns some/all σ such that KB = Sσ AI Slides (5e) c Lin Zuoquan@PKU

35 Application of FOL Facts hold in situations, rather than eternally E.g., Holding(Gold, N ow) rather than just Holding(Gold) Situation calculus is one way to represent change in FOL: Adds a situation argument to each non-eternal predicate E.g., Now in Holding(Gold,Now) denotes a situation Situations are connected by the Result function Result(a,s) is the situation that results from doing a in s PIT Gold PIT PIT Gold PIT PIT S 1 PIT S 0 Forward AI Slides (5e) c Lin Zuoquan@PKU

36 KB: example The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American. Prove that Col. West is a criminal AI Slides (5e) c Lin Zuoquan@PKU

37 KB: example... it is a crime for an American to sell weapons to hostile nations: AI Slides (5e) c Lin Zuoquan@PKU

38 KB: example... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles AI Slides (5e) c Lin Zuoquan@PKU

39 KB: example... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West AI Slides (5e) c Lin Zuoquan@PKU

40 KB: example... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missiles are weapons: AI Slides (5e) c Lin Zuoquan@PKU

41 KB: example... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missiles are weapons: Missile(x) Weapon(x) An enemy of America counts as hostile : AI Slides (5e) c Lin Zuoquan@PKU

42 KB: example... it is a crime for an American to sell weapons to hostile nations: American(x) W eapon(y) Sells(x, y, z) Hostile(z) Criminal(x) Nono... has some missiles, i.e., x Owns(Nono,x) Missile(x): Owns(Nono,M 1 ) and Missile(M 1 )... all of its missiles were sold to it by Colonel West x Missile(x) Owns(Nono,x) Sells(West,x,Nono) Missiles are weapons: Missile(x) Weapon(x) An enemy of America counts as hostile : Enemy(x,America) Hostile(x) West, who is American... American(W est) The country Nono, an enemy of America... Enemy(N ono, America) AI Slides (5e) c Lin Zuoquan@PKU

43 KB: the wumpus world Perception b,g,t Percept([Smell,b,g],t) Smelt(t) s,b,t Percept([s,b,Glitter],t) AtGold(t) Reflex: t AtGold(t) Action(Grab, t) Reflex with internal state: do we have the gold already? t AtGold(t) Holding(Gold, t) Action(Grab, t) Holding(Gold, t) cannot be observed keeping track of change is essential AI Slides (5e) c Lin Zuoquan@PKU

44 Reasoning Properties of locations: x, t At(Agent, x, t) Smelt(t) Smelly(x) x, t At(Agent, x, t) Breeze(t) Breezy(x) Squares are breezy near a pit: Diagnostic rule infer cause from effect y Breezy(y) x Pit(x) Adjacent(x,y) Causal rule infer effect from cause x,y Pit(x) Adjacent(x,y) Breezy(y) Neither of these is complete e.g., the causal rule doesn t say whether squares far away from pits can be breezy Definition for the Breezy predicate: y Breezy(y) [ x Pit(x) Adjacent(x,y)] AI Slides (5e) c Lin Zuoquan@PKU