Relations and Logic: interpretations
Summary: Interpretations map relation-symbols to an actual relation.
Needing Interpretations to Evaluate Formulas
You might have noticed something funny: we said safe(a) depended on the board, but that prime(18) was false. Why are some some relations different than others? To add to the puzzling, there was a caveat in some fine-print from the previous section: “prime(18) is false under the standard interpretation of prime”. Why these weasel-words? Everybody knows what prime is, don't they? Well, if our domain is matrices of integers (instead of just integers), we might suddenly want a different idea “prime”.
Consider the formula
-
for the domain of integers where
E E x x x -
for the domain
2 4 6 8 E E x x x -
for the domain of integers where
E E x x x x -
for the domain of people where
E E x x x
Thus a formula's truth depends on the interpretation of the (syntactic, meaning-free) relation symbols in the formula.
- Definition 1: Interpretation
- The interpretation of a formula is a domain, together with a mapping from the formula's relation symbols to specific relations on the domain.
Some non-standard interpretations (optional)
prime factorization
Note that there are other possible interpretations of
“
For example: Consider only the numbers F = {1,5,9,13,…} — that is, F = {4k+1 | k in N}. It's easy to verify that multiplying two of these numbers still results in a number of the form 4k+1. Thus it makes sense to talk of factoring such numbers: We'd say that 45 factors into 5⋅9, but 9 is considered prime since it doesn't factor into smaller elements of F.
Interestingly, within F, we lose unique factorization: 441 = 9⋅49 = 21⋅21, where each of 9, 21, and 49 are prime, relative to F! (Mathematicians will then go and look for exactly what property of a multiplication function are needed, to guarantee unique factorization.)
The point is, that all relations in logical formula need to be interpreted. Usually, for numbers, we use a standard interpretation, but one can consider those formulas in different, non-standard interpretations!
The Poincare Disc
A long outstanding problem was that of Euclid's parallel postulate: Given a line and a point not on the line, how many lines parallel to the first go through that point? Euclid took this as an axiom (unable to prove that it followed from his other axioms). Non-Euclidean geometries of Lobachevsky and Riemann took different postulates, and got different geometries. However, it was not clear whether these geometries were sound — whether one could derive two different results that were inconsistent with each other.
Poincare developed a method of showing that certain non-Euclidean geometries were consistent (or at least, as consistent as Euclidean geometry. Remember that in Euclidean geometry, the concepts “point” and “line” are left undefined, and axioms are built on top of them (e.g. “two different lines have at most one point in common”). While it's usually left to common sense to interpret “point”, “line”, and “a point is on a line”, any interpretation which satisfies the axioms means that all theorems of geometry will hold.
The Poincare disc is one such interpretation: “point” is taken to mean “a point in the upper half of the unit disc”, and “line” is taken to mean “a circular arc which meets the unit disc at right angles”. This preserves all the axioms (for instance, two different lines have at most one point in common), except for the parallel postulate. Indeed, for a given line and a point not on it, there are an infinite number of parallel (that is, non-intersecting) lines.
The critical point of his interpretation of a non-Euclidean geometry is this: it is embedded in Euclidean geometry! If there is any inconsistency in this non-Euclidean lines, then that could be parlayed into an inconsistency within Euclidean logic dealing with arcs in the unit disc. Thus, his interpretations gives a proof that the strange non-Euclidean geometry is as sound as our familiar Euclidean geometry.
P vs NP and Oracles
A well-known problem in computer science — “P vs NP” — asks whether (for a given problem) it is truly more difficult to find a short solution (when one exists) (“NP”), than it is to verify a short purported solution handed to you (“P”). For example, “Given a set of people and how strong each person is, can you partition them into two tug-of-war teams which are exactly evenly matched?” Certainly it seems easier to check that a pair of proposed rosters has equal strength (and, verify that everybody really is on one team or the other) than to have to come up with two perfectly-matched teams. But conceivably, the two tasks might be equally-difficult up to some acceptable (polynomial time) overhead. While everyone assumes that P is easier than NP, nobody has been able to prove it.
An interesting variant of the problem lets both the problem-solver and the purported-answer-verifier each have access to a particular oracle — a program that will gives instant yes/no answers to some other problem (say, “given any set of numbers, yes or no: is there an even-sized subset whose total is exactly the same as some odd sized subset?”).
It has been shown that there is some oracle which makes the problem-solver's job provably tougher than the proof-verifier's job, and also there is some other oracle problem-solver's job provably no-tougher than the proof-verifier's job.
This means that any proof of P being different from NP has to be subtle enough so that when P and NP are re-interpreted as “P and NP with respect to a particular oracle”, the proof will no longer go through. Unfortunately, this eliminates all the routine methods of proof; we know that solving this problem will take some new attack.
Using Formulas to Classify Interpretations (Optional)
In the previous section, having a formula was rather useless until we had a particular interpretation for it. But we can view that same idea backwards: Given a formula, what are all the interpretations for which the formula is true?
For instance, consider a formula expressing that an array
is sorted ascendingly:
For all numbers
Similarly, mathematicians create some formulas about functions being associative and having an identity element, and then look at all structures which have those properties; this is how they define (say) Groups.
Encoding functions as Relations
What about adding functions, to our language, in addition to relations? Well, functions are just a way of relating input(s) to an output. For example, 3 and 9 are related by the square function, as are 9 and 81, and 0,0. Is any binary relation a function? No — for instance {(9,81), (9,17)} is not a function, because there is no unique output related to the input 9.
How can we enforce uniqueness?
The following sentence asserts that for each element
Exercise 1
We just used a binary relation to model a unary function.
Carry on this idea, by using a ternary relation to
start to model a binary function.
In particular, write a formula stating that
for every pair of elements
Solution 1
For all
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This work is licensed by Ian Barland, John Greiner, Phokion Kolaitis, Moshe Vardi, and Matthias Felleisen under a Creative Commons Attribution License (CC-BY 1.0).
Last edited by John Greiner on Nov 16, 2007 12:08 pm US/Central.