Ex-identification and scientific discovery

ABSTRACT

Consider the empirical laws established in the History of Science. The process of scientific discovery can be roughly explained by (Popper-Kuhn's) cycle which starts with collecting evidence, introducing a model to explain observations, predicting further observations, and confronting them with experimentation, reinforcing or disproving the model. We attempt to prove that the empirical evidence modeled by arithmetical operations can be automatically discovered with a small amount of information, i.e. with very short Blums' locking sequences, due to the suitable enumerations of arithmetical functions.

KEYWORDS:

• Robot scientists
• automatic code generation
• empirical laws
• scientific discovery

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The operation $\dot{-}$ corresponds to the subtraction for the natural numbers, i.e. we have that $x\dot{-}y=max\{x-y,0\}$

2 We use $\mathbf{x}$ to denote tuples $(x_1,\dots ,x_n)\in {\mathbb{N}}^n$.

3 Note that $\left(\right(x\dot{-}0\left)\dot{-}0\right)+1=x+1$; for 0<z<y we have $\left(\right(x\dot{-}z\left)\dot{-}\mathrm{xz}\right)+1=0+1=1$.

4 $\mathtt{\text{{Ex}}}$ comes from Explain.

5 In the standard context of learning theory, we take $l=+\mathrm{\infty }$, and we have $\mathcal{M}$ converging to e on $\psi \left[t\right]$ and, for all $t\in \mathbb{N}$, ${\varphi }_e=\psi$.

6 The algorithm used in Section 6.2 will present a better computational time. Since the description for the distance function is larger than the one for the product we will not proceed to experiment with that function.

7 The source code of these projects on automata can be found in https://github.com/gamatos/gold and https://github.com/chengandre/Reg, respectively.

Additional information

Funding

Paula Gouveia acknowledges the support of Fundação para a Ciência e Tecnologia, Instituto de Telecomunicações Research Unit, ref. UIDB/50008/2020.