Encoding and Solving a Problem
Food Chains
Food chains describe the feeding relationships between species in a biotic
community.
They show the transfer of material and energy from one species to
another within an ecosystem.
If one species eats another, there is an food link from the
eaten of the food link to the eater of the foodlink.
Apex predators are those species that are not eaten by any species.
Primary producers (also known as autotrophs) are those species
that do not eat any species - they are capable of producing complex organic
substances (essentially "food") from an energy source and inorganic materials.
A sequence of food links from a species to a species to a species,
etc, forms a food chain.
An apex predator is dependent on all the species in all the food
chains that lead to the apex.
A species is in a food chain if it is not the apex predator
of the chain.
The food chain environment can be modeled in first order logic, and various
theorems can then be proved.
Here the axioms of the system are given in English, interspersed with
theorems that can be proved from the preceding axioms.
- Axiom: The eaten and eater of a food link are species.
- Axiom: Something eats another thing iff there is a food link from
the eaten of the link to the eater of the link.
- Axiom: Every species eats something or is eaten by something.
- Axiom: The eaten and eater of a food link are not the same (i.e., no
cannibalism).
- Axiom: Something is an apex predator iff it is a species and there is no
species that eats it.
- Theorem: If something is an apex predator then there is no species and
food link such that the apex predator is the eaten of the food link and
the species is the eater of the food link.
- Theorem: If a species is not a apex predator then there is another
species that eats it.
- Theorem: Every apex predator eats some species.
- Axiom: There is a food chain from one thing to another if both are
species, and there is a food link such that the first thing is the
eaten of the food link and one or both of the following conditions hold:
(i) the second thing is the eater of the food link, and the food chain
is the food link, or (ii) there is another food chain from the eater of
the food link to the second thing, and the overall food chain is the
concatenation of the food link and this food chain.
- Axiom: If there is a food chain from one thing to another, then
both are species, each is the only one at their end of the food chain,
there is no food chain from the second back to the first, and either
(i) the second is a apex predator, or (ii) there is a food chain from
the second to an apex predator.
- Theorem: If a species is not a apex predator then there is a food chain
from the species to some species.
- Theorem: If there is a food chain from one thing to another, then
the two are different.
- Theorem: If there is a food chain from one thing to another, then
there is no food link such that the first thing is the eater of the
food link and the second thing is the eaten of the food link.
- Theorem: There is no food chain from one thing back to itself.
- Axiom: Something depends on another iff the first is an apex predator,
and there is a food chain from the second to the first.
- Theorem: If a species is not a apex predator then there is an apex
predator that depends on it.
- Axiom: Something is in a food chain iff it is a species, there are two
other things distinct from the first thing, there is a food chain
from the first other to the second other, there is a food chain from
the first other to the thing, and there is a food chain from
the thing to the second other.
- Theorem: If an apex predator depends on a species, then the apex
predator depends on everything in all food chains from the species to
the apex predator.
You must ...
- Formulate the axioms and theorems in FOF in TPTP syntax.
- Ensure that the axioms are consistent using an ATP system.
- Prove the theorems using an ATP system.
Answers