Prolog Theory

Under Construction

As previously introduced, Prolog is based on formal logic and mathematics (predicate calculus) and with this basis, the fundamental axioms of numbers and set theory assumed to be true.  Theorems are additional propositions which may be inferred from initial set.

Theorem Proving

2 Approaches

Forward Chaining – this is bottom up resolution where the system begins with facts/rules and attempts to find a sequence of matching propositions that lead to goal.  This works well when the number of possible answers is large as backward chaining introduced below would require a large number of matches to get thae answer.

Backward Chaining – this is top down resolution where the system begins with a goal and attempts to find set of matching propositions that lead to set of facts in database.  This is Prolog’s mechanism and works well when there is reasonably small set of candidate answers.

Propositions are logical statements that may or may not be true and they consist of objects and relationships of objects to each other.  These propositions are checked (through inference) to determine validity.  Propositions may contain:

Facts defined to be true

Queries a query is also called a goal and its truth is to be determined through matching and inference engine.  To prove goal is true the inference engine must find a chain of inference rules and facts in database to establish the validity of the goal.  Note if a goal is a compound proposition, each proposition called a subgoal.

Matching is the process of proving goals/subgoals through proposition matching (e.g. checking facts and instantiating variables).  Proving a subgoal/goal is called satisfying the subgoal/goal.

Variables assumed to be universally quantified (read as ‘FOR ALL’ ?) or existentially quantified ( read as ‘THERE EXISTS’ ?) if the variable appears only in body.  Example:

hasachild(X) :- parent(X, Y)  %may be read in two ways

1. For all X and Y, If X is a parent of Y then X has a child (you may think of this as beginning with the body/antecedent)

2. For all X, X has a child if there exists a Y such that (st.) X is a parent of Y (you may think of this as beginning with the goal/consequent)

Closed World Assumption

Prolog is based on a closed world assumption where Prolog only reports truths which may be proved using its database.  In other words, Prolog has no knowledge beyond what is contained in its database and this may be misleading. As an example, if there is insufficient information Prolog will report “False”.


Recursive programming is fundamental to Prolog programming and is required to solve complex tasks.




data objects

/       \

simple objects     structures

/ \

constants variables

/   \

atoms numbers






 PROCEDURE – set of clauses about logically related relations


Question/Query to prolog – sequence of 1 for more goals

Prolog tries to satisfy all goals (e.g. demonstrate the goal is true)

Goal is true (satisfiable) if it logically follows from facts and rules of program

If question contains variables prolog finds instantiations of particular objects to achieve goals

If conclusion proved, instantiated variables are shown

If conclusion can not be proved, prolog returns no


Prolog accepts facts and rules as set of axioms and tautologies

User’s questions are conjectured theorem

Prolog attempts to logically derive theorem from given axioms

Sequence of steps called proof sequence

Prolog finds proof sequence in inverse order

Starts with goals and using rules substitutes current goals with new goals

Until new goals happen to be simple facts


1. predecessor(X, Z) :- parent(X, Z).

goal is predecessor(tom, pat)

variables in rule instantiated as follows, X = tom, Z = pat

original goal predecessor(tom, pat) replaced by parent(tom, pat)

there is no clause in program whose head matches goal parent(tom, pat)

therefore goal fails

Prolog back tracks to original goal to try another alternative

Prolog attempts second clause


predecessor(X, Z) :-

parent(X, Y),

predecessor(Y, Z).


variables X and Z become instantiated as X = tom, Z = pat

Y is not instantiated yet

Top goal predecessor(tom, pat) replaced by two goals:

Parent(tom, Y),

Predecessor(Y, pat)

Prolog tries to satisfy them in order they are written

First one easy => matches one of facts and program

Matching forces Y to become instantiated to bob

Thus first goal satisfied remaining goal becomes

Predecessor(bob, pat)

Prolog tries to satisfy this goal using first rule

Notes this application of rule independent of previous application

Uses new set of variables

Predecessor(X’, Z’) :- parent(X’, Z’)

Head must match current goal predecessor(bob, pat)

X’ = bob, Z’ = pat

Current goal replaced by parent(bob,pat)

goal immediately satisfied => appears as fact in program

** note ** proof sequence looks like a tree

?- trace. % invokes trace mechanism and shows instantiations




Concerned only with relations defined by program

Declarative meaning determines what will be output of program



Determines how output is obtained

How are relations actually evaluated by PROLOG system



Prolog’s ability to work out its own procedural details considered an important advantage

Encourages programmers to consider declarative meaning of programs

independently of procedural meaning

Results of program are in principal determined by declarative meaning

Programmer concentrates on declarative meaning e.g. not distracted by execution of details



CTRL-C aborts execution

In conjunction with a on debug line


Prolog Logic Theory `



Proposition is logical statement that may or may not be true

Consists of objects and relationships of objects to each other

Formal logic developed to provide method describing propositions

Formally stated propositions then checked for validity


2 modes

FACT proposition stated and defined to be true

QUERY proposition stated and truth is to be determined




Objects in propositions represented by single terms

2 types




CONSTANT is symbol that represents an object


VARIABLE is symbol that may represent different objects at different times

Much closer to variable in mathematical formalism

than imperative programming language

Not associated with memory cell




Simplest proposition consisting of compound terms




One element of mathematical relation

Written in form that has appearance of mathematical function notation

Mathematical function is mapping

Represented as either

an expression

a table

a list of tuples


Composed of two parts

FUNCTOR function symbol that names the relation

ordered list of parameters

referred to as tuples



2 or more atomic propositions


Connected by logical connectors for operators

(Same manner as constructing logic expressions in imperative languages)




Quantifiers ?,?

Negation ¬

Conjunction ?

Disjunction ?

Equivalence ?

Implication ?


Variables appear in propositions only when introduced by quantifiers

2 quantifiers exist in predicate calculus


Scope extends to all attached propositions and may be extended with parentheses


UNIVERSAL (?X.P) meaning for all X, P is true

EXISTENTIAL (?X.P) meaning there exists a value for X to make P true

A period between X and P syntactically separates variable from proposition




?X.(woman(X) ? human(X))

States for any value X, if X is a woman, X is human


?X.(mother(mary,X)  ? male(X))

states there exists a value X such that Mary is mother of X

and X is male



There are many different ways of stating propositions that have same meaning


This could possibly lead to redundancy

Not a problem for logicians but it is a problem for automated/computerized system


Proposition in clausal form as following general syntax

B1 ? B2 ? …. ? Bn ? A1 ? A2 ? …? An

A’s and B’s are terms

If all A’s are true then at least 1 B is true

Note the similarity to prolog semantics




Existential quantifiers are not required


Universal quantifiers are implicit in variables in atomic propositions


No operators other than conjunction/disjunction required

(they also need not appear in order above)


all predicate calculus propositions may be algorithmically converted to clausal form



Right hand side of clausal form proposition



Left-hand side of clausal form

Consequence of antecedent





Inference rule that allows deferred propositions to be computed from given propositions

Provides method for automatic theorem proving

Devised/developed to be applied to propositions in clausal form

Allows discovery of new theorems inferred from known axioms and theorems




P1 ? P2

Q1 ? Q2

States P2 implies P1 and Q2 implies Q1

Suppose Q1 is equivalent to P2

If Q2 is true then Q1 and P2 are true and truth of P1 may be inferred

(now we know basis of axiomatic semantics)




Terms of left sides of two propositions ANDed together to make left side of new proposition

Same thing is done to get right side of new proposition

Terms that appear on both sides of new proposition removed from both sides




Process of determining useful values for variables




Temporary assigning of values to variables to allow unification

Often variables instantiated with value that fails

(Fails to complete subsequent required matching)

This requires the variable to be unbound and the procedural mechanism to BACKTRACK





PROLOG is declarative language (logic programming is declarative)


Program consists of declarations

Declarations are statements or propositions in symbolic logic


Adheres to strict Mathematical formalism

Given proposition may be concisely determined from statement itself

Logic programming has no side effects

Prolog is not procedural => describe form of result

imperative semantics requires 

examination of local declarations 

knowledge of scoping/referencing environment

execution trace to determine values of variables






Begins with goal and attempts to prove sub goals

Sub goal becomes new goal which in turn may be comprised of sub goals

Prolog uses left to right, depth first order of the evaluation to prove sub goals

One must be careful not to cause infinite loop

Programmer must be aware inferencing process




Searches complete sequence of propositions

Completes first subgoal before working on others

note this is a recursive definition/description

prolog designers chose depth first because requires fewer computer resources

Prolog fails as a pure logical programming paradigm

Depth first search requires proper ordering of clauses



Works on all sub goals in parallel

Requires substantial memory and computer resources




When system fails to show truth of subgoal

System abandons subgoal it could not prove

Unbinds instantiated variables and backtracks to reconsider previous proven sub goals

Creates new instantiations and new traversal of tree

Time consuming process




Theorem ? propositions

negate theorem -> GOAL

prove theorem by finding inconsistency in propositions -> HYPOTHESES

May not be efficient

Resolution is finite process if set of propositions is finite

Time required to find inconsistency in large database may be huge (large # of propositions)

Simplify resolution process using HORN CLAUSES




Special kind of propositions

2 forms

single atomic proposition on left side


empty left side


left side of clausal form called the head

right side of clausal form called the body


Under Construction

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