Characterizing visual proofs

Question: What would be a nice way to characterize which assertions have such visual proofs? What definitions would one need?

I am not sure if this helps but you might want to check out Visualizing Inequalities by Alsina and Nelsen. Roger Nelsen is also the author of two other books (Proofs without Words) which is a collection of many of those columns in the Mathematics magazine.

Coming back to the book, it gives no definite characterization of visual proofs but it does list various methods by which inequalities can be represented through geometric figures. It provides the beginnings of a "Representation theory for inequalities " (!) if you may call it that.

It discusses how the circumcircle and the in-circle (Chapter 4) can be used to represent an inequality between the radius of circle and the length, perimeter or area of n-gons. This doesn't always work for n-gons with n > 3 since they may not have both the circumcircle and the in-circle.

To visually prove geometric inequalities, one can use isometric transformations such as reflection or rotation (Chap 5 and 6). One can also use non-isometric transformations (Chap 7) such as

1. Similarity of figures which preserve shape but change measure
2. Measure-preserving transformations which change shape
3. Projections which change shape.

I particularly liked Figures 1.4, 1.7, 1.14, 1.17, 2.6, 2.17, 4.4, 7.8, 7.9, 8.17, 8.18, 8.20.

one will have to define a computational model for the ``visual verifier''

Regarding this part of the question, there is prior work on Geometric Theorem Proving. (Is that what you meant?) A couple of books which come to mind are Mechanical Geometry Theorem Proving by Chou and Wu's Mathematics Mechanization. Also see the GEO-Prover

Some relevant papers are

1. Kapur, D. 1986. Using Grobner bases to reason about geometry problems. J. Symb. Comput. 2, 4 (Dec. 1986), 399-408. DOI= http://dx.doi.org/10.1006/jsco.1995.1056
2. Kutzler, B. and Stifter, S. 1986. On the application of Buchberger's algorithm to automated geometry theorem proving. J. Symb. Comput. 2, 4 (Dec. 1986), 389-397.’
3. Kutzler Stifter Automated geometry theorem proving using Buchberger's algorithm http://doi.acm.org/10.1145/32439.32480

Here is a complexity theory perspective. Be warned that it may differ wildly from someone whose primary focus is logic.

I think the appropriate definition of a "visual proof" would boil down to giving an appropriate definition of what a verifier does with such a proof. Proof systems in complexity theory are measured by (a) how much the prover and verifier "interact", (b) the allowed lengths of potential proofs, and (c) the power of the verifier.

In the framework you are proposing, the prover simply gives the verifier a proof and walks away. So (a) is already determined. Also, the visual proof is presumably written on a small sheet of paper, so (b) is essentially determined (let's say the visual proof can be encoded in length that is at most a fixed polynomial in the length of the claim).

That leaves (c), which is where proof complexity gets interesting. It turns out that verifiers can be surprisingly weak and still verify the proof of any statement which has short proofs (where "short" is "fixed polynomial"). For example, if you require that proofs be written on a two-dimensional grid, then for every theorem with short proofs, there are proofs of the theorem which can be verified by a two-dimensional finite automaton, see

J. Hartmanis, D. Ranjan R. Chang, and P. Rohatyi. On IP = PSPACE and theorems with narrow proofs. EATCS-Bulletin, 41:166–174, 1990.

A verifier could be a "streaming" algorithm: it could be randomized, go over the proof in just one pass, and use a tiny amount of workspace relative to the length of the proof, see

Richard J. Lipton: Efficient Checking of Computations. STACS 1990: 207-215

The famous PCP theorem (of Arora et al.) tells us that the verifier could even be a "spot checker" which is randomized and only probes the proof at a constant number of points (which depend on the random coin tosses).

All of these are effectively different ways of characterizing the class NP: the polynomial time verifier in the definition of NP can be replaced with verifiers of the above kind.

So I believe that a good characterization of "visual proof" would turn out to give yet another way in which a simple verifier can check the proof of a theorem. However it is natural to think that maybe not all theorems with short proofs should have short visual proofs, so perhaps it is too ambitious to think that all of NP should have "visual proofs". Hence your definition problem will be a delicate combination of figuring out what the verifier should be able to do in a visual proof, and what kinds of true statements should admit such proofs. Good luck!

Addendum (added 7/1/10). The following neat paper on "Approximate Testing of Visual Properties" by Sofya Raskhodnikova looks very relevant:

http://www.cse.psu.edu/~sofya/pixels.pdf

This may not directly answer to your question. A different MO question about Resources for graphical languages accumulated quite a few references for proofs based on string diagrams. These have precise semantics generally in terms of monoidal categories and can be used to prove results about quantum groups among other things.