First-Order Predicate Calculus

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1 First-Order Predicate Calculus Deepak Kumar November 2017 Propositional Logic - Syntax Sentences Well-formed formulas (wffs) Any atom is a wff [Atomic Sentences] e.g. P, Q, R, R3 Complex Sentences If ω 1 and ω 2 are wffs, then so are ω 1 ω 1 ω 2 disjunction ω 1 ω 2 conjunction ω 1 ω 2 implication ω 1 negation There are no other wffs. Literals are atomic wffs Examples P (P Q) P P Q P P P X etc. P, P 2 1

2 Propositional Logic - Semantics How symbols relate to the domain/world Atomic Sentences Interpretation of symbols & constants True is True False is False P I : Bryn Mawr is in Pennsylvania Meaning of a symbol is its truth value, given its interpretation. P is True, given P I Complex Sentences Use meaning of connectives. 3 Propositional Logic - Key Ideas (un)satisfiability Validity Entailment (Δ ω) Inference & Theorem Proving (Δ R ω) Rules of Inference Soundness (Δ R ω implies Δ ω) Completeness (Whenever Δ ω, Δ R ω) Reductio ad Absurdum (if Δ has a model, but Δ { ω} does not, it must be that Δ ω ) Resolution rule of inference (sound, but not complete) Resolution with Reductio is sound and complete. Wffs as clauses (also, Definite and Horn clauses) Forward & Backward Chaining Inference 4 2

3 Propositional Logic - Key Ideas (un)satisfiability Validity Entailment (Δ ω) Inference & Theorem Proving (Δ R ω) Rules of Inference Soundness (Δ R ω implies Δ ω) Completeness (Whenever Δ ω, Δ R ω) Reductio ad Absurdum (if Δ has a model, but Δ { ω} does not, it must be that Δ ω ) Resolution rule of inference (sound, but not complete) Resolution with Reductio is sound and complete. Wffs as clauses (also, Definite and Horn clauses) Forward & Backward Chaining Inference Propositional Logic is not very expressive. But useful for learning the fundamentals. 5 First-Order Predicate Calculus (FOPC) Components Object constants: A, B, Deepak, etc. Function Constants: fatherof, colorof, etc. Relation Constants: Parent, On, Clear, Sibling, etc. Variables: x, y, z, u, v, w, etc. Connectives:,,, Quantifiers:, Delimiters: (, ), [, ] 6 3

4 Objects, Functions, Relations Object constants: these are objects or individuals in the domain A, B, Deepak, Red, Car54, etc. Function Constants: operate on objects and denote other objects fatherof 1, colorof 1, distance 2 etc. Relation Constants: Denote properties on, between objects Parent 2, On 2, Clear 1, Sibling 2, etc. 7 FOPC - Syntax Terms An object is a term A function constant of arity, n followed by n terms is a term (functional expression) Wffs represent propositions e.g. motherof(sally), colorof A Atomic wffs relation-constant n (term 1, term 2, term n ) e.g. Propositional wffs If ω 1 and ω 2 are wffs, then so are ω 1 ω 1 ω 2 disjunction ω 1 ω 2 conjunction ω 1 ω 2 implication ω 1 negation Parent(Ginny, Lily) 8 4

5 FOPC - Syntax Terms An object is a term A function constant of arity, n followed by n terms is a term (functional expression) Wffs represent propositions Atomic wffs e.g. Propositional wffs If ω 1 and ω 2 are wffs, then so are ω 1 e.g. motherof(sally), colorof A relation-constant( ) ω 1 ω 2 disjunction ω 1 ω 2 conjunction ω 1 ω 2 implication ω 1 negation Parent(Ginny, Lily) Save for later Variables: x, y, z, u, v, w, etc. Quantifiers:, 9 FOPC - Semantics Objects The world can have infinite objects/individuals - concrete - abstract - fictional As long as it is a name and we need to say something about it. Functions Relations Denote objects, we may not have a name for f n (o 1, o 2,, o n ) -> object Denote properties Heavy(x) On(A, B) Big(y) Clear(A) Human(Deepak) Block(A) Interpretation An atomic wff is True/False just in case the relation denoted by it holds for its arguments. 10 5

6 FOPC Example - Blocksworld Objects A, B, C, Table Relations On 2, Clear 1 On(C, Table) On(A, C) On(B, A) Clear(B) Clear(A) Clear(C) Clear(Table) Example Knowledge Base Δ = {On(C, Table), On(A, C), On(B, A), Clear(B)} B A C Table 11 FOPC Example - Blocksworld Objects A, B, C, Table Relations On 2, Clear 1 On(C, Table) On(A, C) On(B, A) Clear(B) Clear(A) Clear(C) Clear(Table) B A C Table Example Knowledge Base But, how to make general Statements about the world? Δ = {On(C, Table), On(A, C), On(B, A), Clear(B)} 12 6

7 Variables & Quantifiers Variables A variable is a term. i.e. it denotes/can denote an object. Universal Quantifier ( - for-all ) if ω is a wff and x is a variable then ( x) ω x (ω) is a wff is a wff x [ω] is a wff ω is the scope of the variable. E.g. x [ P(x) R(x) ] Existential Quantifier ( - there exists ) if ω is a wff and x is a variable then e.g. ( x) ω is a wff x [ P(x) y [ R(x, y) S(f(x) ] ] 13 Semantics of and In a given domain x ω(x) has value True Just in case ω(x) has value True for all assignments of x to objects in the domain. e.g. On 2, Clear 1 x [ On(x, C) Clear(C) ] Just in case x=a/b/c/table are True. 14 7

8 Semantics of and In a given domain x ω(x) has value True Just in case ω(x) has value True for all assignments of x to objects in the domain. e.g. On 2, Clear 1 x [ On(x, C) Clear(c) ] Just in case x=a/b/c/table are True. Similarly, x ω(x) has value True Just in case ω(x) is True for atleast one assignment for x in the domain. 15 Example Tell King(John) Tell Person(Richard) Tell x [ King(x) Person(x) ] Queries 1. King(John)? 2. x Person(x)? But, we want to know who is a person! 3. ASK-VARS(KB, Person(x)) 16 8

9 Example Tell King(John) Tell Person(Richard) Tell x [ King(x) Person(x) ] Queries 1. King(John)? True 2. x Person(x)? True But, we want to know who is a person! 3. ASK-VARS(KB, Person(x)) 17 Example Tell King(John) Tell Person(Richard) Tell x [ King(x) Person(x) ] Queries 1. King(John)? True 2. x Person(x)? True But, we want to know who is a person! 3. ASK-VARS(KB, Person(x)) x = Richard x = John 18 9

10 Example Tell King(John) Tell Person(Richard) Tell x [ King(x) Person(x) ] Queries 1. King(John)? True 2. x Person(x)? True Tell x [ King(x) Greedy(x) Evil(x) ] Tell King(John) Tell Greedy(John) Query Evil(John)? True But, we want to know who is a person! 3. ASK-VARS(KB, Person(x)) x = Richard x = John 19 Blocksworld On(x, y) : x is on top of y Clear(x) : x is clear Block(x) : x is a block On(B, A) On(A, C) On(C, Table) Block(A) Block(B) Block(C) x, y [ Block(x) Block(y) On(x, y) Clear(y) ] 20 10

11 Blocksworld On(x, y) : x is on top of y Clear(x) : x is clear Block(x) : x is a block Isa(x, y) : x is a y On(B, A) Isa(A, Block) On(A, C) Isa(B, Block) On(C, Table) Isa(C, Block) Block(A) Block(B) Block(C) x, y [ Block(x) Block(y) On(x, y) Clear(y) ] 21 Blocksworld On(x, y) : x is on top of y Clear(x) : x is clear Block(x) : x is a block Isa(x, y) : x is a y On(B, A) Isa(A, Block) On(A, C) Isa(B, Block) On(C, Table) Isa(C, Block) Block(A) Block(B) Block(C) x, y [ Isa(x, Block) Isa(y, Block) On(x, y) Clear(y) ] 22 11

12 Mrs Dursley Mr Dursley Mrs Evans Mr Evans Mr Potter Mrs Potter Septimus Cedrella Marge Vernon Petunia Lily James Arthur Molly Dudley Harry Ginny Ron George Fred Percy Charlie Bill Hermione James Albus Lily Hugo Rose 23 Mrs Dursley Mr Dursley Mrs Evans Mr Evans Mr Potter Mrs Potter Septimus Cedrella Marge Vernon Petunia Lily James Arthur Molly Predicates Dudley Harry Ginny Ron George Fred Percy Charlie Bill Hermione James Albus Lily Hugo Rose 1. Female(x) : x is female 2. Male(x) : x is male 3. Parent(x, y) : x is a parent of y 4. Married(x, y) : x is married to y 24 12

13 Mrs Dursley Mr Dursley Mrs Evans Mr Evans Mr Potter Mrs Potter Septimus Cedrella Marge Vernon Petunia Lily James Arthur Molly Predicates 1. Female(x) : x is female 2. Male(x) : x is male 3. Parent(x, y) : x is a parent of y 4. Married(x, y) : x is married to y Symbols Marge, Vernon, James1, James2, Albus, Lily, Harry, etc. Dudley Harry Ginny Ron George Fred Percy Charlie Bill James Albus Lily Female(Marge) Female(Petunia) Female(Lily) Male(James1) Male(Vernon) Etc. Hugo Hermione Rose Married(Vernon, Petunia) Married(Lily, James1) Married(Arthur, Molly) Married(Harry, Ginny) Married(Ron, Hermione) Etc. Parent(Harry, James2) Parent(Ginny, James2) Parent(Petunia, Dudley) Parent(Vernon, Dudley) Parent(Molly, Bill) Etc. 25 Mrs Dursley Mr Dursley Mrs Evans Mr Evans Mr Potter Mrs Potter Septimus Cedrella Marge Vernon Petunia Lily James Arthur Molly Predicates 1. Female(x) : x is female 2. Male(x) : x is male 3. Parent(x, y) : x is a parent of y 4. Married(x, y) : x is married to y Symbols Dudley Harry Ginny Ron George Fred Percy Charlie Bill James Albus Lily Female(Marge) Female(Petunia) Female(Lily) Male(James1) Male(Vernon) Etc. Hugo Hermione Rose Married(Vernon, Petunia) Married(Lily, James1) Married(Arthur, Molly) Married(Harry, Ginny) Married(Ron, Hermione) Etc. Parent(Harry, James2) Parent(Ginny, James2) Parent(Petunia, Dudley) Parent(Vernon, Dudley) Parent(Molly, Bill) Etc. Marge, Vernon, James1, James2, Albus, Lily, Harry, etc. Facts 26 13

14 Modeling Relationships Spouse(x, y) : x is a spouse of y Wife(x, y) : x is a wife of y Husband(x, y) Father(x, y), Mother(x, y) Sibling(x, y), Brother(x, y), Sister(x, y) GrandParent/GrandMother/GrandFather GrandChild/GrandDaughter/GrandSon Aunt/Uncle/AuntOrUncle Cousin Niece/Nephew/NieceOrNephew Child/Daughter/Son Etc. 27 Modeling Relationships Spouse(x, y) : x is a spouse of y Husband/Wife Mother/Father 28 14

15 Modeling Relationships Spouse(x, y) : x is a spouse of y x, y [Married(x, y) Spouse(x, y) ] Husband/Wife * x, y [ Female(x) Married(x, y) Wife(x, y) ] Mother/Father x, y [ Female(x) Parent(x, y) Mother(x, y) ] 29 Modeling Relationships Sibling(x, y) : x is a sibling of y x, y [[ z Father(z, x) Father(z, y)] [ w Mother(w, x) Mother(w, y)] (x y) Sibling(x, y) ] 30 15

16 Modeling Relationships Spouse(x, y) : x is a spouse of y Wife(x, y) : x is a wife of y Husband(x, y) Father(x, y), Mother(x, y) Sibling(x, y), Brother(x, y), Sister(x, y) GrandParent/GrandMother/GrandFather GrandChild/GrandDaughter/GrandSon Aunt/Uncle/AuntOrUncle Cousin Niece/Nephew/NieceOrNephew Child/Daughter/Son Etc. 31 Another Example (From R&N) Domain: people, songs, albums, disks (CDs) CopyOf(d, a) Owns(p, d) Sings(p, s, a) Wrote(p, s) : disk d is a copy of album a : person p owns disk d : album a includes recording of song s sung by p : person p wrote song s Constants: McCartney, Gershwin, Joe, EleanorRigby, TheManILove, Revolver, BillieHoliday, Joe 32 16

17 Another Example 1. Gershwin wrote The Man I Love Wrote(Gershwin, TheManILove) 2. Gershwin did not write Eleanor Rigby Wrote(Gershwin, EleanorRigby) 3. Either Gershwin or McCartney wrote The Man I Love Wrote(Gershwin, TheManILove) Wrote(McCartney, TheManILove) 4. Joe has written at least one song s Wrote(Joe, s) 5. Joe owns a copy of Revolver x CopyOf(x, Revolver) Owns(Joe, x) 33 Another Example 6. Every song that McCartney sings on revolver was written by McCartney s [Sings(McCartney, s, Revolver) Wrote(McCartney, s) ] 7. Gershwin did not write any of the songs on Revolver [ s Wrote(Gershwin, s) p Sings(p, s, Revolver) ] 8. Every song that Gershwin wrote has been recorded on some album s [ Wrote(Gershwin, s) p,a Sings(p, s, a) ] 34 17

18 Another Example 9. There is a single album that contains every song that Joe has written a [ s Wrote(Joe, s) p Sings(p, s, a) ] 10. Joe owns a copy of an album that has Billy Holiday singing The Man I Love d, a, s [CopyOf(d, a) Owns(Joe, d) Sings(BillieHoliday, TheManILove, a)] 35 Another Example 11. Joe owns a copy of every album that has a song sung by McCartney a [ s Sings(McCartney, s, a) d CopyOf(d, a) Owns(Joe, d) ] 12. Joe owns a copy of every album on which all the songs are sung by Billie Holiday a [ [ s, p Sings(p, s, a) Sings(BillieHolliday, s, a)] d CopyOf(d, a) Owns(Joe, d) ] 36 18

19 FOPC Exercise 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. Use the following relations: American(x) : x is an American Enemy(x, y) : x is an enemy of y Hostile(x) : x is hostile Criminal(x) : x is a criminal Missile(x) : x is a missile Weapon(x) : x is a weapon Owns(x, y) : x owns y Sells(x, y, z) : x sells y to z To write FOPC wffs for the following: 1. it is a crime for an American to sell weapons to hostile nations: 2. Nono has some missiles: 3. all of its missiles were sold to it by Colonel West: 4. Also, Missiles are weapons: 5. And, an enemy of America is hostile: 6. West, who is American: 7. The country Nono, an enemy of America: 37 19

Inference in First-Order Predicate Calculus

Inference in First-Order Predicate Calculus Deepak Kumar November 2017 Knowledge Engineering in FOPC Identify the task Assemble relevant knowledge Decide on a vocabulary of predicates, functions, and constants