1. 74.406 Natural Language Processing - Formal Logic - Propositional Calculus/Logic (PropLog) First-Order Predicate Logic/Calculus (FOL or FOPL) Formal Language (Syntax of formulae; wff) Inference System Semantics through Interpretation Function

2. 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.

3. Propositional Calculus

4. Propositional Logic (PL) Propositional Logic: symbols for facts, statements (propositions) logical connectives AND, OR, NOT, ,  "Rules" – condition, consequence; implications Example: “Dog Mood” tongue_out  thirsty growl OR bark  angry ears_back AND tail_in  anxious

5. Propositional Logic - Syntax Propositional Logic (Symbols, Terminals):  propositional symbols P, p, Q, q, r,...  logical connectives , , , ,   brackets (, ) Inductive Definition of well-formed formulae (wff): 1.Propositional symbols P, Q,... are wffs. 2.If P is a wff, then also  (P). 3.If P and Q are formulae then also (P  Q); (P  Q); (P  Q); (P  Q)

6. Propositional Logic - Semantics assign truth values to atomic formulae (propositions) determine truth values for complex formulae (composed from basic propositions using connectives) pq p  qp  q ppp  q  p  q FFFFTTT FTFTTTT TFFTFFF TTTTFTT Truth table

7. Propositional Logic – Example Example: “Dog Moods” tongue_out  thirsty growl OR bark  angry ears_back AND tail_in  anxious Exercise: Set-up a truth-table for “Dog Moods” Write in the left-side columns the observable propositional symbols (growl, bark,...) and in the right columns the derived propositions (anxious,...).

8. Example, Exercise – Truth table Example: “Dog Moods” tongue_out  thirsty growl OR bark  angry ears_back AND tail_in  anxious tongthirstgrowlbarkgrowl  bark angryearstailears  tail anx FFFFFFFFFF FFTFTTTFFF TTFTTTFTFF TTTTTTTTTT

9. Example – Truth table for  Example: If I win the lottery, every CS420 student gets $1.000. I win the lottery  every CS420 student gets $1.000 pq p  q  p  q p  q FFTT FTTT TFFF TTTT

10. First-Order Predicate Logic Syntax and Semantics

11. Syntax of FOPL - Example Predicate SymbolsP, Q, married,... Function Symbolsf, g, father-of,... Variablesx, y, z,... ConstantsSally, John, block-1, c,... Connectives , , , ,  Quantifiers ,  Termsx, Sally, father-of (Sally) Sentencesmarried (Sally, John), P (c) (atomic, complex)  x: married (Sally, x),  x  y: P (x, y)  Q (x)  R (y)

12. FOPL as Formal Language: Symbols NT Non-Terminals Formula, atomic- Formula, complex- Formula, Term, Connective, Quantifier, Predicate, Function, Variable, Constant T Terminals Predicate SymbolsP, Q, married,..., T, F Function Symbolsf, g, father-of,... Variablesx, y, z,... ConstantsSally, block-1, c (Binary) Connectives , , ,  Negation Symbol  (Unary Connective) Quantifiers ,  Equality Symbol= Other Symbols(, ), :

13. FOPL as Formal Language - Rules 1 Non-terminal Rules Formula ::= complex-Formula | atomic-Formula | T | F atomic-Formula ::= Predicate (Term,...) | Term = Term complex-Formula ::= Formula Connective Formula | (Quantifier Variable)*... : Formula |  Formula | (Formula) Term ::= Function (Term,...) | Variable | Constant Terminal Rules Connective ::=  |  |  |  Quantifier ::=  |  Note: The Notation... in the rules above indicates a list, e.g. a sequence of Quantifier-Variable combinations, or of Terms.

14. General Production Rules Formula ::= complex-Formula | atomic-Formula | T | F complex-Formula ::= Formula Connective Formula | Quantifier Variable... : Formula |  Formula | (Formula) atomic-Formula ::= Predicate (Term,...) | Term = Term Term ::= Function (Term,...) | Variable | Constant Connective ::=  |  |  |  Quantifier ::=  |  Domain Specific Production Rules Predicate ::= brothers(_,_) | sisters(_,_) | is-mother-of(_,_) |... Function ::= gender(_) | age(_) |... Variable ::= x | y | Constant ::= Sally | John | Bill | Mary FOPL as Formal Language - Rules 2

15. Notes on FOPL Syntax The term well-formed formula (wff) is often used. equivalent to the term ‘sentence’. wffs are sentences if all their variables are bound by quantifiers. bound variable  x: married (Sally, x) open formula: a variable in the formula is not bound, it is free  x: married (Sally, x)  happy (y) closed formula: all variables in the formula are bound  x  y: married (x, y)  happy (x)  happy (y) scope of a quantifier: all occurrences of quantified variables in formulae until over-ruled by new quantifier

16. Equivalence of Formulae  x:  (x)   x:  (x)  x:  (x)   x:  (x)  x:  (x)   y:  (y)

17. Predicate Logic - Semantics An Interpretation function determines the semantics of Predicate Logic formulae. Based on a “Domain” or “Universe” which models “the world”, consists of a set of Individuals (Objects, Constants) with Relations (Roles, Relations, Predicates) among them and Functions (Features, Attributes, Functions). An Interpretation assigns values to terms and formulae: Termsconstants, variables, function-expressions Formulaepredicate expressions, formulae connected logical connectives, quantified formulae

18. FOPL: Semantics 1 Define the Semantics of FOPL: 1.Interpretation – Mapping of symbols of the formal language (predicates, functions, variables, constants) onto the modeled domain (formal: Domain, relational Structure, or Universe) 2.Valuation - Determine the bindings of variables 3.Constructive Semantics – Determine the semantics of complex expressions inductively based on the definition of the semantics of basic expressions Note: Simpler definitions of semantics exist without explicit Valuation function or explicit notation of the interpretation of predicates, functions, constants, and variables in the domain.

19. FOPL: Semantics 2 Interpretation constantsI(c)  D (0-ary function) predicates I(P)  D n for P n-ary predicate functions I(f)  D n →Dfor f n-ary function variablesI(x)  D (see valuation) ------------------------------------------------------------------- determine constructively based on syntax and above Interpretation: termI(t)  D sentenceI(α)  {T,F}

20. FOPL: Semantics 3 Interpretation term I(f(t 1,...,t n )) = I(f)(I(t 1 ),...,I(t n ))  D atomic sentenceI(P(t 1,...,t n )) = T if (I(t 1 ),...,I(t n ))  I(P) complex sentence I(  α) = T if I(α)=F  |  |  I(α  β) = T if I(α)=T and I(β)=T I(α  β) = T if I(α)=T or I(β)=T I(α  β) = T if I(α)=F or I(β)=T  |  I(  x: α(x)) = Tif I(α(x))=T for at least one d  I(x) I(  x: α(x)) = Tif I(α(x))=T for all d  I(x)

21. FOPL: Semantics 3b Formulae with multiple / nested quantifiers: Evaluate / Interpret formula from left to right / from outside to inside. I(  x: α(x)) = Tif I(α(x))=T for at least one d  I(x) I(  x: α(x)) = Tif I(α(x))=T for all d  I(x) Easier: Substitute x with constant c  C, and later use I(c) instead of I(x). Task: Interpret the following formulae:  x  y: P(x,y)  y  x: P(x,y) What is the difference between them?

22. FOPL: Semantics 4 Interpretation of open formulae and Satisfiability Regard complex sentence α with (free) variable x: α(x) choose valuation function and determine satisfiability: valuation function V: V(x) = d  D α(x)satisfiable if there is a valuation V with wrt I,VV(x)=d such that I(α(d))=T α(x)true / validif for every valuation V with wrt I V(x)=d, d  D I(α(d))=T

23. FOPL: Semantics 5 Model: Given a set of formulae  and a structure D with an Interpretation I, and a valuation V, then D is a model of  iff I(  ) = T for all  

24. FOPL: Semantics 6 Semantic-based consequence: Given a set of formulae  and a formula α, and an Interpretation I into a Structure D, we say that α is a logical consequence of  iff ifI(  ) = T for all   thenI(α) = T Notation:  |= α

25. FOPL: Inference System Inference in FOPL: Derive new formulae by syntactic manipulation of existing formulae (through applying inference rules): given a set of formulae  apply inference rule (based on  ) new formula α is derived; α is a Theorem of . add new formula to . The set of valid formulae is now  α. Notation:  |– α α is inferred or derived from .

26. FOPL: Axioms The start-set  for inferences in FOPL are the axioms of FOPL. Axioms describe the general features of a logic, and are always assumed to be valid formulae in this logic.

27. FOPL Axioms A1      A2      A3        A4(    )  ((    )  (    )) A5  x:  (x)   (y) A6  (x)   y:  (y) based on Frost (1986)

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

29. Inference Rule UG Universal Generalization Universal Generalization:  (x)  x:  (x) where  (x) is a formula containing the free variable x.

30. Inference Rules - Universal Quantifier Introduction Introducing the Universal Quantifier:    (x)  x:  (x)  (x) is a formula containing the free variable x, which is bound in the conclusion by the universal quantifier.

31. Inference Rules - Existential Quantifier Introduction Introducing the Existential Quantifier:  (x)    x:  (x)  (x) is a formula containing the free variable x, which is bound in the conclusion by the existential quantifier.

32. Inference Rules - UI Universal Instantiation:  x:  (x)  (c) where  (x) is any formula containing the quantified variable x, and  (c) is the same as formula  (x) but every occurrence of the variable x is substituted with the arbitrary constant c.

33. Inference Rules - EG Existential Generalization :  (c)  x:  (x) where  (c) is a formula containing the arbitrary constant c but not an unbound occurrence of x, and  (x) is the same formula as  (c) but with every occurrence of the constant c replaced by a variable x. (If x occurs unbound in , use other variable- name.)

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

35. FOPL: Semantics and Inference In First-Order Predicate Logic, there is a correspondence (regarding the truth status) between formulae derived through logical Inference and their semantic Interpretation. In other words: Any formula derived by inference* is true if and only if it is true in the semantic interpretation. Notation:  |– αiff  |= α * in a sound and complete inference system

36. Inference Systems - Soundness and Completeness Soundness An Inference System is sound iff if  |– α then  |= α Every formula which is derived by formal inference, is semantically true. Completeness An Inference System is complete iff if  |= α then  |– α Every formula which is semantically true can be derived by formal inference.

37. Semantics - Example 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 (or: {(Uncle-Bill, Aunt-Sally), (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)

38. Semantics and Inference -Example 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) = 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)  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)

39. Additional References Frost, Richard: Introduction to Knowledge Base Systems. Collins Professional and Technical Books, William Collins Sons & Co. Ltd, London, 1986. Nilsson, Nils J.: Artificial Intelligence - A new synthesis. Morgan Kaufmann Publishers, San Francisco, CA, 1998.