SECTION 1.5: LOGIC PROGRAMMING

Transcription

1 SECTION 1.5: LOGIC PROGRAMMING William DeMeo University of South Carolina February 7, 2013

2 SPECIFYING FACTS AND RULES We populate our Prolog database with facts using predicates, e.g., E(b, fi) E(b, fo) E(d, g) A(b) A(fi) A(fo) A(d) P(g)

3 SPECIFYING FACTS AND RULES We populate our Prolog database with facts using predicates, e.g., E(b, fi) E(b, fo) E(d, g) A(b) A(fi) A(fo) A(d) P(g) where b = bear fi = fish fo = fox d = deer g = grass E(x, y) means x eats y A(x) means x is an animal P(x) means x is a plant

4 SPECIFYING FACTS AND RULES We populate our Prolog database with facts using predicates, e.g., E(b, fi) E(b, fo) E(d, g) A(b) A(fi) A(fo) A(d) P(g) We can specify rules using wffs, e.g., E(y, x) A(x) Pr(x). where b = bear fi = fish fo = fox d = deer g = grass E(x, y) means x eats y A(x) means x is an animal P(x) means x is a plant This determines the elements of our domain that are prey (those x for which Pr(x) holds).

5 SPECIFYING FACTS AND RULES We populate our Prolog database with facts using predicates, e.g., E(b, fi) E(b, fo) E(d, g) A(b) A(fi) A(fo) A(d) P(g) We can specify rules using wffs, e.g., E(y, x) A(x) Pr(x). where b = bear fi = fish fo = fox d = deer g = grass E(x, y) means x eats y A(x) means x is an animal P(x) means x is a plant This determines the elements of our domain that are prey (those x for which Pr(x) holds). Prolog treats rules as universally quantified and uses universal instantiation to strip off universal quantifiers. The rule above is interpreted as ( y)( x)[e(y, x) A(x) Pr(x)]

6 HORN CLAUSES A Horn Clause is a wff consisting of predicates connected by disjunction,, where all but at most one predicate is negated. Example: P 1 P 2 P n Q (1.1)

7 HORN CLAUSES A Horn Clause is a wff consisting of predicates connected by disjunction,, where all but at most one predicate is negated. Example: P 1 P 2 P n Q (1.1) A Horn Clause specifies an implication. Indeed, by DeMorgan s Law, (1.1) is equivalent to which is equivalent to (P 1 P 2 P n) Q, (P 1 P 2 P n) Q.

8 HORN CLAUSES A Horn Clause is a wff consisting of predicates connected by disjunction,, where all but at most one predicate is negated. Example: P 1 P 2 P n Q (1.1) A Horn Clause specifies an implication. Indeed, by DeMorgan s Law, (1.1) is equivalent to which is equivalent to (Recall A B A B.) (P 1 P 2 P n) Q, (P 1 P 2 P n) Q.

9 HORN CLAUSES A Horn Clause is a wff consisting of predicates connected by disjunction,, where all but at most one predicate is negated. Example: P 1 P 2 P n Q (1.1) A Horn Clause specifies an implication. Indeed, by DeMorgan s Law, (1.1) is equivalent to which is equivalent to Example: The rule above, is specified as the Horn Clause (P 1 P 2 P n) Q, (P 1 P 2 P n) Q. E(y, x) A(x) Pr(x), [E(y, x)] [A(x)] Pr(x).

10 RESOLUTION The rule of inference used by Prolog is called resolution.

11 RESOLUTION The rule of inference used by Prolog is called resolution. Two Horn clauses in a Prolog database are resolved to a new Horn clause if one clause contains an unnegated predicate matching a negated predicate in the other. EXAMPLE The pair of Horn clauses A(a) [A(a)] B(b) is resolved by Prolog to B(b).

12 RESOLUTION The rule of inference used by Prolog is called resolution. Two Horn clauses in a Prolog database are resolved to a new Horn clause if one clause contains an unnegated predicate matching a negated predicate in the other. EXAMPLE The pair of Horn clauses A(a) [A(a)] B(b) is resolved by Prolog to B(b). This is just modus ponens, so Prolog s rule of inference includes modus ponens as a special case.

13 RESOLUTION EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [E(b, fi)]

14 RESOLUTION EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [E(b, fi)] resolves to [A(fi)] P(fi).

15 RESOLUTION EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [E(b, fi)] resolves to [A(fi)] P(fi). [A(fi)] P(fi) A(fi)

16 RESOLUTION EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [E(b, fi)] resolves to [A(fi)] P(fi). [A(fi)] P(fi) A(fi) resolves to P(fi). Conclusion: fish are prey.

17 RESOLUTION EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [E(b, fi)] resolves to [A(fi)] P(fi). [A(fi)] P(fi) A(fi) resolves to P(fi). Conclusion: fish are prey. EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [A(b)]

18 RESOLUTION EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [E(b, fi)] resolves to [A(fi)] P(fi). [A(fi)] P(fi) A(fi) resolves to P(fi). Conclusion: fish are prey. EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [A(b)] resolves to [E(y, b)] Pr(b).

19 RESOLUTION EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [E(b, fi)] resolves to [A(fi)] P(fi). [A(fi)] P(fi) A(fi) resolves to P(fi). Conclusion: fish are prey. EXAMPLE The pair of Horn clauses [E(y, x)] [A(x)] Pr(x) [A(b)] resolves to [E(y, b)] Pr(b)....but this time, when we search the database for E(a, b), for some a, we don t find it, so can t conclude that bears are prey.

20 RECURSION Prolog rules are implications. The antecedents may depend on facts or other rules.

21 RECURSION Prolog rules are implications. The antecedents may depend on facts or other rules. The antecedent of a rule may also depend on that rule itself, in which case the rule is defined in terms of itself. This is a recursive definition.

22 RECURSION Prolog rules are implications. The antecedents may depend on facts or other rules. The antecedent of a rule may also depend on that rule itself, in which case the rule is defined in terms of itself. This is a recursive definition. EXAMPLE Consider the binary relation in-food-chain(x, y), meaning y is in x s food chain. This means 1 x eats y, or 2 x eats something that eats something that eats something... that eats y; i.e., x eats z and y is in z s food chain: eats(x, z) in-food-chain(z, y)

23 RECURSION Case (1), x eats y, is simple to test, but without Case (2), in-food-chain(x, y) is no different from eat(x, y).

24 RECURSION Case (1), x eats y, is simple to test, but without Case (2), in-food-chain(x, y) is no different from eat(x, y). OTOH, without (1) we have a rule describing an infinitely descending food chain, which never terminates, and never resolves to True.

25 RECURSION Case (1), x eats y, is simple to test, but without Case (2), in-food-chain(x, y) is no different from eat(x, y). OTOH, without (1) we have a rule describing an infinitely descending food chain, which never terminates, and never resolves to True. Recursive definitions always need a stopping point.

26 RECURSION Case (1), x eats y, is simple to test, but without Case (2), in-food-chain(x, y) is no different from eat(x, y). OTOH, without (1) we have a rule describing an infinitely descending food chain, which never terminates, and never resolves to True. Recursive definitions always need a stopping point. The Prolog rule for in-food-chain: in-food-chain(x, y) if ( ) eat(x, y) or eat(x, z) and in-food-chain(z, y) This is a recursive rule because it defines the predicate in-food-chain in terms of itself.

27 EXPERT SYSTEMS Many interesting applications programs have been developed, in Prolog and similar logic programming languages, that gather a database of facts and rules about some domain and use this database to draw conclusions.

28 EXPERT SYSTEMS Many interesting applications programs have been developed, in Prolog and similar logic programming languages, that gather a database of facts and rules about some domain and use this database to draw conclusions. Such programs are known as expert systems, knowledge-based systems, or rule-based systems.

29 EXPERT SYSTEMS Many interesting applications programs have been developed, in Prolog and similar logic programming languages, that gather a database of facts and rules about some domain and use this database to draw conclusions. Such programs are known as expert systems, knowledge-based systems, or rule-based systems. The database in an expert system attempts to capture the knowledge, or elicit the expertise, of a human expert in a particular field. This includes both the facts known to the expert and the expert s reasoning path in reaching conclusions from those facts.