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The domain axioms of WaterWorld in first-order logic.

We summarize the details of how we choose to model WaterWorld boards in first-order logic: exactly what relations we make up, and the formaldomain axioms which capture the game's rules.

This will follow almost exactly the same pattern as our WaterWorld model in propositional logic . However, we will take advantage of the additional flexibility providedby first-order logic.

Rather than modeling only the default 64 WaterWorld board;, we will be able to model any board representable by our relations.This will allow boards of any size and configuration, with one major constrainteach location can have at most three neighboring pirates.

Domain and relations

Our domain is simply the set of all board locations. This set can be arbitrarily largeeven infinite!

The board configuration is given by the binaryneighborrelation nhbr .

The next relations correspond directly to the propositions in the propositional logic model.

  • Whether or not a location contains a pirate: safe . This is a unary relation.
    We choose not to include a redundant relation unsafe .
  • Unary relations indicating the number of neighboring pirates: has0 , has1 , has2 , and has3 .
    Thus, we have our restriction to three unsafe neighbors. This will also be reflected in our domain axioms below. See also this problem for a discussion of how to avoid this restriction.

In addition, to have encode the domain axioms for an arbitrary domain, we also need an equality relation over our domain of locations.As is traditional, we will use infix notation for this relation, for example, x y . Furthermore, we will allow ourselves to write x y as shorthand for x y . Thus, we do not need a distinct inequality relation.

Note that these relations describe the state of the underlying boardthe modeland not our particular view of it. Our particular view will be reflected in which formulaswe'll accept as premises. So we'll accept has2 A as a premise only when A has been exposed and shows a 2.

The domain axioms

Many of our axioms correspond directly, albeit much more succinctly, with those of the propositional model. In addition, we have axioms that specify that our neighbor and equalityrelations are self-consistent.

Axioms asserting that the neighbor relation is anti-reflexive and symmetric:

  • x nhbr x x
  • x y nhbr x y nhbr y x

Axioms asserting that=truly is an equality relation, i.e. , it is reflexive, symmetric, and transitive.

  • x x x
  • x y x y y x
  • x y z x y y z x z

Axioms asserting that the neighbor counts are correct. Each of these is of the formif location x has n neighboring pirates, then there are n distinct unsafe neighbors of x , and any other distinct neighbor x is safe.We use the equality relation to specify the distinctness of each neighbor.

  • x has0 x y nhbr x y safe y
  • x has1 x a nhbr x a safe a y nhbr x y a y safe y
  • x has2 x a b nhbr x a nhbr x b a b safe a safe b y nhbr x y a y b y safe y
  • x has3 x a b c nhbr x a nhbr x b nhbr x c a b a c b c safe a safe b safe c y nhbr x y a y b y c y safe y

In addition, we want the implications to go the opposite way. Otherwise, each of has0 , has1 , has2 , and has3 could always be false, while still satisfying the above!For brevity, we elide the details in the following list:

  • x y nhbr x y safe y has0 x
  • x has1 x
  • x has2 x
  • x has3 x

Axioms asserting that the neighbor counts are consistent. While redundant, including axioms like the following can be convenient.

  • x has0 x has1 x has2 x has3 x
  • x has1 x has0 x has2 x has3 x
  • x has2 x has0 x has1 x has3 x
  • x has3 x has0 x has1 x has2 x

Note that this set of axioms is not quite complete, as explored in an exercise .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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