Monday, 25 September 2017

Fermat's Little Theorem.

Introduction.

Prime Numbers have uses in Cryptography, important part of Computer Sciences.

Functional Paradigm uses Mathematics for Computing extensively, therefore this article.


Fermat's Little Theorem.

If p is a prime number and a is a natural number, then:

ap≡a (mod p).

Furthermore, if p does not divide a, then there exists some smallest exponent d such that:

ad-1≡0 (mod p)

and d divides p-1. Hence,

a(p-1)-1≡0 (mod p).

The theorem is sometimes also simply known as 'Fermat's theorem' (Hardy and Wright 1979, p. 63).

This is a generalization of the Chinese hypothesis and a special case of Euler's totient theorem. It is sometimes called Fermat's primality test and is a necessary but not sufficient test for primality. Although it was presumably proved (but suppressed) by Fermat, the first proof was published by Euler in 1749. It is unclear when the term 'Fermat's little theorem' was first used to describe the theorem, but it was used in a German textbook by Hensel (1913) and appears in Mac Lane (1940) and Kaplansky (1945).

The theorem is easily proved using mathematical induction on a. Suppose p|a^p-a (i.e., p divides a^p-a). Then examine:

(a+1)p-(a+1).

From the binomial theorem,



Rewriting,



But p divides the right side, so it also divides the left side. Combining with the induction hypothesis gives that p divides the sum:



as assumed, so the hypothesis is true for any a. The theorem is sometimes called Fermat's simple theorem. Wilson's theorem follows as a corollary of Fermat's little theorem.

Fermat's little theorem shows that, if p is prime, there does not exist a base a < p with (a,p)=1 such that a(p-1)-1 possesses a nonzero residue modulo p. If such base a exists, p is therefore guaranteed to be composite. However, the lack of a nonzero residue in Fermat's little theorem does not guarantee that p is prime. The property of unambiguously certifying composite numbers while passing some primes make Fermat's little theorem a compositeness test which is sometimes called the Fermat compositeness test. A number satisfying Fermat's little theorem for some nontrivial base and which is not known to be composite is called a probable prime.

Composite numbers known as Fermat pseudoprimes (or sometimes simply 'pseudoprimes') have zero residue for some as and so are not identified as composite. Worse still, there exist numbers known as Carmichael numbers (the smallest of which is 561) which give zero residue for any choice of the base a relatively prime to p. However, Fermat's little theorem converse provides a criterion for certifying the primality of a number. A table of the smallest pseudoprimes P for the first 100 bases a follows (OEIS A007535; Beiler 1966, p. 42 with typos corrected).


aPaPaPaPaP
234122694220562638291
391233343776334183105
4152425444564658485
5124252845766511285129
63526274613366918687
7252765476567858791
892845484968698891
9282935496669858999
103330495051701699091
1115314951657110591115
12653233528572859293
1321338553657311193301
14153435545574759495
1534135515563759195141
165136915657767796133
1745374557657724797105
1825383958133783419899
194539955987799199145
20214091603418081100153
21554110561918185




































Source: Wolfram MathWorld.

Congruence.

Introduction.

Prime Numbers have uses in Cryptography, important part of Computer Sciences.

Functional Paradigm uses Mathematics for Computing extensively, therefore this article.


Congruence.

If two numbers b and c have the property that their difference b-c is integrally divisible by a number m (i.e., (b-c)/m is an integer), then b and c are said to be 'congruent modulo m.' The number m is called the modulus, and the statement 'b is congruent to c (modulo m)' is written mathematically as:





If b-c is not integrally divisible by m, then it is said that 'b is not congruent to c (modulo m),' which is written:



The explicit '(mod m)' is sometimes omitted when the modulus m is understood by context, so in such cases, care must be taken not to confuse the symbol ≡ with the equivalence sign.

The quantity b is sometimes called the 'base,' and the quantity c is called the residue or remainder. There are several types of residues. The common residue defined to be nonnegative and smaller than m, while the minimal residue is c or c-m, whichever is smaller in absolute value.



Congruence arithmetic is perhaps most familiar as a generalization of the arithmetic of the clock. Since there are 60 minutes in an hour, 'minute arithmetic' uses a modulus of m=60. If one starts at 40 minutes past the hour and then waits another 35 minutes, 40+35≡15 (mod 60), so the current time would be 15 minutes past the (next) hour.



Similarly, 'hour arithmetic' on a 12-hour clock uses a modulus of m=12, so 10 o'clock (a.m.) plus five hours gives 10+5≡3 (mod 12), or 3 o'clock (p.m.)

Congruences satisfy a number of important properties, and are extremely useful in many areas of number theory. Using congruences, simple divisibility tests to check whether a given number is divisible by another number can sometimes be derived. For example, if the sum of a number's digits is divisible by 3 (9), then the original number is divisible by 3 (9).

Congruences also have their limitations. For example, if a≡b and c≡d (mod n), then it follows that ax≡bx, but usually not that xc≡xd or ac≡bd. In addition, by 'rolling over,' congruences discard absolute information. For example, knowing the number of minutes past the hour is useful, but knowing the hour the minutes are past is often more useful still.

Let a≡a' (mod m) and b≡b' (mod m), then important properties of congruences include the following, where => means 'implies':

1. Equivalence: a≡b (mod 0)=>a≡b (which can be regarded as a definition).

2. Determination: either or .

3. Reflexivity: a≡a (mod m).

4. Symmetry: a≡b (mod m) => b≡a (mod m).

5. Transitivity: a≡b (mod m) and b≡c (mod m) => a≡c (mod m).

6. a+b≡a'+b' (mod m).

7. a-b≡a'-b' (mod m).

8. ab≡a'b' (mod m).

9. a≡b (mod m) => ka≡kb (mod m).

10. a≡b (mod m) => an≡bn (mod m).

11. a≡b (mod m1) and a≡b (mod m2) => a ≡b (mod [m1,m2]), where [m1,m2] is the least common multiple.

12. , where (k,m) is the greatest common divisor.

13. If a≡b (mod m), then P(a)≡P(b) (mod m), for P(x) a polynomial.

Properties (6-8) can be proved simply by defining:

a=a'+rm
b=b'+sm,

where r and s are integers. Then:

a+b=a'+b'+(r+s)m
a-b=a'-b'+(r-s)m
ab=a'b'+(a's+b'r+rsm)m,

so the properties are true.

Congruences also apply to fractions. For example, note that:

2×4≡1 3×3≡2 6×6≡1 (mod 7),

so:

1/2≡4 1/4≡2 2/3≡3 1/6≡6 (mod 7).

To find p/q (mod m) where (q,m)=1 (i.e., q and m are relatively prime), use an algorithm similar to the greedy algorithm. Let q0≡q and find:



where is the ceiling function, then compute:

q1≡q0p0 (mod m).

Iterate until qn=1, then:



This method always works for m prime, and sometimes even for m composite. However, for a composite m, the method can fail by reaching 0 (Conway and Guy 1996).

Finding a fractional congruence is equivalent to solving a corresponding linear congruence equation:

ax≡b (mod m).

A fractional congruence of a unit fraction is known as a modular inverse.

Source: Wolfram MathWorld.

Friday, 16 September 2016

Predicate Logic Basics.

Introduction.

This article joins the 'functional programming paradigm' with 'programming in logic', an another important & very well known programming paradigm.

We'll be using both a little of the 'Sets Theory', as well as 'Mathematical Analysis' & 'Logic'.


Predicates.

In mathematics, a predicate is commonly understood to be a Boolean-valued function P: X → {true, false}, called the predicate on X.


Predicate Language Syntax.

Logical formulas of the 'Predicate Calculus' are built from symbols, set of these symbols is called 'alphabet'.

To fully specify 'alphabet' we would need to include all of the symbols that belong to it.

For simplicity, we'll provide a definition that will specify only types of symbols, that might be included in 'alphabet'.


def. 1. A 'predicate calculus alphabet' consists of following symbols:
- constants,
- variables,
- function symbols,
- predicate symbols,
- logical functors,
- parentheses.


def. 2. For a given 'predicate calculus alphabet' the 'terms set' is defined as follows:
- each of constants is a 'term',
- each of variables is a 'term',
- if 'f' is a 'm-argument function symbol' and t1, t2, ... , tm are terms,
    then f(t1, t2, ... , tm) is a 'term'.


The 'terms' in the 'predicate calculus' are arguments of the 'predicate symbols'.

Predicate symbols assign each of the 'term argument' a boolean value from a set of {true, false}.

Assigning a 'predicate symbol' to a 'term argument' gives in a result the simplest of 'logical formulas', called also: 'atomic formula'.


def. 3. For each m-argument 'predicate symbol': 'R', if t1, t2, ... , tm are 'terms',
    then R(t1, t2, ... , tm) is an 'atomic formula'.


The 'complex formulas' are created by joining 'atomic formulas' using 'logical functors' and parentheses. Possibilities of creating 'complex formulas' are specified by the following definition.

def. 4. For a given 'alphabet', the 'predicate calculus formulas set' is defined as follows:
- each of the 'atomic formula' is a 'formula',
- for any 'formula' α, the (α) and ¬α are 'formulas',
- for any 'formulas' α and β, the α∧β, α∨β, α→β, α↔β are 'formulas',
- for any 'formula' α with a variable v, there are 'formulas': (∀v)α and (∃v)α.


Predicate Language Semantics.

Semantics for a 'predicate calculus' are defined by specifying a 'model', in which 'formulas' are interpreted.


def. 5. A 'model' is a pair <X, ι>, where X is a non-empty set called a 'domain' and ι is 'interpretation function' which:
- assigns for each of constants a certain 'domain' element,
- assigns for each of 'm-argument function symbol' a certain mapping from Xm on X,
- assigns for each of 'm-argument predicate symbol' a certain mapping from Xm on {true, false}.


In this context, predicates are treated as the 'relation names', for whose an 'interpretation function' assigns the 'relations on domain'.


def. 6. A 'valuation' of variables for a domain X is any function with a domain containing all of variables, with values in a set X.


For a given model and variables valuation, formulas can be valuated to specify their 'boolean value'.

We'll give only a definition for 'atomic formulas', for the 'complex formulas' can be specified using the known interpretation of the 'logical functors'.


def. 7. For a model <X,ι> and a variable valuation v, the values of the terms are specified as follows:
- for any constant x: VvX, ι(v) = ι(x);
- for any variable v: VvX, ι(v) = v(v);
- for any m-argument function symbol f and any terms t1, ... , tm:
     VvX, ι(f(t1, ... , tm)) = ι(v)(VvX, ι(t1), ... , VvX, ι(tm)).

Let's notice that a value of ι(v) is a function, with a value in a 'functions space', as defined in def. 5.

Then we interpret ι(v)(VvX, ι(t1), ... , VvX, ι(tm)) as a complex function composition.


def. 8. For a model <X,ι> and a variable valuation v,
    the value of an 'atomic formula' R(t1, ... , tm), where:
      R is an m-argument predicate symbol,
      t1, t2, ... , tm are terms,
        is specified as follows:
          VvX, ι(R(t1, ... , tm)) = ι(R)(VvX, ι(t1), ... , VvX, ι(tm)).

Again a value of ι(v) is a function, we have a complex function composition here as well.


Formulas' logical value are basis to specify: satisfiedness, satisfiability & truthfulness of the boolean formulas.


def. 9. Any formula α is satisfied in a model with a variable valuation v, when:
    VvX, ι(α) = 'true'.

def. 10. Any formula α is satisfiable, if there exists a model and a variable valuation, for which α is satisfied.

def. 11. Any formula α is true, when for any model and a variable valuation:
    α is satisfied.


Symetrically we define falsifiedness, falisfiability & falseness of the boolean formulas.

The 'falseness' property can also be defined using formulas' sets.

def. 12. Formulas set FS is false (self-contradictory), when conjunction of all formulas from this set FS is not satisfiable.


Entailment & Inference in a Predicate Calculus.

There are two types of consequences, as far as the 'entailment & inference in a predicate calculus' reaches:
- semantic consequence (logical consequence, entailment),
- syntactic consequence (inference).


def. 13. Formula α is a semantic consequence of a formulas set { β1, β2, ... , βm }, if for all models and variable valuations, for which formula β1 ∧ β2 ∧ ... ∧ βm is satisfied, formula α is also satisfied. We can note it as: { β1, β2, ... , βm } ⊧ α.


Syntactic consequence is defined, by assuming a certain inference system, consisting of a logical axioms set & of a inference rules set. Axioms are logical formulas, rules specify a way of deriving formulas from different formulas.


An inference rule noted as follows:



Means that based on formulas: β1, β2, ... , βm (that are premises), we can infer in a single step the formula α (that is a conclusion).


def. 14. A proof of a formula α based on a formulas set S is:
    a serie of formulas:
      β1, β2, ... , βm = α,
        in which for every i = 1, 2, ..., m :
          a formula βi is:
            - an axiom,
            - or a S-set element,
            - or a result of using a certain inference rule on rules preceding it in a proof.


deft. 15. A formula α is a syntactic consequence of a formulas set S, in a certain inference system, if in this a system exists a proof of this formula based on S. We can note it as: S ⊦ α.


def. 16. Inference system is correct, if for any formulas set S and an α formula, a syntactic consequence S ⊦ α entails semantic consequence S ⊧ α.


def. 17. Inference system is complete, if for any formulas set S and an α formula, a semantic consequence S ⊧ α entails syntactic consequence S ⊦ α.


There are many different (assuming different axioms & inference rules) correct & complete inference systems for the predicate calculus.

Tuesday, 19 July 2016

Derivative of a Function.

Definition & Notation.

def. Derivative of a Function y = f(x) in point x is limit to which closes the ratio of increment of a function Δy to an increment of a variable Δx, when increment of a variable Δx closes to zero.



if such limit does not exist, then function has no derivative in this point.

derivative of a function y = f(x) we can note as:

, , , , .



Geometrical interpretation.

Geometrically, a derivative of a function is equal to a tanget of angle α, between the tangent line touching the graph of the function in a point x and positive direction of
the OX axis.




Additional notes.

Finding a derivative of a function we can call 'function differentiation'.

From definition we can see that derivative of a function is a quickness of the change of a function f(x), when x changes.


Calculations.

Let's calculate a few of function derivatives, using definiton.

1. y = sin(x).



We've used one of trigonometrical formulas (last one, for the sines difference):



... as well, as that is:



Similarly: (cos x)' = - sin x;

2. y = ax3, where a is any constant..



We've used Newton's Formula:



3. y = xn, n - natural number.



We've used Newton's Formula again.

The same formula for derivative (xk)' = kxk-1 we can use for any k.


(an unfinished article, to be continued).