Principles of Programming Languages Lecture 13: Paradigms: Logic Programming.

Transcription

1 Principles of Programming Languages Lecture 13: Paradigms: Logic Programming. Andrei Arusoaie 1 1 Department of Computer Science January 16, 2018

2 Outline Paradigms

3 Outline Paradigms Logic Programming

4 Paradigms Imperative Programming:

5 Paradigms Imperative Programming: Object Oriented Programming:

6 Paradigms Imperative Programming: Object Oriented Programming: Functional Programming:

7 Paradigms Imperative Programming: Object Oriented Programming: Functional Programming: Logic Programming

8 General facts about PLs

9 General facts about PLs We have seen PLs where the computations are based on:

10 General facts about PLs We have seen PLs where the computations are based on: state + modifiable variable + assignment

11 General facts about PLs We have seen PLs where the computations are based on: state + modifiable variable + assignment higher-order + recursion

12 General facts about PLs We have seen PLs where the computations are based on: state + modifiable variable + assignment higher-order + recursion Are there other approaches?

13 General facts about PLs We have seen PLs where the computations are based on: state + modifiable variable + assignment higher-order + recursion Are there other approaches? Yes

14 General facts about PLs We have seen PLs where the computations are based on: state + modifiable variable + assignment higher-order + recursion Are there other approaches? Yes: Logic programming

15 Some background

16 Some background Computation vs. Deduction

17 Some background Computation vs. Deduction Computation: = 10 (using some predefined rules, i.e., n + 0 = 0)

18 Some background Computation vs. Deduction Computation: = 10 (using some predefined rules, i.e., n + 0 = 0) Deduction: requires more than automatic application of rules

19 Some background Computation vs. Deduction Computation: = 10 (using some predefined rules, i.e., n + 0 = 0) Deduction: requires more than automatic application of rules, i.e., needs human ingenuity to complete

20 Some background Computation vs. Deduction Computation: = 10 (using some predefined rules, i.e., n + 0 = 0) Deduction: requires more than automatic application of rules, i.e., needs human ingenuity to complete Example: a 2 + b 2 = c 2

21 Some background Computation vs. Deduction Computation: = 10 (using some predefined rules, i.e., n + 0 = 0) Deduction: requires more than automatic application of rules, i.e., needs human ingenuity to complete Example: a 2 + b 2 = c 2... Pythagoras first proof

22 Some background: FOL

23 Some background: FOL Reminder:

24 Some background: FOL Reminder: Terms: f (t1,..., t n ) where f has arity n, and can contain variables

25 Some background: FOL Reminder: Terms: f (t1,..., t n ) where f has arity n, and can contain variables variables

26 Some background: FOL Reminder: Terms: f (t1,..., t n ) where f has arity n, and can contain variables variables Predicates: p(t 1,..., t n ) where p is a predicate of arity n

27 Some background: FOL Reminder: Terms: f (t1,..., t n ) where f has arity n, and can contain variables variables Predicates: p(t 1,..., t n ) where p is a predicate of arity n Formulas: φ := p(t 1,..., t n ) φ φ φ x.φ

28 Some background

29 Some background Reminder:

30 Some background Reminder: Inference rules: C 1 C 2... C n H RULENAME

31 Some background Reminder: Inference rules: C 1 C 2... C n H RULENAME Examples: nat(0) ZERO nat(n) nat(s(n)) SUCC

32 Some background Reminder: Inference rules: C 1 C 2... C n H RULENAME Examples: nat(0) ZERO even(0) EVEN0 nat(n) nat(s(n)) SUCC even(n) even(s(s(n))) EVENSUCC

33 Proofs Proof that even(s(s(s(s(0)))) holds: even(0) EVEN0 even(s(s(0))) EVENSUCC even(s(s(s(s(0))))) EVENSUCC

34 Proofs Proof that even(s(s(s(s(0)))) holds: even(0) EVEN0 even(s(s(0))) EVENSUCC even(s(s(s(s(0))))) EVENSUCC Proof strategies: goal-directed vs. forward-reasoning

35 Proofs Proof that even(s(s(s(s(0)))) holds: even(0) EVEN0 even(s(s(0))) EVENSUCC even(s(s(s(s(0))))) EVENSUCC Proof strategies: goal-directed vs. forward-reasoning Logic programming uses goal-directed: for the current goal, search the rule that might have been applied to arrive at this conclusion even(0) EVEN0 even(n) even(s(s(n))) EVENSUCC

36 Logic programming: general facts This paradigm is based on formal logic

37 Logic programming: general facts This paradigm is based on formal logic Programs are sets of sentences in the logical form of clauses (referred to as rules):

38 Logic programming: general facts This paradigm is based on formal logic Programs are sets of sentences in the logical form of clauses (referred to as rules): H : C1,..., C 2.

39 Logic programming: general facts This paradigm is based on formal logic Programs are sets of sentences in the logical form of clauses (referred to as rules): H : C1,..., C 2. H is the head of the clause

40 Logic programming: general facts This paradigm is based on formal logic Programs are sets of sentences in the logical form of clauses (referred to as rules): H : C1,..., C 2. H is the head of the clause C 1,..., C 2 is the body

41 Logic programming: general facts This paradigm is based on formal logic Programs are sets of sentences in the logical form of clauses (referred to as rules): H : C1,..., C 2. H is the head of the clause C 1,..., C 2 is the body H. denotes a clause without body, a.k.a. facts

42 Logic programming: general facts This paradigm is based on formal logic Programs are sets of sentences in the logical form of clauses (referred to as rules): H : C1,..., C 2. H is the head of the clause C 1,..., C 2 is the body H. denotes a clause without body, a.k.a. facts Logic programming can be seen as controlled deduction

43 Logic programming: general facts This paradigm is based on formal logic Programs are sets of sentences in the logical form of clauses (referred to as rules): H : C1,..., C 2. H is the head of the clause C 1,..., C 2 is the body H. denotes a clause without body, a.k.a. facts Logic programming can be seen as controlled deduction Logic programming languages (or families of PL): Prolog, ASP (Answer set programming), Datalog

44 Logic programming: how it works Prolog: early 1970, Alain Colmerauer and Robert Kowalski Demo in Prolog: goals. nat(0) ZERO nat(n) nat(s(n)) SUCC

45 Logic programming: how it works Prolog: early 1970, Alain Colmerauer and Robert Kowalski Demo in Prolog: goals. nat(0) ZERO nat(n) nat(s(n)) SUCC even(0) EVEN0 even(n) even(s(s(n))) EVENSUCC Proof search: goal-directed

46 Substitutions σ = {x 3, y y, z f (a, b)} - substitution

47 Substitutions σ = {x 3, y y, z f (a, b)} - substitution Let T 1 = g(h(z, y), x, y) be a term

48 Substitutions σ = {x 3, y y, z f (a, b)} - substitution Let T 1 = g(h(z, y), x, y) be a term Then, σ(t 1 ) = σ(g(h(z, y), x, y)) = g(h(f (a, b), y ), 3, y )).

49 Unification When proving, the conclusion of an inference rule matches the current goal.

50 Unification When proving, the conclusion of an inference rule matches the current goal. How? Unification A substitution σ is the unifier of t 1 and t 2 if σ(t 1 ) = σ(t 2 ). Various algorithms for unification (Robinson,... )

51 Answer substitutions Unification variables in goals, i.e, even(s(s(x)).

52 Answer substitutions Unification variables in goals, i.e, even(s(s(x)). Demo in Prolog: queries. even(0) EVEN0 even(n) even(s(s(n))) EVENSUCC

53 Answer substitutions Unification variables in goals, i.e, even(s(s(x)). Demo in Prolog: queries. even(0) EVEN0 even(n) even(s(s(n))) EVENSUCC Other aspects:

54 Answer substitutions Unification variables in goals, i.e, even(s(s(x)). Demo in Prolog: queries. even(0) EVEN0 even(n) even(s(s(n))) EVENSUCC Other aspects: Backtracking: more than one possible rule applicable

55 Answer substitutions Unification variables in goals, i.e, even(s(s(x)). Demo in Prolog: queries. even(0) EVEN0 even(n) even(s(s(n))) EVENSUCC Other aspects: Backtracking: more than one possible rule applicable Subgoal order: which of the premises to consider?

56 Demo Light meal problem Facts: meat(steak, 10). meat(pork, 12). fish(tuna, 2). fish(trout, 4). starter(salad, 1). starter(soup, 3). desert(fruit, 1). desert(cake, 10). desert(icecream, 4).

57 Demo Light meal problem Main course: main_course(n, K) :- meat(n, K);fish(N, K).

58 Demo Light meal problem Main course: main_course(n, K) :- meat(n, K);fish(N, K). A simple meal: meal(s, M, D) :- starter(s, SK), main_course(m, MK), desert(d, DK).

59 Demo Light meal problem Main course: main_course(n, K) :- meat(n, K);fish(N, K). A simple meal: meal(s, M, D) :- starter(s, SK), main_course(m, MK), desert(d, DK). A light meal: light_meal(s, M, D) :- starter(s, SK), main_course(m, MK), desert(d, DK), SK + MK + DK < 10.

60 Resources Reading material: Chapter/Lecture 1: ~dg/teaching/lis2014/modules/lp-fp-07.pdf