16
Logic Chapter Programming Languages
ISBN 0-321-19362-8
Chapter 16 Topics
Introduction A Brief Introduction to Predicate Calculus Predicate Calculus and Proving Theorems An Overview of Logic Programming The Origins of Prolog The Basic Elements of Prolog Deficiencies of Prolog Applications of Logic Programming
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Copyright 2004 Pearson Addison-Wesley. All rights reserved. Westmont College
Introduction
Logic programming language or declarative programming language Express programs in a form of symbolic logic Use a logical inferencing process to produce results Declarative rather that procedural:
only specification of results are stated (not detailed procedures for producing them)
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Introduction to Predicate Calculus
Proposition: a logical statement that may or may not be true
Consists of objects and relationships of objects to each other
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Introduction to Predicate Calculus
Symbolic logic can be used for the basic needs of formal logic:
express propositions express relationships between propositions describe how new propositions can be inferred from other propositions
Particular form of symbolic logic used for logic programming called predicate calculus
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Propositions
Objects in propositions are represented by simple terms: either constants or variables Constant: a symbol that represents an object Variable: a symbol that can represent different objects at different times
different from variables in imperative languages
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Propositions
Atomic propositions consist of compound terms Compound term: one element of a mathematical relation, written like a mathematical function
Mathematical function is a mapping Can be written as a table
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Propositions
Compound term composed of two parts
Functor: function symbol that names the relationship Ordered list of parameters (tuple)
Examples: student(jon) like(seth, OSX) like(nick, windows) like(jim, linux)
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Propositions
Propositions can be stated in two forms:
Fact: proposition is assumed to be true Query: truth of proposition is to be determined
Compound proposition:
Have two or more atomic propositions Propositions are connected by operators
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Logical Operators
Name negation conjunction disjunction equivalence implication Symbol Example a ab ab a b a b a b Meaning not a a and b a or b a is equivalent to b a implies b b implies a
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Quantifiers
Name universal existential
Example X.P X.P
Meaning For all X, P is true There exists a value of X such that P is true
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Clausal Form
Too many ways to state the same thing Use a standard form for propositions Clausal form: B1 B2 ... Bn A1 A2 ... Am means if all the As are true, then at least one B is true Antecedent: right side Consequent: left side
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Predicate Calculus and Proving Theorems
A use of propositions is to discover new theorems that can be inferred from known axioms and theorems Resolution: an inference principle that allows inferred propositions to be computed from given propositions
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Resolution
Unification: finding values for variables in propositions that allows matching process to succeed Instantiation: assigning temporary values to variables to allow unification to succeed After instantiating a variable with a value, if matching fails, may need to backtrack and instantiate with a different value
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Theorem Proving
Use proof by contradiction Hypotheses: a set of pertinent propositions Goal: negation of theorem stated as a proposition Theorem is proved by finding an inconsistency
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Theorem Proving
Basis for logic programming When propositions used for resolution, only restricted form can be used Horn clause - can have only two forms
Headed: single atomic proposition on left side Headless: empty left side (used to state facts)
Most propositions can be stated as Horn clauses
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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
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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) j such that 1 j < n, list(j) list (j+1)
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The Origins of Prolog
University of Aix-Marseille
Natural language processing
University of Edinburgh
Automated theorem proving
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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 characters ASCII delimited by apostrophes
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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)
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Fact Statements
Used for the hypotheses Headless Horn clauses
student(jonathan). sophomore(ben). brother(tyler, cj).
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Rule Statements
Used for the hypotheses Headed Horn clause Right side: antecedent (if part)
May be single term or conjunction
Left side: consequent (then part)
Must be single term
Conjunction: multiple terms separated by logical AND operations (implied)
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Rule Statements
parent(kim,kathy):- mother(kim,kathy).
Can use variables (universal objects) to generalize meaning:
parent(X,Y):- mother(X,Y). sibling(X,Y):- mother(M,X), mother(M,Y), father(F,X), father(F,Y).
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Goal Statements
For theorem proving, theorem is in form of proposition that we want system to prove or disprove goal statement Same format as headless Horn
student(james)
Conjunctive propositions and propositions with variables also legal goals
father(X,joe)
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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
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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
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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
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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
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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!
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