University of Aberdeen, Computing Science CS3518 Lecture on Models and Satisfiability

1. University of Aberdeen, Computing ScienceCS3518Lecture on Models and Satisfiability Further details: K.Rosen, Discrete Mathematics & Its Applications(any edition), chapter on logic Michael Frank / Kees van Deemter

2. These slides are not meant as a general-purpose introduction, but as preparation for • a proof that monadic Predicate Logic is decidable • a proof that full Predicate is undecidable • The slides contain questions Q1, Q2, etc. These questions will be addressed in the Practical Michael Frank / Kees van Deemter

3. First Propositional Logic Michael Frank / Kees van Deemter

4. The syntax of propositional logic • Atoms: p1, p2, p3, .. • Formulas: • All atoms are formulas • For all  , if  is a formula then ¬  is a formula • For all  and , if  and  are formulas then the following are formulas: (  ), (  ), (  ) (etc.) CS2013 Univ of Aberdeen

5. Subformulas • A formula that is a substring of a formula ψ is a subformula of ψ • Example: (p  q) has these subformulas: Michael Frank / Kees van Deemter

6. Subformulas • A formula that is a substring of a formula ψ is a subformula of ψ • Example: (p  q) has these subformulas:(p  q), (p  q),p, q, p, q Michael Frank / Kees van Deemter

7. Tautologies A tautology is a compound proposition that is trueno matter what the truth values of its atomic propositions are! Ex.p  p[What is its truth table?] CS2013 Univ of Aberdeen

8. Tautologies • When every row of the truth table gives T. • Example: p p T T FT F T TF CS2013 Univ of Aberdeen

9. Contradictions A contradiction is a compound proposition that is false no matter what! Ex.p  p [Truth table?] CS2013 Univ of Aberdeen

10. Contradictions • When every row of the truth table gives F • Example: p p T F FT F F TF CS2013 Univ of Aberdeen

11. Propositional Equivalence Two syntactically (i.e., textually) different compound propositions may be semantically identical (i.e., have the same meaning). Here semantically identical means just that they have the same truth table for input truth values of the propositions. We call them logically equivalent. Notation: …  … CS2013 Univ of Aberdeen

12. Logical Equivalence Compound proposition p is logically equivalent to compound proposition q, written pq, IFFp and q contain the same truth values in all rows of their truth tables CS2013 Univ of Aberdeen

13. Proving Equivalencevia Truth Tables Ex. Prove that pq  (p  q). Q1 CS2013 Univ of Aberdeen

14. Proving Equivalencevia Truth Tables Ex. Prove that pq  (p  q). F T T T F T T F F T T F T F T T F F F T Shows that OR is equivalent to a combination of NOT and AND. CS2013 Univ of Aberdeen

15. Logical consequence • Formula φ has formula ψ as a logical consequence if ψ “follows from φ” • Formally: every assignment of truth values that makes φ true makes ψ true as well • Notation: φ |= ψ • φ and ψ are logically equivalent iff φ |= ψ and ψ |= φ Michael Frank / Kees van Deemter

16. Example of one-sided logical consequence Ex. Prove that p&q p Q2 CS2013 Univ of Aberdeen

17. Example of one-sided logical consequence Ex. Prove that p&q p Each row that has T for p&q has T for p . CS2013 Univ of Aberdeen

18. Logical consequence Observe: φ |= ψ iff φ & −ψ is a contradiction Michael Frank / Kees van Deemter

19. Topic #3 – Predicate Logic First-Order Predicate logic (FOPL) • An extension of Propositional Logic • Sometimes called quantificational logic, because it allows us to quantify over objects in a domain • E.g. saying that all/some objects have a property • Some odd numbers are prime • Futher extensions exist, e.g. second-order logic, which quantifies over properties, e.g. • 4 and 2+2 share all their semantic properties Michael Frank / Kees van Deemter

20. Topic #3 – Predicate Logic Predicates/relations with n places • Full FOPL permits predicates of any finite number of arguments. E.g.: S(x) = “x is even” (1 arg.) R(x,y) = “x adores y” (2 arg.) P(x,y,z) = “x gave y the grade z” (3 arg.) Q(x,y,z,u)= “x*(y+z)=u ”(4 arg.) Michael Frank / Kees van Deemter

21. Syntax of predicate logic (for 1- and 2-place predicates) • Variable: x,y,z,… Constants: a,b,c,… • 1-place predicates: P,Q,… • 2-place predicates: R,S,… Atomic formulas: If  is a 1-place predicate and  a var or constant then () is an atomic formula. If  is a 2-place predicate and  and  are var or constants, (,) is an atomic formula. If each of  and  is a var or constant then = is an atomic formula Michael Frank / Kees van Deemter

22. Syntax of predicate logic (for 1- and 2-place predicates) (Wellformed) Formulas: • All atomic formulas are formulas • If  and  are formulas then , ( ), (), ( ) are formulas. • If  is a formula then x and x are formulas. (Likewise, y and y,and so on.) Michael Frank / Kees van Deemter

23. Topic #3 – Predicate Logic Meaning of Quantified Expressions First, informally: • xP(x) means for all x in the u.d., P holds. • x P(x) means there exist x in the u.d. (that is, 1 or more) such thatP(x) is true. Michael Frank / Kees van Deemter

24. Topic #3 – Predicate Logic Free and Bound Variables • An expression like P(x) is said to have a free variablex (i.e., x is not “defined”). • A quantifier (either  or ) operates on an expression having one or more free variables, and binds one or more of those variables, to produce an expression having one or more boundvariables. Michael Frank / Kees van Deemter

25. Topic #3 – Predicate Logic Example of Binding • P(x,y) has 2 free variables, x and y. • xP(x,y) has 1 free variable, and one bound variable. • An expression with zero free variables is called a sentence. Michael Frank / Kees van Deemter

26. Topic #3 – Predicate Logic Free variables, defined formally • The free-variable occurrences in an Atom are: all the variable occurrences in that Atom • The free-variable occurrences in  are: the free-variable occurrences in  • The free-variable occurrences in ( connective ) are: the free-variable occurrences in  plus the free-variable occurrences in  • The free-variable occurrences in xand xare: the free-variable occurrences in  except for all/any occurrences of x. Michael Frank / Kees van Deemter

27. Topic #3 – Predicate Logic A more precise definition of the truth/falsity of quantified formulas • Formulation is simplified somewhat because we assume that every object in the u.d. D has a `name` (i.e., a constant referring to it). • Notation: (x:=a) is the result of substituting all free occurrences of the variable x in  by the constant a Michael Frank / Kees van Deemter

28. Topic #3 – Predicate Logic A more precise definition • Let  be a formula. Then x is true in D if at least one expression (x:=a) is true in D, and false otherwise. • A simple example: = P(x) P(x) is a formula, hence x P(x) is true in D if at least one expression of the form P(a) is true in D, and false otherwise. Michael Frank / Kees van Deemter

29. Topic #3 – Predicate Logic Similarly for  • Let  be a formula. Then the propositionx is true in D if every expression of the form (x:=a) is true in D, and false otherwise. • A simple example: = P(x) P(x) is a formula, hence x P(x) is true in D if every expression P(a) is true in D, and false otherwise. Michael Frank / Kees van Deemter

30. Predicate logic • A formula of FOPL is True (False) with respect to a model • Model = domain of objects plus specification of the “meanings” of the predicates, e.g., by • listing explicitly which objects the predicate is true of Michael Frank / Kees van Deemter

31. Interactions between quantifiers and connectives Let the u.d. be parking spaces at UF. Let P(x) be “x is occupied.” Let Q(x) be “x is free of charge.” • x (Q(x)  P(x)) • x (Q(x)  P(x)) • x (Q(x) P(x)) • x (Q(x)  P(x)) Michael Frank / Kees van Deemter

32. I. Construct English paraphrases Let the u.d. be parking spaces at UF. Let P(x) be “x is occupied.” Let Q(x) be “x is free of charge.” • x (Q(x)  P(x)) • x (Q(x)  P(x)) • x (Q(x) P(x)) • x (Q(x)  P(x)) Michael Frank / Kees van Deemter

33. I. Construct English paraphrases • x (Q(x)  P(x)) Some places are free of charge and occupied • x (Q(x)  P(x)) All places are free of charge and occupied • x (Q(x) P(x)) All places that are free of charge are occupied • x (Q(x)  P(x)) For some places x, if x is free of charge then x is occupied Michael Frank / Kees van Deemter

34. II. Q3: Construct a model where 1 and 4 are true, while 2 and 3 are false Let the u.d. be parking spaces at UF. Let P(x) be “x is occupied.” Let Q(x) be “x is free of charge.” • x (Q(x)  P(x)) • x (Q(x)  P(x)) • x (Q(x) P(x)) • x (Q(x)  P(x)) Michael Frank / Kees van Deemter

35. II. Construct a model where 1 and 4 are true, while 2 and 3 are false • x (Q(x)  P(x)) (true for place a below) • x (Q(x)  P(x)) (false for places b below) • x (Q(x) P(x)) (false for place b below) • x (Q(x)  P(x)) (true for place a below) One solution: a model with exactly two objects in it. One object has the property Q and the property P; the other object has the property Q but not the property P. In a diagram: a: Q P b: Q not-P Michael Frank / Kees van Deemter

36. III. Q4: Construct a model where 1 and 3 and 4 are true, but 2 is false • x (Q(x)  P(x)) • x (Q(x)  P(x)) • x (Q(x) P(x)) • x (Q(x)  P(x)) Michael Frank / Kees van Deemter

37. III. Construct a model where 1 and 3 and 4 are true, but 2 is false • x (Q(x)  P(x)) • x (Q(x)  P(x)) • x (Q(x) P(x)) • x (Q(x)  P(x)) Here is such a model (using a diagram). It has just two objects in its u.d., called a and b: a: Q P b: not-Q P Michael Frank / Kees van Deemter

38. Formally what is a model? • A model M is a pair <D,I> where D is a set (an u.d.), and I is a suitable ‘interpretation’ function. Informally speaking, a given M defines one way of giving a concrete ‘meaning’ to the symbols (other than ) in your formulas. • If a is an individual constant then I(a)D • If P is a 1-place predicate then I(P)D • If R is a 2-place predicate then I(P) is a set of pairs (,) such that  D, and  D Michael Frank / Kees van Deemter

39. An example of a model • Suppose M = (D,I) where D={John,Mary,Bill} and I(j)=John, I(m)=Mary, I(b)=Bill, I(B)={John,Bill}I(G)={Mary}I(A)={(John,Mary),(Bill,Mary)} • Note: John is a person (i.e., a part of the world)j is an individual constant (i.e., a part of the language of predicate logic).j“names”John. Michael Frank / Kees van Deemter

40. A specially defined relation: = • Some things can only be expressed in FOPL if we add a 2-place relation “=“ • We use infix notition: a=b instead of =(a,b) • Other 2-place relations do not have a fixed interpretation (think of the relation A on the previous page) • However, = is always interpreted as equality Michael Frank / Kees van Deemter

41. Formally (where M = <D,I>) atomic formulas Here is the start of the new truth definition: • A formula of the form P(a) is true with respect to M iffI(a)  I(P). • A formula of the form R(a,b) is true with respect to M iff(I(a),I(b))  I(R). • A formula of the form = is true with respect to M iffI(a) = I(b) Michael Frank / Kees van Deemter

42. Formally (where M = <D,I>) formulas with propositional connectives • A formula of the form  is true wrt M iff  is false wrt M • A formula of the form  is true wrt M iff  is true wrt M or  is true wrt M or both • A formula of the form  is true wrt M iff  is true wrt M and  is true wrt M • A formula of the form   is true wrt M iff  is false wrt M or  is true wrt M or both Michael Frank / Kees van Deemter

43. Formally (where M = <D,I>) quantified formulas • A formula of the form x is true wrt M iff at least one expression (x:=a) is true wrt M. (More precisely: there is an a such that ...) • A formula of the form x is true wrt M iff all expressions (x:=a) are true wrt M. (More precisely: for all a, ...) • (End of truth definition.) Michael Frank / Kees van Deemter

44. models: some illustrations • Suppose M= (D,I) , where D={John,Mary,Bill}, andI(j)=John, I(m)=Mary, I(b)=Bill, andI(B)={John,Bill}I(G)={Mary}I(A)={(John,Mary),(Bill,Mary)} • Is B(j) true wrt M? Michael Frank / Kees van Deemter

45. models: some illustrations • Suppose M=( D,I) , where D={John,Mary,Bill}, andI(j)=John, I(m)=Mary, I(b)=Bill, andI(B)={John,Bill}I(G)={Mary}I(A)={(John,Mary),(Bill,Mary)} • Is B(j) true wrt M? Yes Michael Frank / Kees van Deemter

46. models: some illustrations • Suppose M=(D,I), where D={John,Mary,Bill}, andI(j)=John, I(m)=Mary, I(b)=Bill, andI(B)={John,Bill}I(G)={Mary}I(A)={(John,Mary),(Bill,Mary)} • Is B(j) true wrt M? Yes • Is B(j)B(b)A(m,m) true wrt M? Michael Frank / Kees van Deemter

47. models: some illustrations • Suppose M=(D,I), where D={John,Mary,Bill}, andI(j)=John, I(m)=Mary, I(b)=Bill, andI(B)={John,Bill}I(G)={Mary}I(A)={(John,Mary),(Bill,Mary)} • Is B(j) true wrt M? Yes • Is B(j)B(b)A(m,m) true wrt M? No Michael Frank / Kees van Deemter

48. models: some illustrations • Suppose M=(D,I), where D={John,Mary,Bill}, andI(j)=John, I(m)=Mary, I(b)=Bill, andI(B)={John,Bill}I(G)={Mary}I(A)={(John,Mary),(Bill,Mary)} • Is B(j) true wrt M? Yes • Is B(j)B(b)A(m,m) true wrt M? No • Q5: Is x(B(x)A(x,m)) true wrt M? Michael Frank / Kees van Deemter

49. models: some illustrations • Suppose M=(D,I), where D={John,Mary,Bill}, andI(j)=John, I(m)=Mary, I(b)=Bill, andI(B)={John,Bill}I(G)={Mary}I(A)={(John,Mary),(Bill,Mary)} • Is B(j) true wrt M? Yes • Is B(j)B(b)A(m,m) true wrt M? No • Is x(B(x)A(x,m)) true wrt M? Yes Michael Frank / Kees van Deemter

50. models: some illustrations x(B(x)A(x,m)) true wrt M iff all expressions B(a) A(a,m) are true wrt M. These are the following: B(j) A(j,m) B(b) A(b,m) B(m) A(m,m) Michael Frank / Kees van Deemter