## The Fourth Law of Robotics - Part II

Written by Sam Vaknin

Note - Godel's Theorems

The work of an important, though eccentric, Czech-Austrian mathematical logician, Kurt Gödel (1906-1978) dealt with completeness and consistency of logical systems. A passing acquaintance with his two theorems would have saved architect a lot of time.

Gödel's First Incompleteness Theorem states that every consistent axiomatic logical system, sufficient to express arithmetic, contains true but unprovable ("not decidable") sentences. In certain cases (when system is omega-consistent), both said sentences and their negation are unprovable. The system is consistent and true - but not "complete" because not all its sentences can be decided as true or false by either being proved or by being refuted.

The Second Incompleteness Theorem is even more earth-shattering. It says that no consistent formal logical system can prove its own consistency. The system may be complete - but then we are unable to show, using its axioms and inference laws, that it is consistent

In other words, a computational system can either be complete and inconsistent - or consistent and incomplete. By trying to construct a system both complete and consistent, a robotics engineer would run afoul of Gödel's theorem.

Note - Turing Machines

In 1936 an American (Alonzo Church) and a Briton (Alan M. Turing) published independently (as is often case in science) basics of a new branch in Mathematics (and logic): computability or recursive functions (later to be developed into Automata Theory).

The authors confined themselves to dealing with computations which involved "effective" or "mechanical" methods for finding results (which could also be expressed as solutions (values) to formulae). These methods were so called because they could, in principle, be performed by simple machines (or human-computers or human-calculators, to use Turing's unfortunate phrases). The emphasis was on finiteness: a finite number of instructions, a finite number of symbols in each instruction, a finite number of steps to result. This is why these methods were usable by humans without aid of an apparatus (with exception of pencil and paper as memory aids). Moreover: no insight or ingenuity were allowed to "interfere" or to be part of solution seeking process.

What Church and Turing did was to construct a set of all functions whose values could be obtained by applying effective or mechanical calculation methods. Turing went further down Church's road and designed "Turing Machine" – a machine which can calculate values of all functions whose values can be found using effective or mechanical methods. Thus, program running TM (=Turing Machine in rest of this text) was really an effective or mechanical method. For initiated readers: Church solved decision-problem for propositional calculus and Turing proved that there is no solution to decision problem relating to predicate calculus. Put more simply, it is possible to "prove" truth value (or theorem status) of an expression in propositional calculus – but not in predicate calculus. Later it was shown that many functions (even in number theory itself) were not recursive, meaning that they could not be solved by a Turing Machine.

## Ringtones Explained - Monophonic, Polyphonic or Whatever Phonic Rings Your Bell

Written by Lillian Fuller

The cell phone and their various ringtones have become status symbols. Ringtones show that you’re up to date and happening. You can even take a test that defines which ringtone best suits your personality. You can truly display your lifestyle and certainly your musical taste on your sleeve.

This article will describe briefly what ring tones are, what’s currently available and what future holds. It will also suggest a caveat to consider prior to downloading multitude of formats of available on market. Prices are low and sky’s limit, so read on.

• A Caveat Before You Buy!
• What’s Available?
• A Brief Glimpse At Ancient History.
• Ringtones – Where Do We Go From Here?
• Other Products Available On Ringtone Websites
• What Can You Say About Ringtones

Ringtones are sound that a regular or cell phone makes when it’s rings. Early ring tones were simple chimes and on later models, a combination of chimes. Limited numbers of ringtones were available and mostly consisted of different patterns of tones or chirps. An example of one of most infamous ringtones was ominous” ring ring…ring ring” that is reminiscent of Todd Beamer’s last telephone call on 9/11. You can hear that ringtone featured as an intro to Neil Young’s famous song “Let’s Roll.”

Ringtone technology developed quickly and has progressed a lot since that song was recorded. Originally, you purchased a cell phone and had a choice of monophonic ring tones that produced a one note song. The number of songs was extremely limited and downloadable ring tones were simply an Engineering student’s good idea.

Because of lightning speed technology and numerous service providers, one of latest and greatest features available today is downloadable ringtones. When you’re surfing ‘Net you’ll find plenty of sites offering tones by today’s hottest musicians; you can pick monophonic, polyphonic or true tone formats. More on that later.

What’s Available?

Ringtones are divided into three formats.

1. Monophonic
2. Polyphonic
3. True Tone or Real Tone
To understand this type of technology, a person needs to have a physics/mathematics background and understand musical composition. Luckily, all that is needed to know is what types of ringtones are available and what they sound like. The following is a description of three types of ringtones, monophonic, polyphonic and true tone. Most websites offer samples of each and once you hear an example, you’ll know what type you’ll want to use on your cell phone.

A) Monophonic Ringtones Some of first cell phones came equipped with capability of having a one tone ringer. “Monophonic” is simplest of ring tone technology that uses a midi format. This “one note wonder” changed ringtones. Gone were chirps and chimes; now you could play a song. A monophonic ringtone is single notes playing a song. You can recognize your favorite song but it sounds a little lame. Most monophonic tones are offered free of charge because of more full bodied tones now available, but if you’re trying to make a statement, however, read on.

B) Polyphonic Ringtones Polyphonic ring tones also use midi technology but tonal quality is richer and has more depth. This is because polyphonic ringtones can play up to 40 notes or more. A quick example is sound produced when Windows XP® starts up. It reproduces sound of an orchestra playing a note rather than one instrument playing same note.

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