# 6.5 First-order axioms for waterworld

 Page 1 / 1
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 $\mathrm{nhbr}$ .

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

• Whether or not a location contains a pirate: $\mathrm{safe}$ . This is a unary relation.
We choose not to include a redundant relation $\mathrm{unsafe}$ .
• Unary relations indicating the number of neighboring pirates: $\mathrm{has0}$ , $\mathrm{has1}$ , $\mathrm{has2}$ , and $\mathrm{has3}$ .
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 $x\neq y$ as shorthand for $\neg (x=y)$ . 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 $\mathrm{has2}(A)$ 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:

• $\forall x\colon \neg \mathrm{nhbr}(x, x)$
• $\forall x\colon \forall y\colon \mathrm{nhbr}(x, y)\implies \mathrm{nhbr}(y, x)$

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

• $\forall x\colon x=x$
• $\forall x\colon \forall y\colon (x=y)\implies (y=x)$
• $\forall x\colon \forall y\colon \forall z\colon ((x=y)\land (y=z))\implies (x=z)$

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.

• $\forall x\colon \mathrm{has0}(x)\implies \forall y\colon \mathrm{nhbr}(x, y)\implies \mathrm{safe}(y)$
• $\forall x\colon \mathrm{has1}(x)\implies \exists a\colon \mathrm{nhbr}(x, a)\land \neg \mathrm{safe}(a)\land \forall y\colon (\mathrm{nhbr}(x, y)\land (a\neq y))\implies \mathrm{safe}(y)$
• $\forall x\colon \mathrm{has2}(x)\implies \exists a\colon \exists b\colon \mathrm{nhbr}(x, a)\land \mathrm{nhbr}(x, b)\land (a\neq b)\land \neg \mathrm{safe}(a)\land \neg \mathrm{safe}(b)\land \forall y\colon (\mathrm{nhbr}(x, y)\land (a\neq y)\land (b\neq y))\implies \mathrm{safe}(y)$
• $\forall x\colon \mathrm{has3}(x)\implies \exists a\colon \exists b\colon \exists c\colon \mathrm{nhbr}(x, a)\land \mathrm{nhbr}(x, b)\land \mathrm{nhbr}(x, c)\land (a\neq b)\land (a\neq c)\land (b\neq c)\land \neg \mathrm{safe}(a)\land \neg \mathrm{safe}(b)\land \neg \mathrm{safe}(c)\land \forall y\colon (\mathrm{nhbr}(x, y)\land (a\neq y)\land (b\neq y)\land (c\neq y))\implies \mathrm{safe}(y)$

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

• $\forall x\colon \forall y\colon \mathrm{nhbr}(x, y)\implies \mathrm{safe}(y)\implies \mathrm{has0}(x)$
• $\forall x\colon \text{}\implies \mathrm{has1}(x)$
• $\forall x\colon \text{}\implies \mathrm{has2}(x)$
• $\forall x\colon \text{}\implies \mathrm{has3}(x)$

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

• $\forall x\colon \mathrm{has0}(x)\implies \neg (\mathrm{has1}(x)\lor \mathrm{has2}(x)\lor \mathrm{has3}(x))$
• $\forall x\colon \mathrm{has1}(x)\implies \neg (\mathrm{has0}(x)\lor \mathrm{has2}(x)\lor \mathrm{has3}(x))$
• $\forall x\colon \mathrm{has2}(x)\implies \neg (\mathrm{has0}(x)\lor \mathrm{has1}(x)\lor \mathrm{has3}(x))$
• $\forall x\colon \mathrm{has3}(x)\implies \neg (\mathrm{has0}(x)\lor \mathrm{has1}(x)\lor \mathrm{has2}(x))$

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

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!