Abstract
In this paper we propose a novel semi-supervised active machine-learning method, based on two recursive higher-order functions that can inductively synthesize a functional computer program. Based on properties formulated using abstract algebra terms, the method uses two combined strategies: to reduce the dimensionality of the Boolean algebra where a target function lies and to combine known operations belonging to the algebra, using them as a basis to build a program that emulates the target function. The method queries for data on specific points of the problem input space and build a program that exactly fits the data. Applications of this method include all sorts of systems based on bitwise operations. Any functional computer program can be emulated using this approach. Combinatorial circuit design, model acquisition from sensor data, reverse engineering of existing computer programs are all fields where the proposed method can be useful.
Keywords
- Boolean Function
- Boolean Algebra
- Input Space
- Target Function
- Inductive Logic Programming
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Notes
- 1.
Function-level programming, as proposed by Backus [14], is a particular, constrained type of functional programming where a program is built directly from programs that are given at the outset, by combining them with program-forming operations or functionals.
- 2.
Throughout this paper we will use indistinctively 0, F or false for the binary number 0 and 1, T or true for the binary number 1.
- 3.
Bit strings can represent any complex data type. Consequently, our functional setting includes any functional computer program having fixed length input and output.
- 4.
The dimension of the Boolean algebra will also determine the minimum number of monomials required to define each of its Boolean polynomials in algebraic normal form.
- 5.
We will use the overline notation to distinguish between modules (Boolean functions and Boolean spaces) belonging to the target function subspace (without overline) and modules belonging to a lower dimension linear subspace generated by the drill linear mapping (with overline).
- 6.
\(F_r\) is a set containing Boolean functions belonging to a Boolean algebra of dimension \(r\).
- 7.
Note that in Eq. 8 we are considering the target function as having one single vector input \(X\) while in 2 the same target function has two vector inputs \(X\) and \(Y\). These different notations for the same target function can be understood as \(X\) in Eq. 8 being the concatenation of \(X\) and \(Y\) from Eq. 2 or \(X\) and \(Y\) in Eq. 2 being a split of \(X\) from Eq. 8 in two vector subspaces. It follows that drill can be applied only to target Boolean functions having at least two input variables while join can be applied to target Boolean functions of any arity.
- 8.
We will use the double overline notation to distinguish between modules (Boolean functions and Boolean spaces) belonging to the target function subspace (without overline) and modules belonging to a lower dimension linear subspace generated by the join linear mapping (with double overline).
- 9.
The notation \(\overline{f_i}\) indicates a function resulting from the i-th recursive drill linear mapping.
- 10.
The notation \(\overline{\overline{f_i}}\) indicates a function resulting from the i-th recursive join linear mapping.
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Acknowledgments
The author would like to thank Pierre-Jean Laurent from the Laboratoire de Modelisation et Calcul- LMC-IMAG at the Universite Joseph Fourier, Grenoble, France for his contributions concerning the mathematical proofs of the proposed method and Emmanuel Mazer from the Institut National De Recherche en Informatique et en Automatique- INRIA- Rhne Alpes, France for his assistance and helpful contributions to this research.
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Appendices
Appendices
1.1 Details of the Fibonacci Sequence Program Synthesis
The translation between bit lists and integers is made using the following routines:
The Fibonacci sequence can be implemented in Lisp as:
Calling the synthesis procedure and then testing the generated program:
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Balaniuk, R. (2015). Drill and Join: A Method for Exact Inductive Program Synthesis. In: Proietti, M., Seki, H. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2014. Lecture Notes in Computer Science(), vol 8981. Springer, Cham. https://doi.org/10.1007/978-3-319-17822-6_13
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DOI: https://doi.org/10.1007/978-3-319-17822-6_13
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