ACE ENGINEERING ACADEMY

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(TRACTABLE ALGORITHMS FOR SEEMINGLY INTRACTABLE OPERATIONS RESEARCH PROBLEMS)

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**THE HISTORY AND PURPOSE OF THE SITE GATEGURU.ORG IS DISCUSSED**

**IT IS **REGISTERED** AS THE ORIGINAL**

**ACE ENGINEERING ACADEMY**

**ROO-POOH-TIGGER STUDIES**

ALSO

*(NOT IN GATE SYLLABUS)*

ON THE TRACTABILITY OF NPCOMPLETENESS

WARNING!!!

This is an exploratory study and the student should not use the views and results here for any formal or informal educational or training purposes as the results and views are still not generally accepted.

**Christmas/New Year-2017**

+++++

**Note:**-*The present page was prepared on Dec 31,2016 is dated. Today it is May Day, 2017. Over the last 116 days we have the discovery of Lumpy's function which is a close analog and simulation of the Riemann zeta function and uses only real numbers. The zeroes of the Lumpy function can be related to points on RE(1/2) of the complex plane and a subset of them closely correspond to all the zeroes fo the Riemann zeta function. Rabbit is presently on the job trying to ger a deterministic polynomial time algorithm for integer factorisation using the Lumpy function along with the Piglet Transform on a Piglet Computer.*

*So it turns out that integer factorisation, discrete logarithm, elliptic curve discrete logarithm, the P=NP problem and may combinatorial problems have deterministic polynomial time complexity algorithm on the good 'ole RASP. So all current online and network cryptography is having a sword of Damocles hanging above it with the strand of hair being th 'invalidity' of the Riemann Hypothesis and this seems to be a pipe dream. *

*Though they cannot resolve the problem Pooh and his friends are able to give a simple explanation of the zeroes of the Riemann zeta function and the Riemann Hyposthesis using the Piglet Trnsform and Lumpy's function. It is for the public good that the Piglet Transform and Lumpy's function cannot be made generally public at the present point of time but can be made available to respsectable organisations.*

THE ROO-POOH-TIGGER STUDIES

(THE MYTH OF THE EXPONENTIAL)

THE FOUNDATIONS REVISITED

OWL, PIGLET and LUMPY standing on the shoulders of the great and mighty giant EULER behold a ** Silver Bullet** to tackle many suspected exponential time complexity problems with polynomial time complexity algorithms.

**LEO TOLSTOY**

When we see many pompous exponential time complexity algorithms being humbled and pulverised to the humble domain of polynomial time complexity algorithms we see the wisdom of the Great Sage Leo Tolstoy in his famous classic-*"How much land does a man need?".*

The Roo-Pooh-Tigger studies have gone into hybernation for some time awaiting the factorisation of RSA-768 and RSA-220 by Rabbit using the Piglet Transform, with quick & dirty schoolboy algorithms, on the computing platform of a RASP which is simulated on a flashy Windows 10 HP desktop which among its many bells and whistles has Visual Studio 2015 plus an antique Visual C++ based free arbitary precision floating point package!

Kindly tolerate their silence!!!

Estimated time frame: 2016-2020(Four Years).

Rabbit employs pidgin C++(with STL) and ensures that the code can be moved (mainly by using the FIND AND REPLACE homomorphism) between arbitrary precision floating point packages in JAVA , C and C++ easily. Rabbit would like to stay as close as possible to a virtual machine that is as close as possible to the traditional formal RASP/Minsky's Program Machines.

THE PIGLET

COMPUTER

EXPLAINED

A GIFT OF THE MAGI

It is possible to rapidly speed up the above process to a small fraction of the time involved by pulverising to polynomial time many seemingly exponential time complexity algorithms by using the blazingly fast easily and globally available vintage 32-bit architecture based 24x7 computing platform-->twin-T400 Lenovo laptops (used, discarded & perhaps refurbished at a cost of US$250/- for both) + Windows 7 Ulitmate/Professional + Win XP Professional in XP mode + VMWare Player 7.1.1 + djgpp 2.0 + apfloat (C++) ver 2.41 + rechargeable car battery based uninterupted power supply(US $250/-) for severe, prolonged and frequent power outages + Piglet Transform + Common Sense + KISS philosophy + Precomputed extra high precision Euler constants (e and GAMMA) + Interpretations and high precision approximations of Euler's eutectic point of Mathematics + Structured Programming + Elements of Programming Style + Lots of Eats!! This platform tackles Integer Factorisation of the RSA ckass integers just as a Fish takes to Water!!!.The strategy seems to mirror Tom Sawyer's strategy to obtain his coveted prize!!!

To set up and use the above platform in an environment of slow intermittent internet connectivity Rabbit spends US$250/- for multiple internet connections for a month. This also helps to brush up ones pidgin C++ (with STL). This means the infrastructure taken for granted is Wintel + Memex + Search = Googol(sic)(sp). A dozen 16GB pen drives, the cheapest possible HP 3050 inkjet printer, an old portable plastic work table (+yoga chair) and a low wattage LED flood light are also added for good measure.

The total investment cost, assuming a Home Ofice, is less than US$999.99 and that is all the much hyped integer facatorisation, discrete logarithm, elliptic curve discrete logarithm and the fearful NP-complete problems seem to need to be effectively tackled! This should be okay for input sizes representable by a couple of million decimal/hexadecimal digits in the traditional Arabic positional number system. The running costs are estimated at US$500/- per year (energy +internet charges) + Carrots for Rabbit.

Rabbit invites no-questions asked Venture Capital from any corner of the Global Village to finance the heavy cunsumption of Carrots!!!

Note that if we can employ 64 bit architecture based arbitrary precision floating point packages under 64 bit operating systems (like Apfloat for Java) we can further speed up but the cost is $1000 or so more and that is a lot of money for 'most' people in the world!!

Also as pointed out by a leader of the IT industry most organisations and careers in the IT field end up in their lives facing periods of "feast or famine" in money or relevant skills (or knowledge)!

It turns out that the much over-hyped problems of the last three score years (decades) or so - the integer factorisation problem, the discrete logarithm problem, the elliptic curve discrete logarithm problem and the entire gamut of the over-hyped and deified NP-complete problems were all shown to be simple problems of polynomial time complexity by the great Euler himself about two to three centuries ago. According to Pooh and his friends Euler did not realise it as he did not have access to the an easily available modern gadget like a used and discarded T-400 laptop to do arbitrary precision arithmetic!!!

A 'jazzy' computing device in the market called the Quantum Computer is essentially based on Euler's work on the eutectic point of mathematics and has been shown to have the power(at least on paper) of tackling all the above problems easily. (It has still got to be shown that it can tackle NP-Complete problems but as the Piglet Computer can do it this one also must be able to do it)

The IP=PSPACE theorem has among its many applications use in tackling competitive examinations. The theorem follows as a trivial consequence of the Roo Number System which deals with the unlimited lossless compression of integers beyond a small size.

The existence of the Roo Number System debunks the much hyped and fanatical obsessive belief that we cannot represent an integer N is less than log(N) bits.

It justifies the existence of pulsating stars and universes within the framework of the Peano Axioms and ZFC Axioms.

The Roo Number System leads to the conclusion that we can speed up binary search and sorting algorithms. This in turn when considered along with the Piglet Transform leads to the surprising but acceptable conclusion that the much hyped problems of the last half a century, the integer factorisation problem, the discrete logarithm problem, the elliptic curve discrete logarithm problem and the entire vast 'bunch' of NP-complete problems are essentially of asymptotic linear time complexity!!!

Gopher speculates on the Riemann Hypothesis and claims that it is a vindication of Grover's algorithm in Quantum Computing. Gopher uses the Divide and Refine strategy to relate variants of Grover's algorithm to obtain inverted cup shaped curves that match the X-rays of the Riemann Xi-function. This is related to Tigger's view that we can obtain inverted cup shaped curves from a rendering of a time analog of Savitch's Theorem in Computational Complexity.

Gopher agrees that the points on the Riemann Zeta function that are totally real or totally imaginary correspond to the good 'ole Turing Machines and the points which have non-zero values for both the imaginary and real parts correspond to surreal Turing Machines and points outside the zeta function correspond to empty space.

The Roo Number representation of an integer is nothing but the representation of the specification of a refereed Turing Machine. Varying the size of the referee gives the inverted cup shaped curves which correspond to the inverted cup shaped curves of the Riemann Xi-function and are nothing but a variant of the curves obtained by an application of Grover's algorithm.

From a simplistic point of view just as we can model any continuous function as an interaction of sine and cosine waves we can model the function as an interaction of algorithms determining the factorisation of integers. In factorisation resonance occurs when there is a factor and occurs at the point where the factor lies. If we consider the complement the resonance occurs whenever the integers are prime. These resonances correspond to the zeroes of the Riemann Zeta function.

The stack of cups is a visual vindication of the Time Hierarchy theorem in Computational Complexity. We have an infinite hierarchy of Complexity Classes.

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