Artificial Intelligence First-Order Predicate Logic - First-Order Predicate Logic (FOL or FOPL), also called First-Order Predicate Calculus Formal Language Semantics through Interpretation Function Axioms Inference System

FOPL - Formal Language / Syntax -

Formal Language A Formal Language is specified as L = (NT, T, P, S) NT Set of Non-Terminal Symbols T Set of Terminal Symbols P Set of Production or Grammar Rules S Start Symbol (top-level node in syntax tree / parse tree) A formal language specifies the syntactically correct or well-formed expressions of a language.

Terminals and Non-Terminals NT Non-Terminals wff (well-formed formula), atomic-formula; Predicate, Term, Function, Constant, Variable; Quantifier, Connective T Terminals Predicate (Symbols)P, Q, married,..., T, F Function (Symbols)f, g, father-of,... Variablesx, y, z,... ConstantsSally, block-1, c Connectives , , ,  Negation Symbol  Quantifiers ,  Equality Symbol= Parentheses(, ) Other Symbols: Domain Specifc General

Non-terminal Rules wff ::= atomic-formula | (wff) |  wff | wff Connective wff | Quantifier Variable wff atomic-formula ::= Predicate (Term,...)* | Term = Term Term ::= Function (Term,...)* | Variable | Constant Terminal Rules Connective ::=  |  |  |  Quantifier ::=  |  *Note: n-ary functions and predicates go with n terms Production / Grammar Rules

Terminal Rules for the specific Domain Predicate ::= on(_,_) | near(_,_) |... Function ::= distance(_,_) | location(_) |... Variable ::= x | y |... Constant ::= Flakey | John-Bear | Karen | Alan-Alder | The-File | Kurt Domain-Specific Terminal Rules

Quantifiers and Binding A variable in a formula can be bound by a quantifier. bound variable  x: married (Sally, x) open formula: a variable in the formula is not bound by a quantifier  x: married (Sally, x)  happy (y) closed formula: all variables in the formula are bound by quantifiers:  x  y: married (x, y) Most authors regard quantified formulas only as wffs if all quantified variables appear in the formula. Some authors regard quantified formulas only as wffs if all variables are bound by quantifiers.

FOPL - Semantics / Interpretation -

Semantics - Overview Define the Semantics of FOPL expressions (formulae): 1. Interpretation – Maps symbols of the formal language (predicates, functions, variables, constants) onto objects, relations, and functions of the “world” (formally: Domain, relational Structure, or Universe) 2.Valuation - Assigns domain objects to variables* 3.Constructive Semantics – Determines the semantics of complex expressions inductively, starting with basic expressions * The Valuation function can be used for describing value assignments and constraints in case of nested quantifiers. The Valuation function otherwise determines the satisfaction of a formula only in case of open formulae.

Semantics – Domain, Interpretation I Domain, relational Structure, Universe Dset of Objects R, S,...set of Relations over D f, g,...set of Functions in D Basic Mapping / Interpretation constantsI [c] = d  D functions I [f] = F: D n →DFunction predicates I [P] = R  D n Relation (set of n-tuples) Valuation V variablesV(x) = d  D

Semantics –Interpretation Next, determine the semantics for complex terms and formulae constructively - regarding the syntax - from the basic interpretation above.

Semantics - Interpretation II Term with function I [f(t 1,...,t n )) = I [f] (I [t 1 ],..., I [t n ]) = F(I [t 1 ],..., I [t n ])  D atomic Formula I [P(t 1,...,t n )]true if (I [t 1 ],..., I [t n ])  I [P] = R negated Formula I [  ]true if I [  ] is not true complex Formula I [  ] true if I [  ] or I [  ] true I [  ] true if I [  ] and I [  ] true I [  ]if I [  ] not true or I [  ] true

Semantics - Interpretation III quantified Formula(relative to Valuation function) I [  x.  ] true if  is true with V’(x)=d for some d  D where V’ is otherwise identical to the prior V. I [  x.  ] true if  is true with V’(x)=d for all d  D and where V’ is otherwise identical to the prior V. Note: The order of quantifiers plays a role for the semantic interpretation and evaluation:  x  y.  is different from  y  x.  In the first case, we go through all values for x, and for each value of x we pick a suitable value for y. In the second case, we have to find one value for y which is good for all values of x.

Semantics - Model Model Given a set of formulae  and a domain D with an interpretation I. Then D is a model of  if I[  ] is true for all   * That means the interpretation I into the domain D makes every formula  in  true. * for every possible valuation, in case  has open formulae.

Semantics – Logical Consequence Logical Consequence Given a set of formulae  and a formula α. α is a logical consequence of  if α is true in every model of . Notation:  |= α That means that for every model (interpretation into a domain) in which  is true, α must also be true.

FOPL - Inference System - Axioms & Inference Rules

FOPL Axioms A1      A2      A3        A4(    )  ((    )  (    )) A5  x:  (x)   (y) A6  (x)   y:  (y)

Formal Inference - Overview Derive new formulae by syntactic manipulation of existing formulae: given set of formulae   describes your KB, or a Theory,... (FOPL axioms + your own "proper" axioms) apply inference rule (based on  ) new formula α is derived add new formula to KB or Theory new KB or Theory is  α

Formal Inference Formal Inference, Theorem Given a set of formulae  and a set of inference rules IR. A new formula α can be generated based on  using inference rules in IR. We say that α is formally inferred or derived from  or α is a Theorem (of  ) Notation:  |– α

IR Modus Ponens Modus Ponens   ,   States that  can be concluded provided we know that the formulae    and  are true in our knowledge base.

IR Universal Instantiation Universal Instantiation  x:  (x)  (c) where  (x) is any formula containing the variable x, and  (c) is the formula  (x) where every occurrence of the quantified variable x is substituted with the arbitrary constant c.

IR Existential Generalization Existential Generalization  (c)  x:  (x) where  (c) is any formula containing the arbitrary constant c, and  (x) is the same formula as  (c) but with every occurrence of the constant c replaced by a variable x.

Replacement Rules                 (    )  (    )    IR Replacement Rules

FOPL Inference System The Axioms and the Inference Rules above constitute a formal inference system for FOPL. This system - we call it FS1 - is complete and sound.

Soundness and Completeness Soundness An Inference System is sound iff  |– α   |= α every formula which can be derived by formal inference from  is a also logical consequence of . Completeness An Inference System is complete iff  |= α   |– α every formula which is a logical consequence of  can be derived by formal inference from .

FOPL - Sound and Complete 2 The above inference system for FOPL is sound and complete. Thus, every formula which can be derived in FOPL using FS1 (  |– α) is also a logical consequence of the given axioms (  |= α) :  |– αiff  |= α Thus, there is a correspondence between formal Inference and semantic Interpretation.

Semantics - Example A1 Predicate Logic Language constantsBill-1, John-3, Sally-1, Mary-1, Mary-2 predicateshappy-together, hate-each-other Structure D objects: Uncle-Bill, Uncle-John, Aunt-Sally, The-woman-I-don't-like, Mary relations: Married, Divorced (Uncle-Bill, Aunt-Sally)  Married, (Uncle-John, Mary)  Married (Uncle-John, The-woman-I-don't-like)  Divorced Interpretation I(Bill-1)=Uncle-Bill, I(John-3)=Uncle-John, I(Sally-1)=Aunt-Sally, I(Mary-1)=The- woman-I-don't-like, I(Mary-2)=Mary I(happy-together)=Married, I(hate-each-other)=Divorced True or false? hate-each-other (Bill-1, John-3) happy-together(Bill-1, Sally-1) hate-each-other(John-3, Mary-1) happy-together(John-3, Mary-2)

Semantics -Example A2 Structure D objects: Uncle-Bill, Uncle-John, Aunt-Sally, The-woman-I-don't-like, Mary relations: Married, Divorced (Uncle-John, The-woman-I-don't-like)  Divorced (Uncle-Bill, Aunt-Sally)  Married, (Uncle-John, Mary)  Married (or: {(Uncle-Bill, Aunt-Sally), (Uncle-John, Mary)} = Married) Interpretation I I(Bill-1) = Uncle-Bill, I(John-3) = Uncle-John, I(Sally-1) = Aunt-Sally, I(Mary-1) = Mary, I(Mary-2) = The-woman-I-don't-like I(happy-together) = Married, I(hate-each-other) = Divorced True or false? hate-each-other (Bill-1, John-3)  hate-each-other (John-3, Mary-1) happy-together (Bill-1, Sally-1)  happy-together (John-3, Mary-2)  x: happy-together(Uncle-Bill, x))  x,y,z: happy-together(x,y)  hate-each-other (x,z) What if you want to add a formula?  x,y: happy-together(x,y)  happy-together(y,x)

Additional References Frost, Richard: Introduction to Knowledge Base Systems. Collins Professional and Technical Books, William Collins Sons & Co. Ltd, London, Nilsson, Nils J.: Artificial Intelligence - A new synthesis. Morgan Kaufmann Publishers, San Francisco, CA, 1998.