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Introduction to Logic and Prolog

Sabu Francis, B.Arch (Hons)

Turing’s discovery

Faced with the complex problem of decoding encrypted Nazi messages, Alan Turing invented the modern computer in the forties.

The breakthrough came when he realized that any algorithm can be carried out using a linear process involving instructions that manipulated symbols

Tracking the instruction pointer

While developing a program, programmers implicitly track the movement of the instruction pointer linearly through the code.

Algorithm = Logic + Control

If you look at the entire sequence of arrows, you will find that these arrows will always be stitched up together in one large unbroken sequence

In short, the programmer has to work on how the logic of the program is to be controlled

Can we separate logic from control?

If a programmer can disconnect the logic of the code from the way that logic is to be controlled …

The source code becomes smaller and the time spent stitching up the instruction pointer becomes lesser. In short: Fewer bugs!

Prolog = Programming in Logic

When using Prolog to solve problems, it is important to discard all notions of procedural programming

Issues such as stitching up the instruction pointer is no longer your concern. The Prolog system will do it for you. Just have faith, and worry only about the logic(and a few other things which we’ll come to later)But before we jump to Prolog ….

What is logic?

The words logic, logical, etc. has common interpretations. We need to be more rigorous than that and recognize that logic as used in Prolog is purely an important part of mathematics.

For example; sometimes the word logical is used to represent something that is carefully thought out – which is reasonable or acceptable to the world around.

All aspects of logic are equally useable

It is well known that the mathematical symbols “1”, “2”, “3”, etc are equally weighted. That means, none of them have “superiority” over the others.

Equivalently, in the field of logic; you would encounter concepts such as true, false, if->then, not, and, or … etc. None of them are more “superior” than the other. The context determines the outcome and they are all equally useful.

Logic is the tool for correct reasoning. However, not all logical operations may lead to scientifically valid conclusions. Or even things that are useful to us. It all depends on core premises that may have been made in the said operation.

For example, consider the following sequence of logical statements that constitute an “inference”:

a) Some real estate will increase in value.b)Anything that will increase in value is a good investment.c)Therefore, some real estate is a good investment.

Is the above inference right?

This inference is logically correct, because the conclusion “some real estate is a good investment” necessarily follows once we accept the premises “some real estate will increase in value” and “anything that will increase in value is a good investment”. Yet this same inference may not be a demonstration of its conclusion, because one or both of the premises may be faulty. Thus logic can help us to clarify our reasoning, but it can only go so far. The real issue in this particular inference is ultimately one of finance and economics, not logic.

If the above inference was used in Navi Mumbai during 1999, it would have been a disaster. Because the premise “some real estate will increase in value” was incorrect during that period. Many investors lost considerable money during that period because of this. Some even lost their lives!

Of all the sentences that are used by humans to yield meaning, logic concerns itself with only those that yield truth values. Such sentences are known as statementsor by some authors as propositions. The English teacher calls them declarative sentences.This may be the reason why Prolog is often called a declarative language.

Examples:

“The fruit is an apple” is a proposition(true if the indicated fruit is actually an apple, false otherwise)But

“Is the fruit an apple?” is not. It is just a question that humans ask, and logic by itself cannot provide an answer that results in a truth value. Similarly, “Oh my God!” is just an exclamation. No logic there.

All truth values are useful in reasoning

In arithmetic, we are allowed to write expressions like this:2+3 = 5 (and the maths has no clue what in the world does “5” represent. It cannot even determine whether that expression is useful to us)... and this expression would be incorrect,

2+3= 7

If we now regard 2+3=5 and 2+3=7 as two logicalstatements (i.e. From the field of logic and not arithmetic), then both are allowed. The truth value of the first statement is “true” and the second one “false”. Hence both are logical statements and therefore allowed in the field of logic.In fact, the field of mathematics is famous for using the principle of reductio-ad-absurdumwhere theorems are proved by inferring a premise that is logically false, which will make the starting premise null and void. (E.g. The theorem for proving the existence of an infinite number of primes)

Which of these are propositions?

1) X=X2) Everyone belongs here3) All humans fly4) Take a book from this pile

5) I am requesting you to take this book

A proposition contains a subject and a predicate. The subject is the thing for which the truth value is being determined. The predicateis something that is being told about the subject (property, mode of existence of the subject, etc.) When we examine the predicate in context of the subject we can determine the truth value of the proposition.

Aristotle gave three fundamental rules of predication:

a) A is A = The principle of identity(E.g. A baby boy will grow to become a man and not an oak!)b) A is not A is impossible = The principle of non-contradiction(E.g. An honest man cannot be a thief)c) Either A or non-A = The principle of Either-orA given predicate either belongs or does not belong to a given subject in a given respect at a given time. (E.g. A society must be either free or not free.)

The basic unit of reasoning is a syllogism, as per Aristotelian logic. For example, the real estate inference which was presented above is a syllogism. It is of the form:

a) Some A is B. b) All B is C. c) Therefore, some A is C.

Here A denotes real estate, B denotes increase in value, and C denotes a good investment. Just as in the case of this example, every syllogism consists of two premises and one conclusion. The premises and conclusion are all propositions (i.e. Statements yielding truth values).

All propositions fall in 4 classes

Each of the premises and the conclusion is of one of four types:

a) universal affirmative:All A is B.

b) universal negative:No A is B.

c) particular affirmative:Some A is B.

d) particular negative:Some A is not B

The letters A, B, C are known as terms. Every syllogism contains three terms. The two premises always share a common term which does not appear in the conclusion. This is known as the middle term. In our real estate example, the middle term is B, i.e., that which increases in value.

pons asinorum: Bridge of Asses

In order to classify the various types of syllogisms, one must understand that certain premises are symmetric and therefore equivalent. In particular, “no A is B” and “no B is A” are equivalent, as are “some A is B” and “some B is A”. Furthermore, the order of the two premises in a syllogism does not matter. Allowing for these symmetries, we can enumerate a total of 126 possible syllogistic forms. Of these 126, only 11 represent correct inferences. For example, the form

all A is B, all B is C, therefore all A is Crepresents a correct inference, while

all A is B, all C is B, therefore some A is C does not.

Medieval thinkers developed ingenious mnemonics to aid in distinguishing the correct forms from the incorrect ones. This culminated in the famous pons asinorum (``bridge of asses''), an intricate diagram which illustrates all of the syllogistic forms by means of a contrast between the good and the pleasurable.

The first hurdle in logical reasoning

Apart from using the wrong syllogism for inferences, classifying propositions into the wrong category is an even more basic issue that affect our reasoning.

For example; we sometimes put a proposition into the universal affirmative instead of the particular affirmative.

This often happens in inductive arguments, where we take an empirical observation and assume that it is a universal truth.

This is a classification error, which can lead to serious problems whether it is writing logical code using Prolog or simply ... arguing!

Often simple propositions may not be sufficient to declare a premise. We may need to stitch up several propositions into one compound proposition. This is done using connectives.

1. Conjunctions- (P and Q) - for proposition to be true both P and Q must be true, if either of them is false, proposition is false.

2. Disjunctions- (P or Q) - for proposition to be true only one of the component simple statements need be true.

3. Negations- (NOT P) - in this type of proposition the simple proposition is modified so that the truth value is reversed.

4. Conditionals- (IF P, THEN Q) - in this proposition, the first part is identified as the antecedent, and the second part as the consequent. For this statement to be true, the condition expressed for the truth of Q must not be compromised. The only case when this happens is when P is true and Q is false.5. IF_AND_ONLY_IF or equivalence.

There are symbols that are used as short-forms for the above connectives.

Each compound proposition can be examined against a truth-table to determine the truth value of the compound proposition.

For example; the truth-table for the orconnective is :

T or T = TT or F = T

F or T = T

F or F = F

Other truth-tables can be looked up from any reference book.

Misunderstanding the truth table of a compound proposition is another common source of errors when interpreting premises.

In 1879 the German philosopher Gottlob Frege gave a more powerful logical reasoning system that lead to the development of predicate logic. It overcame some of the problems in representing logical issues using propositional logic.

In the 1970's, the French mathematician, Colmereur developed Prolog that used predicate logic and its extensions to develop a natural language processor. It had a logic control system built-in so the programmer did not have to worry about the controlportion of any algorithm.

But what were the shortcomings of propositional logic?

Shortfalls of Propositional logic

a) A proposition talks about the subject and its predicate. Both have to be known before hand. So propositions cannot be used in all logical reasoning that humans are involved in.For example, the assertion "x is greater than 1", where x is a variable, is not a proposition because you can not tell whether it is true or false unless you know the value of x. Thus the propositional logic can not deal with such sentences. However, such assertions appear quite often in mathematics and we want to carry out inferences on those assertions.

Shortfalls of Propositional logic

b) Propositional logic cannot be used for some equivalence determination. For example; the pattern involved in the following logical equivalences can not be captured by the propositional logic: "Not all birds fly" is equivalent to "Some birds don't fly". "Not all integers are even" is equivalent to "Some integers are not even". "Not all cars are expensive" is equivalent to "Some cars are not expensive"

If only propositional logic is used and an inference is sought, each of these equivalences must be listed individually and acted upon. A general formula that covers all these equivalences collectively and instantiating it as they become necessary would have been a better approach ... and that is exactly what predicate logic gives!

Predicates as proposition builders

We had learnt earlier that a predicate describes a property of objects, or a relationship among objects.In predicate logic, the predicate acts as a verb template to which the subject and objects are given as arguments.

Now let us use a predicate as a template; something that builds a proposition as required:

For example, the propositions "The car Tom is driving is blue", "The sky is blue", and "The cover of this book is blue" come from the template "is blue" by placing an appropriate noun/noun phrase in front of it. The phrase "is blue" is a predicate and it describes the property of being blue.

Predicates are often given a name. For example any of "is_blue", "Blue" or "B" can be used to represent the predicate "is blue" among others. If we adopt B as the name for the predicate "is_blue", sentences that assert an object is blue can be represented as "B(x)", where x represents an arbitrary object. B(x) reads as "x is blue".

Just like propositions can be stitched up together to form compound propositions using one of the 5 logical operators, similarly simple predicates can also be stitched up.

A simple predicate like the is_blue ( or Bx) predicate is also known as an atomic formula.

If A is an atomic formula, and so is B thena) not(A)b) A and Bc) A or Bd) if A then Be) A is equivalent to Bare known as well formed formula or wff… and obviously, an atomic formula is also well-formed.

A predicate with variables is not a proposition. For example, the statement x > 1 with variable x over the universe of real numbers is neither true nor false since we don't know what x is. It can be true or false depending on the value of x. For x > 1 to be a proposition either we substitute a specific number for x or change it to one of two types:“There is a number x for which x > 1 holds” or “For every number x, x > 1 holds.”

- More generally, a predicate with variables can be made a proposition by applying one of the following two operations to each of its variables:
- assign a value to the variable – a process called bindingin Prolog
- quantify the variable using a quantifier

In general, a quantification is performed on formulas of predicate logic (called wff ), such as x > 1 or P(x), by using quantifiers on variables. There are two types of quantifiers: universal quantifier and existential quantifier.

The universal quantifier turns, for example, the statement x > 1 to "for every object x in the universe, x > 1", which is expressed as " xx > 1". This new statement is true or false in the universe of discourse. Hence it is a proposition once the universe is specified.

Similarly the existential quantifier turns, for example, the statement x > 1 to "for some object x in the universe, x > 1", which is expressed as " x x > 1." Again, it is true or false in the universe of discourse, and hence it is a proposition once the universe is specified.

Quantifiers

Predicates using Prolog syntax

Predicate logic will now onwards be described using the language Prolog, rather than the syntax adopted in mathematics books.

The objective of Prolog is to declare a set of predicates that describe the logical reasoning required to derive an inference. The “is_blue” predicate described earlier can be written down as follows:is_blue(X)

If you note, the variable x is now capitalized. This is how Prolog distinguishes a variable from other parts of the program. Any word with an initial letter capitalized is a variable.

The above predicate can act as proposition builder, provided we give some more information to it. This is done via a horn-clause ...

If we want the template

is_blue(X)

to generate propositions, we need to give more information. What may X stand for? This is done using horn clauses thus:

is_blue(X):- X = “Tom's Car”.

Note the :- that demarcates the headofthe horn-clause from the body. Also, note the full-stop at the end of the horn-clause.

The statement X = “Tom's car” is itself a declaration which states that X is no longer a variable, but it has now got bound to the value “Tom's car”.

Though in this example there is only one statement, the clause body can describe a compound set of predicate calls stitched up by conjunctions. The syntax for the and conjunction is the plain ole’ comma.If a horn-clause becomes a wff (well formed formula) using exactly one clause, then it is known as a fact. Just as its real-world meaning, a fact is always true! The Prolog system need not investigate further. In fact, a set of factsis collectively known in as a database(some call it knowledgebase)

/* database has two facts*/

parent(fred, greta). /*fact*/ parent(greta, henry). /*fact*/

/*this is a predicate that has a clause body*/ grandparent(X, Z):-

parent(X, Y), /*and operator*/

parent(Y, Z).

A Prolog program can be roughly be described as follows:

It contains a set of facts, or a knowledgebase

It contains a set of horn-clauses (or rules) that determine how to make deductions

The program starts by a starting horn-clause which it checks whether it is true. In order to do that, it goes through a logical path which may involve examining other horn-clauses and the available process.

Side-effects would be simultaneously performed as each truth statement is evaluated.

At this point, we have given the Prolog system all that is needed to get a simple inference done.

After entering the horn-clause, we can now ask Prolog a question at the prompt (which in many Prolog systems is a question mark):? is_blue(X)At this point, you would see Prolog answering back like this:X= “Tom's car”trueThe first line contained the answer.The second line contained the truth value.(Some Prolog systems may indicate the number ofsolutions that were yielded on the second line)

This means Prolog generated propositions on the fly, using the horn-clauses given to it, and used those proposition to get an inference done.

This is not all. If we now give

? is_blue(“Tom's pants”)At this point, you would see Prolog answering back like this:false(some Prologsystems may answer “No solutions” or signal an error)

This means Prolog generated a proposition on the fly, using the horn-clauses given to it, and used that proposition to get an inference done. In this case, it was not calculating the value of X. But instead, it was using the value that we had supplied and it was checking if such a value existed as an argument in its proposition for is_blue. It could not find any, and therefore it returned false.

What has happened here, is that the simple three line program can be used by Prolog in multiple ways. This happens because a variable in Prolog, unlike in any other procedural language, can either be bound or free.In the first case, the horn-clause is_blue was being used where the variable X was free. (i.e. We did not supply any constant as an argument. Instead we passed on an “empty box” metaphorically speaking)

Prolog then filled that empty box and returned the bound variable back to us. Think of a bound variable as a box which has something inside it.

In the second example, the value “Tom's pants”was passed as an argument to the horn-clause. Prolog implicitly created a variable (a box), and filled it with the words “Tom's pants” (i.e. The variable became bound to “Tom's pants”).This time around, Prolog intelligently deduced that we do not want any value back from the system, but we want the system to determine if such an assertion was true.

It then went around making propositions form the horn-clause template, and found that is_blue(“Tom's pants”) can never be true. Hence it returned false. (Some Prolog systems may return an error)

This is one simple demonstration that shows how Prolog controls the program on your behalf.

Logic

A pause …

Our first Prolog program is ready. Now let us tackle some issues that we had left for later in the earlier slides. For example; we said that in a Prolog program we need to describe the logic and a few other things.

What are those?

from the world outside

from within the computer …

Problem space

Internal data

Algorithm

Logic

What is in our hands?

Not in our hands!

in our hands!

from the world outside

from within the computer …

Not in our hands!

Problem space

Internal data

Algorithm

Interpretation is in our hands!

Problem space

Internal data

One wrong interpretation can destroy your code!Critical thinking is a must at this step!

In architecture, they say: “God is in the details” (Mies Van De Rohe)In computer software, we should say: “God is in the Data”

Theories regarding representing the real word in a logically correct mathematical model has occupied the minds of most philosophers. Fortunately, much of the problem space that programmers encounter is not as complicated as those handled by philosophers! This is because programs written today are for extremely focussed problem spaces, which are actually quite small – even if the programmer may think otherwise. The area of interest to the programmer is also known as the universe of discourse or just universe in some literature.

In Prolog, the outside problem-space is handled by understanding the domains that are needed.

What is a domain?

The concept can be understood if we take a specific example:

When we say sin(X) what do we understand about X ?

When we say sin(X) what implications can we make about X ?X has to be an anglewhich will invariably lie between zero and 2pi. In short, we understand thatof the various things nice, beautiful, ugly, whatever that is out there in the whole wild, wide world, X can only be bound to values from one small corner of it. We therefore have clearly established the domain of X.

Another example: In the is_blue(X) predicate, the variable X happened to be belong to the domain of strings.

It is up to us as programmers to clearly delineate which domain a particular variable ought to belong to, and use that consistently in all the horn-clauses that we set-up in the Prolog system.

Technically speaking Prolog is loosely typed. This means, we do not have to specify the domain for a variable. It remembers the variable type the first time it was used in a horn-clause, and within that horn-clause that data type should not change, else the horn-clause will become false.

Note: In some Prolog systems (e.g. Visual Prolog) the variables are strongly typed, and programmers do have to specify the domain type in a “header” file before a predicate is used. However, they have given some other flexibility that can more or less get back the advantages of loosely typed languages.

How we wish a problem-space can be defined purely by simple data types like strings, integers, etc.

Unfortunately, data representation is not as simple as that. How do we represent stuff which have multiple-components?

For e.g. If we wanted a data type in C that represented the name of a person, the gender and the religion and the person’s age; we could use this typedef of a struct:

typedef struct {

char name [30];

char gender[2];

char religion[30]; int age; } aPersonType ;And whenever we want to use a variable, we can simply use it thus:

main (void) { aPersonType thePerson; /*…. */ }

Complex domains are represented in Prolog using functors.Syntatically, a functor is written quite similarly to a predicate.The earlier “C” example can be written down in Prolog thus:

… X=aperson(Name,Religion,Gender,Age)…

The above is part of a horn-clause, where a variable X is bound to the stated functor.

Note two differences from C: a) We “made up” the functor the moment we felt the need for it – right in the middle of our “code” (not in any header file) and b) There are variables (words starting with capital letters) within the functor which are positionally recognized to the actual real-world data they represent. We do not need struct indirection (using the –> operator ) to use the components of the complex domain. We can simply use the internal functor variables themselves.

Does Prolog really manage control?

There are four important concepts in Prolog that allow it to control the execution of a program

Binding,

unification,

backtracking,

fail

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