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Predicate Logic

Predicate Logic. Lecture Module 12. Limitation of PL. The facts like: “peter is a man”, “paul is a man”, “john is a man” can be symbolized by P, Q and R, respectively in PL. We can not draw any conclusions about similarities between P, Q and R.

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Predicate Logic

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  1. Predicate Logic Lecture Module 12

  2. Limitation of PL • The facts like: • “peter is a man”, “paul is a man”, “john is a man” can be symbolized by P, Q and R, respectively in PL. • We can not draw any conclusions about similarities between P, Q and R. • Better representations of these facts can be • Further, the sentences like “All men are mortal” can not be represented in PL. • MAN(peter), MAN(paul) and MAN(john). • Similarity between these facts that they are all man can be easily derived. • In Predicate Logic (logical extension of PL) such sentences can be easily represented. • The limitations of propositional logic are removed to some extent.

  3. Predicate Logic • It has three more logical notions as compared to PL. • Terms, Predicates and • Quantifiers • Term • a constant (single individual or concept i.e.,5, john etc.), • a variable that stands for different individuals • n-place function f(t1, …, tn) where t1, …, tn are terms. A function is a mapping that maps n terms to a term. • Predicate • a relation that maps n terms to a truth value true (T) or false (F). • Quantifiers • Universal or existential quantifiers i.e.  and  used in conjunction with variables.

  4. Examples • “x loves y” is represented as LOVE(x, y) which maps it to true or false when x and y get instantiated to actual values. • “john’s father loves john” is represented as LOVE(father(john), john). • Here father is a function that maps john to his father. • x is greater than y is represented in predicate calculus as GT(x, y). • It is defined as follows: GT( x, y) = T , if x  y = F , otherwise • Symbols like GT and LOVE are called predicates • Predicates two terms and map to T or F depending upon the values of their terms.

  5. Examples – Cont.. • Translate the sentence "Every man is mortal" into Predicate formula. • Representation of statement in predicate form • “x is a man” and “MAN(x), • x is mortal” by MORTAL(x) • Every man is mortal : (x) (MAN(x)  MORTAL(x))

  6. First Order Predicate Logic (FOPL) • Inference rules are added to Predicate Calculus. • In First Order Predicate Logic (FOPL), quantification is over variables. • Higher order Predicate Logic is one if quantification is on functions or predicates. • Using inference rules one can derive new formula using the existing ones. • Interpretations of Formulae in FOPL • In PL, an interpretation is simply an assignment of truth values to the atoms. • In FOPL, there are variables, so we have to do more than that. • An interpretation I for a formula  in FOPL consists of a non empty domain D and an assignment of values to each constant, function symbol and predicate symbol.

  7. Cont… • A formula  is said to be consistent (satisfiable) if and only if there exists an interpretation I such that I[] = T. • Alternatively we say that I is a model of  or I satisfies . • A formula  is said to be inconsistent(unsatisfiable) if and only if  no interpretation that satisfies  or there exists no model for . • A formula  is valid if and only if for every interpretation I, I[] = T. • A formula  is a logical consequence of a set of formulae {1, 2, ..., n } if and only if • for every interpretation I, if I[1  … n ] = T, then I[] = T.

  8. Prenex Normal Form • In FOPL, there are infinite number of domains and consequently infinite number of interpretations of a formula. • Therefore, unlike PL, it is not possible to verify a validity and inconsistency of a formula by evaluating it under all possible interpretations. • We will discuss the formalism for verifying inconsistency and validity in FOPL. • In FOPL, there is also a third type of normal form called Prenex Normal Form.

  9. Contd.. • A closed formula  of FOL is said to be in Prenex Normal Form(PNF) if and only if  is represented as (q1 x1) (q2 x2) … (qn xn) (M), where, qk , (1  k  n ) quant ( or ) and M is a formula free from quantifiers. • A list of quantifiers [(q1 x1) … (qn xn)] is called prefixand M is called the matrix of a formula . Here M is represented in CNF.

  10. Transformation of Formula into PNF • A formula can be easily transformed into PNF using various equivalence laws. • Following conventions are used: •  [x]  a formula , which contains a variable x. •  a formula without a variable x. • q  Quantifier ( or  ).

  11. Equivalence Laws • In addition to the equivalence laws given for PL, following equivalence laws FOPL available. • (q x)  [x] V  (q x) ( [x] V  ) •  V (q x)  [x]  (q x) ( V  [x]) • (q x)  [x]  (q x) ( [x]  ) •  (q x)  [x]  (q x) ( [x]) • ~((x)  [x])  (x) (~ [x]) • ~((x)  [x])  (x) (~ [x])

  12. Skolemisation (Standard Form) • Prenex normal form of a formula is further transformed into special form called Skolemisation or Standard Form. • This form is used in resolution of clauses. • The process of eliminating existential quantifiers and replacing the corresponding variable by a constant or a function is called skolemisation. • A constant or a function is called skolem constant or function.

  13. Conversion of PNF to its Standard Form • Scan prefix from left to right. • If q1 is the first existential quantifier then • choose a new constant c D & Matrix. Replace all occurrence of x1 appearing in Matrix by c and delete (q1 x1) from the prefix to obtain new prefix and matrix. • If qr is an existential quantifier and q1….qr-1 are universal quantifiers appearing before qr , then • choose a new (r-1) place function symbol 'f ' D & Matrix. Replace all occurrence of xr in Matrix by f(x1, …, xr-1 ) and remove (qr xr). • Repeat the process till all existential quantifiers are removed from matrix.

  14. Cont… • Any formula  in FOPL can be transformed into its standard form. • Matrix of standard formula is in CNF and prefix is free from existential quantifiers. • Formula in standard form is expressed as: (x1)… (xn) (C1… Ck), where Ck ,(1 k m) is formula in disjunctive normal form. • Since all the variables in prefix are universally quantified, we omit prefix part from the standard form for the sake of convenience and write standard form as (C1… Ck). • The standard form of any formula is of the form (C1… Ck), where each Ci , (1  i  m) is a clause. • A clause Ciis a closed formula of the form (L1V … V Lm), where each Li is a literal with all the variables occurring in L1, …, Lm being universally quantified.

  15. Cont… • Let S = { C1, … ,Ck } be a set of clauses that represents a standard form of a formula . • S is said to be unsatisfiable (inconsistent) if and only if there  no interpretation that satisfies all the clauses of S simultaneously. • A formula  is unsatisfiable (inconsistent) if and only if its corresponding set S is unsatisfiable. • S is said to be satisfiable (consistent) if and only if each clause is consistent i.e.,  an interpretation that satisfies all the clauses of S simultaneously. • Alternatively, an interpretation I models S if and only if I models each clause of S.

  16. Resolution Refutation in Predicate Logic • Resolution method is used to test unsatisfiability of a set S of clauses in Predicate Logic. • It is an extension of resolution method for PL. • The resolution principle basically checks whether empty clause is contained or derived from S. • Resolution for the clauses containing no variables is very simple and is similar to PL. • It becomes complicated when clauses contain variables. • In such case, two complementary literals are resolved after proper substitutions so that both the literals have same arguments.

  17. Example • Consider two clauses C1 and C2 as follows: C1 = P(x) V Q(x) C2 = ~ P(f(x)) V R(x) • Substitute 'f(a)' for 'x' in C1 and 'a' for 'x' in C2 , where 'a' is a new constant from the domain, then C3 = P(f(a)) V Q(f(a)) C4 = ~ P(f(a)) V R(a) • Resolvent C of C3 and C4 is [Q(f(a)) V R(a)] • Here C3 and C4 do not have variables. They are called ground instances of C1 and C2 . • In general, if we substitute 'f(x)' for 'x' in C1 , then C'1 = P(f(x)) V Q(f(x)) • Resolvent C' of C'1 and C2 is [Q(f(x)) V R(x)] • We notice that C is an instance of C' .

  18. Theorems • Logical Consequence: L is a logical consequence of S iff {S ~L} = { C1,,… ,Cn , ~ L} is unsatisfiable. • A deduction of an empty clause from a set S of clauses is called a resolution refutation of S. • Soundness and completeness of resolution: There is a resolution refutation of S if and only if S is unsatisfiable (inconsistent). • L is a logical consequence of S if and only if there is a resolution refutation of S  {~L}. • We can summarize that in order to show L to be a logical consequence the of set of formulae {1,…,n }, use the procedure given on next slide.

  19. Procedure • Obtain a set S of all the clauses. • Show that a set S  { ~ L} is unsatisfiable i.e., • the set S  { ~ L} contains either empty clause or empty clause can be derived in finite steps using resolution method. • If so, then report 'Yes' and conclude that L is a logical consequence of S and subsequently of formulae 1, …, n otherwise report 'No'. • Resolution refutation algorithm finds a contradiction if one exists, if clauses to resolve at each step are chosen systematically. • There exist various strategies for making the right choice that can speed up the process considerably.

  20. Useful Tips • Initially choose a clause from the negated goal clauses as one of the parents to be resolved. • This corresponds to intuition that the contradiction we are looking for must be because of the formula to be proved . • Choose a resolvent and some existing clause if both contain complementary literals. • If such clauses do not exists, then resolve any pairs of clauses that contain complementary literals. • Whenever possible, resolve with the clauses with single literal. • Such resolution generate new clauses with fewer literals than the larger of their parent clauses and thus probably algorithm terminates faster. • Eliminate tautologies and clauses that are subsumed by other clauses as soon as they are generated.

  21. Logic Programming • Logic programming is based on FOPL. • Clause in logic programming is special form of FOPL formula. • Program in logic programming is a collection of clauses. • Queries are solved using resolution principle. • A clause in logic programming is represented in a clausal notation as A1, …, Ak B1,…, Bt , where Aj are positive literals and Bk are negative literals.

  22. Conversion of a Clause into Clausal Notation • A clause in FOPL is a closed formula of the form, L1V … V Lm where each Lk , (1  k  n) is a literal and all the variables occurring in L1, …, Lm are universally quantified. • Separate positive and negative literals in the clause as follows: (L1V … V Lm)  (A1 V … VAk V ~ B1 V …V~ Bt), where m = k + t, Aj, (1  j  k)are positive literals and Bj , (1  j  t) are negative literals

  23. Cont… (L1V … V Lm)  (A1 V … VAk V ~ B1 V …V~ Bt),  (A1 V … VAk ) V ~ (B1 …  Bt)  (B1 …  Bt)  (A1 V … VAk ) {since P  Q  ~ P V Q} • Clausal notation is written in the form : (A1V … VAk )  (B1 …  Bt) OR A1, …, Ak B1,…, Bt . Here Aj , (1  j  k) are positive literals and Bi , (1  i  t) are negative literals. • Interpretations of A  B and B  A are same. • In clausal notation, all variables are assumed to be universally quantified. • Bi , (1  i  t)(negative literals) are called antecedents and Aj , (1  j  k) (positive literals) are called consequents. • Commas in antecedent and consequent denote conjunction and disjunction respectively.

  24. Cont… • Applying the results of FOPL to the logic programs, • a goal G with respect to a program P (finite set of clauses) is solved by showing that the set of clauses P  { ~ G} is unsatisfiable or there is a resolution refutation of P  { ~ G}. • If so, then G is logical consequence of a program P. • There are three basic statements. These are special forms of clauses. • facts, • rules and • queries.

  25. Example • Consider the following logic program. GRANDFATHER (x, y)  FATHER (x, z) , PARENT (z, y) PARENT (x, y)  FATHER (x, y) PARENT (x, y)  MOTHER (x, y) FATHER ( abraham, robert)  FATHER ( robert, mike)  • In FOL above program is represented as a set of clauses as: S = { GRANDFATHER (x, y) V ~FATHER (x, z) V ~ PARENT (z, y), PARENT(x, y) V ~ FATHER (x, y), PARENT(x, y) V ~ MOTHER (x, y), FATHER ( abraham, robert), FATHER ( robert, mike) }

  26. Example • Let us number the clauses of S as follows: i.GRANDFATHER(x, y) V ~FATHER(x, z) V ~PARENT(z, y) ii. PARENT(x, y) V ~ FATHER (x, y) iii. PARENT(x, y) V ~ MOTHER (x, y), iv. FATHER ( abraham, robert) v. FATHER ( robert, mike) • Simple queries : • Ground Query Query:“Is abraham a grandfather of mike ?"

  27. Example – Cont… • Ground Query “Is abraham a grandfather of mike ?"  GRANDFATHER (abraham, mike). • In FOPL, ~GRANDFATHER(abraham, mike) is negation of goal { GRANDFATHER (abraham, mike). • Include {~goal} in the set S and show using resolution refutation that S  {~ goal} is unsatisfiable in order to conclude the goal. • Let ~ goal is numbered as ( vi) in continuation of first five clauses of S listed above. vi. ~ GRANDFATHER (abraham, mike) • Resolution tree is given on next slide:

  28. Resolution Tree ( i ) ( vi ) {x / abraham, y / mike} ( iv) ~ FATHER(abraham, z) V ~PARENT(z, mike) {z / robert} ~ PARENT (robert, mike) ( ii) ~ FATHER (robert, mike) (v) Answer: Yes

  29. Different Type of Queries • Non ground queries Query:“Does there exist x such that x is a father of robert ?” {Who is father of robert?}  FATHER (x, robert). Query: “Does there exist x such that abraham is a father of x?" {abraham is father of whom?}  FATHER (abraham, x). Query: “Do there exist x and y such that x is a father of y?" { who is father of whom?}  FATHER (x, y).

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