Da Vinci's 2000 years later. These diagrammatic ... demonstration of how intuitive human reasoning can be used to solve problems. If we remove two ... board with alternative black and white squares, like the chessboard. (in the figure below),.
How can machines reason? Mateja Jamnik1 1 Can machines reason like humans in mathematics? Some of the deepest and greatest insights in reasoning were made using mathematics. It is not surprising therefore that emulating such powerful reasoning on machines – and particularly the way humans use diagrams to “see” an explanation for mathematical theorems – is one of the important aims of artificial intelligence.
can the board still be covered with dominoes (that is, rectangles made out of two squares)? The elegant solution is to colour the checkerboard with alternative black and white squares, like the chessboard (in the figure below),
2 Diagrams for reasoning Drawing pictures and using diagrams to represent a concept is perhaps one of the oldest vehicles of human communication. In mathematics, the use of diagrams to prove theorems has a rich history: as just one example, Pythagoras’ Theorem has yielded several diagrammatic proofs (one is given in the figure below) in the 2500 years following his contribution to mathematics, including that of Leonardo Da Vinci’s 2000 years later. These diagrammatic proofs are so clear, elegant and intuitive that with little help, even a child can understand them [9].
a
a a b b
c
c
b
a
b
b
a
a2 + b2 = c2
and do the same with the dominoes so that a domino is made of one black and one white square. The solution then immediately becomes clear: there are more black squares than white squares, and so the mutilated checkerboard cannot be covered with dominoes. This problem is very easy for people to understand, but no system has yet been implemented that can solve it in such an elegant and intuitive way. Here is another example of an “informal” diagrammatic proof: that of a theorem about the sum of odd natural numbers. Unlike in automated theorem proving, where this problem is tackled symbolically with formal logical tools, the diagrammatic operations of our proof enable us to “see” the solution: a square can be cut into so-called “L” shaped pieces (as in the figure below), and each “L” represents a successive odd number, since both sides of a square of size n are joined (2n), but the joining vertex was counted twice (hence 2n − 1).
The concept of the “mutilated” checkerboard is another useful demonstration of how intuitive human reasoning can be used to solve problems. If we remove two diagonally opposite corner squares (like in the picture of the mutilated checkerboard in the figure below):
n2 = 1 + 3 + 5 + · · · + (2n − 1)
1
University of Cambridge www.cl.cam.ac.uk/˜mj201
Computer
Laboratory,
UK,
Web:
These examples above demonstrate how people use “informal” techniques, like diagrams, when solving problems. As these reasoning techniques can be incredibly powerful, wouldn’t it be exciting if a system could learn such diagrammatic operations automatically? So far, few automated systems have attempted to benefit from their power by imitating them [5, 1, 2, 4]. One explanation for this might
be that we don’t yet have a deep understanding of informal techniques and how humans use them in problem solving. To advance the state of the art of automated reasoning systems, some of these informal human reasoning techniques might have to be integrated with the proven successful formal techniques, such as different types of logic.
3 From intuition to automation There are two approaches to the difficult problem of automating reasoning. The first is cognitive, which aims to devise and experiment with models of human cognition. The second is to approach the problem computationally – attempting to build computational systems that model part of human reasoning. In my research at the University of Cambridge, I have taken steps along the computational approach. While at the University of Edinburgh, I built a system known as Diamond that uses diagrammatic reasoning to prove mathematical concepts [6, 8]. Diamond can prove theorems such as the one above about the odd natural numbers using geometric operations, just like in the example. With Diamond I showed that diagrammatic reasoning about mathematical theorems can be automated. Diamond uses simple and clear concrete rather than ambiguous general diagrams (in the example above, the concrete example is for n = 6). The “inference steps” of the solution in Diamond are intuitive geometric operations rather than complex logical rules. The universal statement of the theorem is showed using concrete cases by means of particular logical rules. The solution in Diamond is represented as a recursive program, called a schematic proof, which given a particular diagram, returns the proof for that diagram [7, 3]. To check that this solution is correct, Diamond carries out in the background a verification test in an abstract theory of diagrams. Diamond currently only tackles theorems which can be expressed as diagrams. However, there are theorems, like the mutilated checkerboard, that might require a combination of symbolic and diagrammatic reasoning steps to prove them, so-called heterogeneous proofs. The figure below demonstrates √ a heterogeneous proof. The theorem states an inequality: a+b ≥ ab where a, b ≥ 0. The first few sym2 bolic steps of the proof are: a+b 2
≥
√
ab
(a + b)2 22
↓
square both sides of ≥
≥
ab
(a + b)2
≥
4ab
a2 + 2ab + b2
≥
4ab
↓
↓
×4 on both sides of ≥
The second part of the proof, which is presented in the figure below, shows diagrammatically the inequality a2 + 2ab + b2 ≥ 4ab.
b
ab
b
ab ab
a
a2 a
b2 b
ab a ab b
ab b
a
In Cambridge, I am now investigating how a system could automatically reason about such proofs. This requires combining diagrammatic reasoning in Diamond with symbolic problem solving in an existing state-of-the-art automated theorem prover. The way forward is to give such a heterogeneous reasoning framework access to intelligent search facilities in the hope that the system will not only find new and more intuitive solutions to known problems, but perhaps also find new and interesting problems. Automated diagrammatic reasoning could be the key not only to making computer reasoning systems more powerful, but also to providing the necessary tools to study and explore the nature of human reasoning. We might then have a means to investigate the amazing ability of the human brain to solve problems.
ACKNOWLEDGEMENTS The research reported in this paper was supported by EPSRC Advanced Research Fellowship GR/R76783.
REFERENCES [1] D. Barker-Plummer and S.C. Bailin, ‘Proofs and pictures: Proving the diamond lemma with the GROVER theorem proving system’, in Working Notes of the AAAI Spring Symposium on Reasoning with Diagrammatic Representations, ed., N.H. Narayanan, Cambridge, MA, (1992). American Association for Artificial Intelligence, AAAI Press. [2] J. Barwise and J. Etchemendy, Hyperproof, CSLI Press, Stanford, CA, 1994. [3] A. Bundy, M. Jamnik, and A. Fugard, ‘What is a proof?’, Philosophical Transactions of Royal Society A: The Nature of Mathematical Proof, 363(1835), 2377–2391, (2005). [4] Diagrammatic Reasoning: Cognitive and Computational Perspectives, eds., B. Chandrasekaran, J. Glasgow, and N.H. Narayanan, AAAI Press/MIT Press, Cambridge, MA, 1995. [5] H. Gelernter, ‘Realization of a geometry theorem-proving machine’, in Computers and Thought, eds., E. Feigenbaum and J. Feldman, 134–152, McGraw Hill, New York, (1963). [6] M. Jamnik, Mathematical Reasoning with Diagrams: From Intuition to Automation, CSLI Press, Stanford, CA, 2001. [7] M. Jamnik and A. Bundy, ‘Psychological validity of schematic proofs’, in Mechanizing Mathematical Reasoning: Essays in Honor of J¨org H. Siekmann on the Occasion of His 60th Birthday, eds., D. Hutter and W. Stephan, number 2605 in Lecture Notes in Artificial Intelligence, 321–341, Springer Verlag, (2005). [8] M. Jamnik, A. Bundy, and I. Green, ‘On automating diagrammatic proofs of arithmetic arguments’, Journal of Logic, Language and Information, 8(3), 297–321, (1999). [9] R.B. Nelsen, Proofs without Words: Exercises in Visual Thinking, Mathematical Association of America, Washington, DC, 1993.
