© Copyright 2008 STI INNSBRUCK Intelligent Systems – Predicate Logic Florian Fischer Partially based on CS188, Berkely by Stuart Russell

2 Overview Motivation Syntax of First Order Logic Semantics of First Order Logic Using FOL Knowledge engineering in FOL

3 Motivation Starting from propositional logic (see last lecture): Propositional logic is declarative –Meaning is Independent of particular evaluation procedure Propositional logic allows partial/disjunctive/negated information –Unlike most data structures and relational database systems Propositional logic is compositional: –meaning of B 1,1  P 1,2 is fully derived from meaning of B 1,1 and of P 1,2 Meaning in propositional logic is context-independent –Statements convey the same meaning every time  Propositional logic has very limited expressive power

Motivation – Limitations of Propositional Logic Propositional logic assumes the world contains facts that are either true or false. In propositional logic the smallest atoms represent whole propositions (propositions are atomic) –Propositional logic does not capture the internal structure of the propositions –It is not possible to work with units smaller than a proposition Example: –“A Mercedes Benz is a Car” and “A car drives” are two individual, unrelated propositions –We cannot conclude “A Mercedes Benz drives” It is possible to represent everything you want in propositional logic But often this is not very efficient 4

5 Predicate Logic / First-order logic Basic idea: A proposition is expressed as predicate about (on or more) objects in the world Propositions are predicates and arguments –I.e. Car(Mercedes Benz). First-order logic (FOL) assumes the world contains: –Objects: people, houses, numbers, colors, baseball games, wars, … –Relations: red, round, prime, brother of, bigger than, part of, comes between, … –Functions: father of, best friend, one more than, plus, … One to one mapping

Predicate Logic / First-order logic Some relations are properties: –They state some fact about a single object: Round(ball), Prime(7). Some relations are n-ary relations: –State facts about two or more objects: Married(John,Mary), Largerthan(3,2). Some relations are functions: –Their value is another object: Plus(2,3), Father(Dan).

Defining First-Order Logic As usual for a Logic, FOL is defined in several parts: –Syntax: Telling us how well-formed formulae can be built from basic syntactic parts –A proof theory, consisting of: Inference rules:Telling us what kind of conclusions are admissable base on a set of propositions ((i.e. Modus ponens. see next lecture) Axioms: To encode basic assumptions (i.e. equality) –Semantics: Telling us which interpretations make a formula true 7

8 Introduction to Artificial Intelligence 2005 – Axel Polleres Syntax of FOL A First-Order language is formed by the following: –A finite or countable set R of relation symbols or predicate symbols, with an integer n associated with symbol. This is called the arity of the symbol. –A finite or countable set F of function symbols. Again each function symbol has a specific arity. –A finite or countable set C of constant symbols. –Quantifiers (existential, universal) –Punctuation ( “(”, “)”, “,”) –Variables –The (already known) usual connectives

9 Introduction to Artificial Intelligence 2005 – Axel Polleres Syntax of FOL - Examples ConstantsKingJohn, 2, UIBK,... Predicate symbolsBrother, >,... Function symbolsSqrt, LeftLegOf,... Variablesx, y, a, b,... Connectives , , , ,  Equality= Quantifiers , 

10 Introduction to Artificial Intelligence 2005 – Axel Polleres Syntax of FOL - Terms From basic syntax we define terms: –Any variable is a term –Any constant symbol (member of C) is a term –If f is an n-ary function symbol and t1,…,tn are terms, then f(t1, …, tn) is a term term =function (term 1,...,term n ) or constant or variable E.g., Brother(KingJohn,RichardTheLionheart) (Length(LeftLegOf(Richard)) = Length(LeftLegOf(KingJohn))) Functions can be viewed as complex names for constants

11 Introduction to Artificial Intelligence 2005 – Axel Polleres Syntax of FOL – Atomic Formulas An atomic formula is any string of the form r(t1, …, tn), where –r is an n-ary relation symbol (a member of R) –t1, …,tn are terms Atomic formula =predicate (term 1,...,term n ) or term 1 = term 2 E.g., Brother(KingJohn, RichardTheLionheart) (Length(LeftLegOf(Richard)) = Length(LeftLegOf(KingJohn))) Equality is a special predicate with a fixed sematnics

Syntax of FOL – Atomic Formulas Formulas in logic state facts that are true or false. Properties and n-ary relations do just that: –LargerThan(2,3) (means 2>3) is false. –Brother(Mary,Pete) is false. Functions do not state facts and form no sentence: –Brother(Pete) refers to the object John (his brother) and is neither true nor false. Brother(Pete, Brother(Pete)) is true. 12 Binary relationFunction

13 Syntax of FOL – Formulas We can again use formulas to form more complex constructs Complex sentences are made from atomic sentences using connectives (just like in propositional logic)  S, S 1  S 2, S 1  S 2, S 1  S 2, S 1  S 2, E.g. : Sibling(KingJohn,Richard)  Sibling(Richard,KingJohn ) >(1,2)  ≤ (1,2) >(1,2)   >(1,2)

Syntax of FOL – Example 14 binary relation function property objects connectives

Semantics of FOL Semantics of a logical sentence are described by relating it to a formal description of a “possible world” called a model –In propositional logic: A set of true and false propositions (think of a truth-table) –In FOL: A set of objects and a set of relations between objects Giving meaning to a logical formula is more difficult as the propositional case: –We must fix the domain (“what” we are talking about) over which quantifiers operate –We must fix the interpretation for constants, function, and relation symbols (w.r.t this domain) –We must give an assignment of values to variables 15

16 Semantic of FOL Sentences are true with respect to a model and an interpretation Model contains objects (domain elements) and relations among them Interpretation specifies referents for constant symbols → objects predicate symbols → relations function symbols →functions 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 For more theoretical (and philosophical!) background on model theory see [4]

Semantic of FOL More precisely, a model is a pair M = where –D is a nonempty set, called the domain of M –I an interpretation that maps constant symbols, function symbols, relations elements in the domain D An assignment in a model M= is a mapping from the set of variables to the set D Given an interpretation (that gives meaning to constant and function symbols) and an assignment (that gives values to variables), we can calculate truth values for arbitrary terms! For a detailed definition see [1] 17

18 Semantics of FOL: Example

19 Semantics in FOL: Example

20 Logical Entailment in FOL Enumerating models to check if a statements logically follows is not a good idea Nonentailment is even undecidable, i.e. cannot be computed  ! Overall FOL is semi-decideable

Quantification Round (ball) is true or false because we give it a single argument (ball). We can be much more flexible if we allow variables which can take on values in a domain. e.g. reals x, all persons P, etc. To construct logical sentences we need a quantifier to make it true or false. 21

Quantification Is the following true or false? To make it true or false we use For all real x, x>2 implies x>3.There exists some real x which square is minus 1.

23 Universal quantification  E.g: "Everyone at UIBK is smart“:  x At(x,UIBK)  Smart(x)  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, UIBK)  Smart(KingJohn)  At(Richard, UIBK)  Smart(Richard)  At(UIBK, UIBK)  Smart(UIBK) ...

24 A common mistake to avoid Typically,  is the main connective with  Common mistake: using  as the main connective with  :  x At(x,UIBK)  Smart(x) means “Everyone is at UIBK and everyone is smart” Correct:  x At(x,UIBK)  Smart(x) As you can see many axioms can be written as rules!

25 Existential quantification  "Someone at UIBK is smart":  x At(x,UIBK)  Smart(x)  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,UIBK)  Smart(KingJohn)  At(Richard,UIBK)  Smart(Richard)  At(UIBK,UIBK)  Smart(UIBK) ...

26 Another common mistake to avoid Typically,  is the main connective with  Common mistake: using  as the main connective with  :  x At(x,UIBK)  Smart(x) is true if there is anyone who is not at UIBK! Usually used in Queries: "Is there someone in UIBK who is smart?" Correct:  x At(x,UIBK)  Smart(x )

Nested Quantifiers Combinations of universal and existential quantification are possible: Which is which?:Everyone is the father of someone. Everyone has everyone as a father There is a person who has everyone as a father. There is a person who has a father There is a person who is the father of everyone. Everyone has a father. Binary relation: “x is a father of y”.

28 Properties of quantifiers  x  y is the same as  y  x  x  y is the same as  y  x  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,IceCream)  x Likes(x,Broccoli)  x  Likes(x,Broccoli)

De Morgan’s Law for Quantifiers De Morgan’s Rule Generalized De Morgan’s Rule Rule is simple: If you bring a negation inside a disjunction or a conjunction, always switch between them (or  and, and  or). Equality symbol: Father(John)=Henry. This relates two objects.

30 Introduction to Artificial Intelligence 2005 – Axel Polleres 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., definition of Sibling in terms of Parent:  x,y Sibling(x,y)  [  (x = y)   m,f  (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)]

Using FOL We want to TELL things to the KB, –e.g. TELL(KB, ) We also want to ASK things to the KB, –e.g. ASK(KB, ) The KB should return the list of x’s for which Person(x) is true: {x/John,x/Richard,...} –A variable binding!

32 Using FOL – Simple Example The family domain: Brothers are siblings  x,y Brother(x,y) ) Sibling(x,y) One's mother is one's female parent  m,c Mother(c) = m  (Female(m)  Parent(m,c)) “Sibling” is symmetric  x,y Sibling(x,y)  Sibling(y,x) Attention! Mother is a function symbol here, whereas female and parent are predicate symbols!

Using Fol More complex example: The set domain ∀ s Set(s) ⇔ (s = {} ) ∨ ( ∃ x,s2 Set(s2) ∧ s = { x | s2 }) ¬ ∃ x,s { x | s } = {} ∀ x,s x ∈ s ⇔ s = { x | s } ∀ x,s x ∈ s ⇔ [ ∃ y,s2} ( s = { y | s2} ∧ (x = y ∨ x ∈ s2 ))] ∀ s1,s2 s1 ⊆ s2 ⇔ ( ∀ x x ∈ s1 ⇒ x ∈ s2) ∀ s1,s2 (s1= s2) ⇔ (s1 ⊆ s2 ∧ s2 ⊆ s1 ) ∀ x, s1,s2 x ∈ (s1 ∩ s2) ⇔ (x ∈ s1 ∧ x ∈ s2 ) ∀ x, s1,s2 x ∈ (s1 ∪ s2) ⇔ (x ∈ s1 ∨ x ∈ s2) 33

34 Representing Knowledge in FOL 1.Identify the task 2.Assemble the relevant knowledge 3.Decide on a vocabulary of predicates, functions, and constants 4.Encode general knowledge about the domain 5.Encode a description of the specific problem instance 6.Pose queries to the inference procedure and get answers 7.Debug the knowledge base

35 Example: The electronic circuits domain One-bit full adder

36 Example: The electronic circuits domain 1.Identify the task –Does the circuit actually add properly? (circuit verification) 2.Assemble the relevant knowledge –Composed of wires and gates; Types of gates (AND, OR, XOR, NOT) –Irrelevant: size, shape, color, cost of gates 3.Decide on a vocabulary/encoding –Alternatives: Type(X 1 ) = XOR Type(X 1, XOR) XOR(X 1 ) …

37 Example: The electronic circuits domain 4.Encode general knowledge of the domain  t 1,t 2 Connected(t 1, t 2 )  Signal(t 1 ) = Signal(t 2 )  t Signal(t) = 1  Signal(t) = 0 1 ≠ 0  t 1,t 2 Connected(t 1, t 2 )  Connected(t 2, t 1 )  g Type(g) = OR  (Signal(Out(1,g)) = 1   n Signal(In(n,g)) = 1)  g Type(g) = AND  (Signal(Out(1,g)) = 0   n Signal(In(n,g)) = 0)  g Type(g) = XOR  (Signal(Out(1,g)) = 1  Signal(In(1,g)) ≠ Signal(In(2,g)))  g Type(g) = NOT  Signal(Out(1,g)) ≠ Signal(In(1,g))

38 Example: The electronic circuits domain 5.Encode the specific problem instance Type(X 1 ) = XOR Type(X 2 ) = XOR Type(A 1 ) = AND Type(A 2 ) = AND Type(O 1 ) = OR Connected(Out(1,X 1 ),In(1,X 2 ))Connected(In(1,C 1 ),In(1,X 1 )) Connected(Out(1,X 1 ),In(2,A 2 ))Connected(In(1,C 1 ),In(1,A 1 )) Connected(Out(1,A 2 ),In(1,O 1 )) Connected(In(2,C 1 ),In(2,X 1 )) Connected(Out(1,A 1 ),In(2,O 1 )) Connected(In(2,C 1 ),In(2,A 1 )) Connected(Out(1,X 2 ),Out(1,C 1 )) Connected(In(3,C 1 ),In(2,X 2 )) Connected(Out(1,O 1 ),Out(2,C 1 )) Connected(In(3,C 1 ),In(1,A 2 ))

39 Example: The electronic circuits domain 6.Pose queries to the inference procedure –What are the possible sets of values of all the terminals for the adder circuit? –  i 1,i 2,i 3,o 1,o 2 Signal( In(1,C 1 ) ) = i 1  Signal( In(2,C 1 ) ) = i 2  Signal( In(3,C 1 ) ) = i 3  Signal( Out(1,C 1 ) ) = o 1  Signal( Out(2,C 1 ) ) = o 2 7.Debug the knowledge base –Maybe you have omitted assertions like 1 ≠ 0, etc?

40 Summary Predicate Logic: –objects, relations and functions are semantic primitives –syntax: constants, function symbols, predicate symbols, equality, quantifiers Increased expressive power over propositional logic –Predicate Logic allows generalisations over objects –It is more economical in many cases Atomic sentences have internal structure –Atomic sentences are decomposed into predicates and terms The semantics of predicate logic: –Model is a set of objects, a set of relations and a set of referential functions –An interpretation maps a language onto a model Next time: Inference in First-Order logic

References and Further Information [1] M. Fitting: First-Order Logic and Automated Theorem Proving, 1996 Springer-Verlag New York (Chapter 5) [2] Michael Huth and Mark Ryan, Logic in Computer Science(second edition), 2004, Cambridge University Press [3] S. Russel and P. Norvig: Artifical Intelligence – Modern Approach, 2002k, Prentice Hall (Web-site: ) [4] Wikipedia: – and particularlyhttp://en.wikipedia.org/wiki/First-order_logic – order_logic for several relevant propertieshttp://en.wikipedia.org/wiki/First-order_logic#Metalogical_theorems_of_first- order_logic 41