CS 363 Comparative Programming Languages Logic Programming Languages

CS 363 Spring 2005 GMU2 Chapter 16 Topics An Overview of Logic Programming The Origins of Prolog The Basic Elements of Prolog Applications of Logic Programming

CS 363 Spring 2005 GMU3 Introduction Logic (declarative) programming languages express programs in a form of symbolic logic Runtime system uses a logical inferencing process to produce results Declarative rather that procedural: –only specification of results are stated (not detailed procedures for producing them)

CS 363 Spring 2005 GMU4 Example: Sorting a List Describe the characteristics of a sorted list, not the process of rearranging a list sort(old_list, new_list)  permute (old_list, new_list)  sorted (new_list) sorted (list)  for all j, 1  j < n, list(j)  list (j+1)

CS 363 Spring 2005 GMU5 Overview of Logic Programming Declarative semantics –There is a simple way to determine the meaning of each statement –Simpler than the semantics of imperative languages Programming is nonprocedural –Programs do not state now a result is to be computed, but rather the form of the result Based on Propositional logic

CS 363 Spring 2005 GMU6 Propositional Logic A proposition is formed using the following rules: Constants true and false are propositions Variables (p,q,r,…) which have values true or false are propositions Operators ^, v, , ≡, and ⌐ (conjunction, disjunction, implication, equilivance and negation) are used to form more complex propositions. p v q ^ r  ⌐ s v t

CS 363 Spring 2005 GMU7 Predicate Logic Expressions Include: –Propositions –Variables in domains such as integer, real –Boolean valued functions Ex: prime(n), 0 <= x + y, –Quantifiers (forall, exists)

CS 363 Spring 2005 GMU8 Properties of Predicate Logic Expressions PropertyMeaning Commutativityp v q ≡ q v pp ^ q ≡ q ^ p Associativity(p v q) v r ≡ p v (q v r )(p ^ q) ^ r ≡ p ^ (q ^ r ) Distributivity(p v q) ^ r ≡ (p ^ r ) v (q ^ r)(p ^ q) v r ≡ (p v r ) ^ (q v r) Idempotencep v p ≡ pp ^ p ≡ p Identity p v ⌐ p ≡ truep ^ ⌐ p ≡ false deMorgan⌐(p v q) ≡ ⌐p ^ ⌐q⌐(p ^ q) ≡ ⌐p v ⌐q Implication p  q ≡ ⌐ p v q Quantification⌐ forall x P(x) ≡ exists x ⌐ P(x)⌐ exists x P(x) ≡ forall x ⌐ P(x)

CS 363 Spring 2005 GMU9 Horn Clauses A Horn clause has a head h (which is a predicate), and a body, which is a list of predicates p 1, p 2, …, p n. Predicate p is true only if p 1, p 2, …, p n are all true. Meaning: p 1 ^ p 2 ^ … ^ p n  h Many, but not all predicates can be translated into Horn clauses using predicate logic expression properties.

CS 363 Spring 2005 GMU10 The Origins of Prolog University of Aix-Marseille –Natural language processing University of Edinburgh –Automated theorem proving

CS 363 Spring 2005 GMU11 The Basic Elements of Prolog Edinburgh Syntax Term: a constant, variable, or structure Constant: an atom or an integer Atom: symbolic value of Prolog Atom consists of either: –a string of letters, digits, and underscores beginning with a lowercase letter –a string of printable ASCII characters delimited by apostrophes

CS 363 Spring 2005 GMU12 The Basic Elements of Prolog Variable: any string of letters, digits, and underscores beginning with an uppercase letter Instantiation: binding of a variable to a value –Lasts only as long as it takes to satisfy one complete goal Structure: represents atomic proposition functor(parameter list)

CS 363 Spring 2005 GMU13 Fact Statements Used for the hypotheses (assumed information): Ex: student(jonathan). sophomore(ben). brother(tyler, cj).

CS 363 Spring 2005 GMU14 Rule Statements Give the rules of implication: –Consequence :- antecedent_expression Right side: antecedent (if part) –May be single term or conjunction –Conjunction: multiple terms separated by logical AND operations (implied) Left side: consequent (then part) –Must be single term

CS 363 Spring 2005 GMU15 Rule Statements Can use variables (universal objects) to generalize meaning: parent(X,Y):- mother(X,Y). parent(X,Y):- father(X,Y). sibling(X,Y):- mother(M,X), mother(M,Y),father(F,X), father(F,Y).

CS 363 Spring 2005 GMU16 Example mother(Susan,Emily). mother(Susan,Lucy). mother(Anne,Susan). mother(Anne,Mary). father(Jim,Emily). father(Jim,Lucy). father(Russell,Susan). father(Russell,Mary). parent(X,Y):-mother(X,Y). parent(X,y):-father(X,y). sibling(X,Y):-mother(M,X),mother(M,Y), father(F,X),father(F,Y). Facts Rules

CS 363 Spring 2005 GMU17 Example speed(ford,100). speed(chevy,105). speed(dodge,95). speed(volvo,80). time(ford,20). time(chevy,21). time(dodge,24). time(volvo,24). distance(X,Y) :- speed(X,Speed), time(X,Time), Y is Speed * Time. Facts Rules

CS 363 Spring 2005 GMU18 Goal Statements For theorem proving, theorem is in form of proposition that we want system to prove or disprove – goal statement Same format as facts student(james). Conjunctive propositions and propositions with variables also legal goals father(X,joe).

CS 363 Spring 2005 GMU19 Example mother(Susan,Emily). mother(Susan,Lucy). mother(Anne,Susan). mother(Anne,Mary). father(Jim,Emily). father(Jim,Lucy). father(Russell,Susan). father(Russell,Mary). parent(X,Y):-mother(X,Y). parent(X,y):-father(X,y). sibling(X,Y):-mother(M,X),mother(M,Y), father(F,X),father(F,Y). mother(Anne,Mary).  T mother(Anne,Emily).  No father(Jim,X).  X = Emily, X = Lucy Facts Rules Goals (queries)

CS 363 Spring 2005 GMU20 Simple Arithmetic Prolog supports integer variables and integer arithmetic is operator: takes an arithmetic expression as right operand and variable as left operand A is B / 10 + C Not the same as an assignment statement! For example, k is k + 1 is not solvable.

CS 363 Spring 2005 GMU21 Example speed(ford,100). speed(chevy,105). speed(dodge,95). speed(volvo,80). time(ford,20). time(chevy,21). time(dodge,24). time(volvo,24). distance(X,Y) :- speed(X,Speed), time(X,Time), Y is Speed * Time. Distance(ford,S). > S = 2000 Facts Rules Goal

CS 363 Spring 2005 GMU22 Inferencing Process of Prolog Queries are called goals If a goal is a compound proposition, each of the facts is a subgoal To prove a goal is true, must find a chain of inference rules and/or facts. For goal Q: B :- A C :- B … Q :- P Process of proving a subgoal called matching, satisfying, or resolution

CS 363 Spring 2005 GMU23 Inferencing Process Bottom-up resolution, forward chaining –Begin with facts and rules of database and attempt to find sequence that leads to goal –works well with a large set of possibly correct answers Top-down resolution, backward chaining –begin with goal and attempt to find sequence that leads to set of facts in database –works well with a small set of possibly correct answers Prolog implementations use backward chaining

CS 363 Spring 2005 GMU24 Inferencing Process When goal has more than one subgoal, can use either –Depth-first search: find a complete proof for the first subgoal before working on others –Breadth-first search: work on all subgoals in parallel Prolog uses depth-first search –Can be done with fewer computer resources

CS 363 Spring 2005 GMU25 Inferencing Process With a goal with multiple subgoals, if fail to show truth of one of subgoals, reconsider previous subgoal to find an alternative solution: backtracking Begin search where previous search left off Can take lots of time and space because may find all possible proofs to every subgoal

CS 363 Spring 2005 GMU26 Example mother(Susan,Emily). mother(Susan,Lucy). mother(Anne,Susan). mother(Anne,Mary). father(Jim,Emily). father(Jim,Lucy). father(Russell,Susan). father(Russell,Mary). parent(X,Y):-mother(X,Y). parent(X,y):-father(X,y). sibling(X,Y):-mother(M,X),mother(M,Y), father(F,X),father(F,Y). sibling(Emily,Y).  Y = Lucy Sibling(Susan,Anne). > no Facts Rules Goals (queries)

CS 363 Spring 2005 GMU27 sibling(Emily,Y). mother(M,Emily),mother(M,Y),father(F,Emily), father(F,Y). After searching the db: mother(Susan,Emily),mother(Susan,Y), father(Jim,Emily),father(Jim,Y). Searching again: mother(Susan,Emily),mother(Susan,Lucy), father(Jim,Emily),father(Jim,Lucy) So Y = Lucy

CS 363 Spring 2005 GMU28 sibling(Susan,Anne). mother(M,Susan),mother(M,Anne), father(F,Susan),father(F,Anne). If we choose M = Anne, we can’t resolve the second mother clause (similar problem with father)  fail

CS 363 Spring 2005 GMU29 List Structures Other basic data structure (besides atomic propositions we have already seen): list List is a sequence of any number of elements Elements can be atoms, atomic propositions, or other terms (including other lists) [apple, prune, grape, kumquat] [] ( empty list) [X | Y] ( head X and tail Y)

CS 363 Spring 2005 GMU30 Example Definition of member function: member(Elem,[Elem | _ ). member(Elem,[ _ | List]) :- member (Elem,List). Anonymous variable

CS 363 Spring 2005 GMU31 member(a,[b,c,d]). member(a,[b,c,d]) member(a,[c,d]) member(a,[d]) member(a,[])  fails

CS 363 Spring 2005 GMU32 Example Definition of append function: append([], List, List). append([Head | List_1], List_2, [Head | List_3]) :- append (List_1, List_2, List_3).

CS 363 Spring 2005 GMU33 How append works append([a,b],[c,d],X). append([a|b],[c,d],_). append([b],[c,d],_). append([],[c,d],[c,d]).  first rule! append([b],[c,d],[b,c,d]). append([a,b],[c,d],[a,b,c,d]). X = [a,b,c,d]

CS 363 Spring 2005 GMU34 Example Definition of reverse function: reverse([], []). reverse([Head | Tail], List) :- reverse (Tail, Result), append (Result, [Head], List).

CS 363 Spring 2005 GMU35 Cut Operator (!) Explicit control of backtracking ! Is a goal that always succeeds but cannot be resatisified by backtracking a,b,!,c,d – if a and b succeed, but c fails, the whole goal fails.

CS 363 Spring 2005 GMU36 Deficiencies of Prolog Resolution Order control –Prolog always starts at the beginning of DB when searching –Prolog always starts trying to resolve the leftmost subgoal (depth first) –  rule/definition order can affect performance!

CS 363 Spring 2005 GMU37 Deficiencies of Prolog Easy to write infinite loops (due to resolution order) –ancestor(X,X). –ancestor(X,Y) :- ancestor(Z,y),parent(X,Z). Reversing rules fixes the problem

CS 363 Spring 2005 GMU38 Deficiencies of Prolog Closed World assumption – Prolog can only prove things to be true (based on the info) but it cannot be used to prove a goal is false –Leads to a problem with negation

CS 363 Spring 2005 GMU39 Deficiencies of Prolog Problems with negation: Given: parent(bill,jake). parent(bill, shelley). sibling(X,Y) :- parent(M,X),parent(M,Y). For goal: sibling(X,y). Prolog will respond with: X=jake, Y=jake –sibling(X,Y) :- parent(M,X),parent(M,Y),not(X=Y).

CS 363 Spring 2005 GMU40 Deficiencies of Prolog Problems with negation: Need to write the rule using negation: –sibling(X,Y) :- parent(M,X),parent(M,Y),not(X=Y). However, must be done carefully – not(not(goal)) is not the same as goal.

CS 363 Spring 2005 GMU41 Trace Built-in structure that displays instantiations at each step Tracing model of execution - four events: –Call (beginning of attempt to satisfy goal) –Exit (when a goal has been satisfied) –Redo (when backtrack occurs) –Fail (when goal fails)

CS 363 Spring 2005 GMU42 Applications of Logic Programming Relational database management systems Expert systems Natural language processing Education

CS 363 Spring 2005 GMU43 Conclusions Advantages: –Prolog programs based on logic, so likely to be more logically organized and written –Processing is naturally parallel, so Prolog interpreters can take advantage of multi- processor machines –Programs are concise, so development time is decreased – good for prototyping