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: The eaten and eater of a food link are not the same (no
cannibalism).
- 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 (or both).
- 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 food link
such that the apex predator is the eaten 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 other species.
- Axiom: There is a food chain from its start to its end if
the start and the end are species, there is a food link such that
the start species is the eaten of the food link, and one of the following
holds:
(i) the end species is the eater of the food link,
or (ii) there is another food chain from the eater of the food link to
the end species.
- Axiom: For every food chain, the start and the end are species,
there is no food chain from the end back to the start (no death spirals),
and one of the following holds:
(i) the end is a apex predator, or
(ii) there is a food chain from the end to an apex predator.
- Theorem: If a species is not a apex predator then there is a food chain
from that species to some other species.
- Theorem: The start and end of a food chain are different.
- Theorem: For every food chain, the start species does not eat the end
species.
- Theorem: There is no food chain from a species back to itself.
- Axiom: Given two species, the first depends on the second iff the first
is an apex predator, and there is a food chain from the second species
to the first.
- Axiom: An apex predator depends on a species if there is a foodchain
from the species to the apex predator.
- Theorem: If a species is not a apex predator then there is an apex
predator that depends on it.
- Theorem: Every species is either an apex predator, or has an apex
predator that depends on it.
- Axiom: A species is in a foodchain iff it is at the start of
the foodchain or there there are component foodchains from the start of
the whole foodchain to that species, and from that species to the end
of the whole foodchain.
- Theorem: An apex predator depends on every species in every food chain
that ends at 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.
- Submit one file containing all the axioms and conjectures, in the order
they are listed above.
Your submission will be graded according to the following:
- Axioms 30%
- Consistent axioms 20% (only if seriously encoded)
- Conjectures 20%
- Provable conjectures 30% (only if seriously encoded and axioms are
consistent)
You must submit your work by email to me, by 19th March.
It is worth 20% of the subject's assessment.
Please review the policies on assessment in
the administration document.
Answers