The domain axioms for WaterWorld in propositional logic.
We summarize the details of how we
choose to model WaterWorld boardsin propositional logic:
exactly what propositions we make up,and the formal domain axioms which capture the game's rules.
The board is fixed at 64,
named
,,
(with I and O omitted).
A Sample WaterWorld board
Propositions
There are a myriad of propositions for WaterWorld,
which can be grouped:
Whether or not a location contains a pirate:
,
,,
.
Whether or not a location contains no pirate:
,
,,
.
Yes, using the intended interpretation,
these are redundant with the previous ones.Some domain axioms below will formalize this.
Propositions indicating
the number of neighboring pirates, to a location:
,
,
,
,
,
,,
,
,
,
,,
,
.
These are all true/false propositions;there are no
explicit numbers in the logic.A domain axiom below will assert that whenever (say)
is true, then
and
are both false.
There is no proposition
since location
has only two neighbors.
Similarly, there is no proposition
.
We
could have chosen to
include those, but under the intended interpretationthey'd always be false.
These propositions describe the state of the underlying
boardthe modeland not our particular view of it. Our particular view will be reflected in which formulas
we'll accept as premises. So we'll accept
as a premise
only when
has been exposed and shows a 2.
The domain axioms
Axioms asserting that the neighbor counts are correct:
Count of 0:
A0:
H0:
Z0:
Count of 1:
A1:
H1:
Z1:
Count of 2:
A2:
H2:
There aren't any such axioms for locations
with only one neighbor.
Count of 3:
H3:
There aren't any such axioms for locations
with only one or two neighbors.
Axioms asserting that the propositions for counting neighbors
are consistent:
Axioms asserting that the safety propositions are consistent:
,
,
,
.
This set of axioms is not quite complete, as explored in
an exercise .
As mentioned, it is redundant to have both
and
as propositions.
Furthermore, having both allows us to express inconsistent states(ones that would contradict the safety axioms).
If implementing this in a program, you might use both as variables,but have a safety-check function to make sure
that a given board representation is consistent.Even better, you could implement WaterWorld so that these
propositions wouldn't be variables,but instead be calls to a lookup (accessor) functions.
These would examine the same internal state,to eliminate the chance of inconsistent data.
Using only true/false propositions;
without recourse to numbers makes these domain axioms unwieldy.Later, we'll see how
relations and
quantifiers help us model the game of WaterWorld more concisely.
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .