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But does that learning generalize to comparing 3 digit numbers? That's the challenge of arithmetic: induction-- learning to predict results on larger problems.

by hopler 8y ago

YeGoblynQueenne 8y ago
Ah, but this is the great thing about ILP, including the δILP system in the article- that the "models" are actually logical theories that can be directly inspected (by reading them!) and verified. So we don't have to wonder whether the model learned by the algorithm will generalise outside its training data- we can just check the learned theory. This is one of the theories learned by the δILP system for the less-than relation between images of numbers, taken from the pdf version of the paper [1]. Using Prolog as the notation (the transformation from the article is trivial): target:- image2(X), pred1(X). pred1(X) :- image1(Y), pred2(Y,X). pred2(X,Y) :- succ(X,Y). pred2(X,Y) :- pred2(Z,Y), pred2(X,Z). succ/2 is the successor relation; succ(X,Y) is true when Y = X+1. image1/1 and image2/1 are the numeric values of the compared images as read from the neural net. So the first two clauses of target/0 collect the two compared images' numeric values, then they pass them on to pred2/2, which actually does the comparison. pred2/2 itself is true when one of the following relations holds: a) Y = X+1 b) Y > Z AND X < Z Where (b) is equivalent to X < Z < Y. In other words, target/0 is always true when the value of image1 (which is the X in pred2/2) is less than the value of image2 (the Y). It's a bit fiddly to follow because pred1/1 inverts the order of the arguments passed to pred2/2, and because the predicate names don't mean anything, but you can see that's what's going on: the above is as general a definition as is possible to have of the less-than relation. So, yes, totally, it will generalise to integers with any number of digits. ________________ [1] https://arxiv.org/pdf/1711.04574.pdf https://arxiv.org/pdf/1711.04574.pdf The learned theory is on page 34.

But does that learning generalize to comparing 3 digit numbers? That's the challenge of arithmetic: induction-- learning to predict results on larger problems.

by hopler 8y ago

But does that learning generalize to comparing 3 digit numbers? That's the challenge of arithmetic: induction-- learning to predict results on larger problems.

YeGoblynQueenne 8y ago

Ah, but this is the great thing about ILP, including the δILP system in the article- that the "models" are actually logical theories that can be directly inspected (by reading them!) and verified. So we don't have to wonder whether the model learned by the algorithm will generalise outside its training data- we can just check the learned theory. This is one of the theories learned by the δILP system for the less-than relation between images of numbers, taken from the pdf version of the paper [1]. Using Prolog as the notation (the transformation from the article is trivial): target:- image2(X), pred1(X). pred1(X) :- image1(Y), pred2(Y,X). pred2(X,Y) :- succ(X,Y). pred2(X,Y) :- pred2(Z,Y), pred2(X,Z). succ/2 is the successor relation; succ(X,Y) is true when Y = X+1. image1/1 and image2/1 are the numeric values of the compared images as read from the neural net. So the first two clauses of target/0 collect the two compared images' numeric values, then they pass them on to pred2/2, which actually does the comparison. pred2/2 itself is true when one of the following relations holds: a) Y = X+1 b) Y > Z AND X < Z Where (b) is equivalent to X < Z < Y. In other words, target/0 is always true when the value of image1 (which is the X in pred2/2) is less than the value of image2 (the Y). It's a bit fiddly to follow because pred1/1 inverts the order of the arguments passed to pred2/2, and because the predicate names don't mean anything, but you can see that's what's going on: the above is as general a definition as is possible to have of the less-than relation. So, yes, totally, it will generalise to integers with any number of digits. ________________ [1] https://arxiv.org/pdf/1711.04574.pdf https://arxiv.org/pdf/1711.04574.pdf The learned theory is on page 34.