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Research on the inductive synthesis of recursive functional programs started in the early 1970s and was brought onto firm theoretical foundations with the seminal THESIS system of Summers<ref>{{cite journal|first1=P.D.|last1=Summers|title=A methodology for LISP program construction from examples|journal=J ACM|volume=24 | issue = 1|pages=161–175|year=1977|doi=10.1145/321992.322002|s2cid=7474210|doi-access=free}}</ref> and work of Biermann.<ref>{{cite journal|first1=A.W.|last1=Biermann|title=The inference of regular LISP programs from examples|journal=IEEE Trans Syst Man Cybern|volume=8 | issue = 8|pages=585–600|year=1978|doi=10.1109/tsmc.1978.4310035|s2cid=15277948}}</ref>
These approaches were split into two phases: first, input-output examples are transformed into non-recursive programs (traces) using a small set of basic operators; second, regularities in the traces are searched for and used to fold them into a recursive program. The main results until the mid
The advent of logic programming brought a new elan but also a new direction in the early 1980s, especially due to the MIS system of Shapiro<ref>{{cite book|first1=E.Y.|last1=Shapiro|title=Algorithmic program debugging|publisher=The MIT Press|year=1983}}</ref> eventually spawning the new field of inductive logic programming (ILP).<ref>{{Cite journal | last1 = Muggleton | first1 = S. | title = Inductive logic programming | doi = 10.1007/BF03037089 | journal = New Generation Computing | volume = 8 | issue = 4 | pages = 295–318 | year = 1991 | citeseerx = 10.1.1.329.5312 | s2cid = 5462416 }}</ref> The early works of Plotkin,<ref>{{cite journal|first1=Gordon D.|last1=Plotkin|title=A Note on Inductive Generalization|editor1-first=B.|editor1-last=Meltzer|editor2-first=D.|editor2-last=Michie|journal=Machine Intelligence|volume=5|pages=153–163|year=1970|url=http://homepages.inf.ed.ac.uk/gdp/publications/MI5_note_ind_gen.pdf}}</ref><ref>{{cite journal|first1=Gordon D.|last1=Plotkin|title=A Further Note on Inductive Generalization|editor1-first=B.|editor1-last=Meltzer|editor2-first=D.|editor2-last=Michie|journal=Machine Intelligence|volume=6|pages=101–124|year=1971}}</ref> and his "''relative least general generalization (rlgg)''", had an enormous impact in inductive logic programming. Most of ILP work addresses a wider class of problems, as the focus is not only on recursive logic programs but on machine learning of symbolic hypotheses from logical representations. However, there were some encouraging results on learning recursive Prolog programs such as quicksort from examples together with suitable background knowledge, for example with GOLEM.<ref>{{cite journal|first1=S.H.|last1=Muggleton|first2=C.|last2=Feng|s2cid=14992676|title=Efficient induction of logic programs|journal=Proceedings of the Workshop on Algorithmic Learning Theory|volume=6|pages=368–381|year=1990}}</ref> But again, after initial success, the community got disappointed by limited progress about the induction of recursive programs<ref>{{cite journal
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|year=2012
|url=http://ilp11.doc.ic.ac.uk/short_papers/ilp2011_submission_62.pdf}}
</ref><ref name="Inducing probabilistic programs by">{{cite arXiv
|first1=H.|last1=Irvin|first2=A.|last2=Stuhlmuller|first3=N.D.|last3=Goodman
|title=Inducing probabilistic programs by Bayesian program merging|eprint=1110.5667
|year=2011|class=cs.AI}}</ref>
One powerful paradigm that has been recently used for the representation of hypotheses in inductive programming (generally in the form of [[generative model]]s) is [[probabilistic programming language|probabilistic programming]] (and related paradigms, such as stochastic logic programs and Bayesian logic programming).<ref>{{cite journal
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|publisher=Springer
|year=2008}}
</ref><ref name="Inducing probabilistic programs by"/><ref name="Reasoning about reasoning by nested conditioning: Modeling theory of mind with probabilistic programs">▼
▲</ref><ref name="Reasoning about reasoning by nested conditioning: Modeling theory of mind with probabilistic programs">
{{cite journal
|first1=A.|last1=Stuhlmuller|first2=N.D.|last2=Goodman
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