Inductive functional programming systems can be characterised by two diametric approaches: Either they apply exhaustive program enumeration which uses input/output examples (IO) as test cases, or they perform an analytical, data-driven structural generalisation of the IO examples. Enumerative approaches ignore the structural information provided with the IO examples, but use type information to guide and restrict the search. They use higher-order functions which capture recursion schemes during their enumeration, but apply them randomly in a uninformed manner. Analytical approaches on the other side heavily exploit this structural information, but have ignored the benefits of a strong type system so far and use only recursion schemes either fixed and built in, or selected by an expert user. In category theory universal constructs, such as natural transformations or type morphisms, describe recursion schemes which can be defined on any inductively defined data type. They can be characterised by specific universal properties. Those type morphisms and related concepts provide a categorical approach to functional programming, which is often called categorical programming. This work shows how categorical programming can be applied to Inductive Programming and how universal constructs, such as catamorphisms, paramorphisms, and type functors, can be used as recursive program schemes for inductive functional programming. The use of program schemes for Inductive Programming is not new. The special appeal and novelty of this work is that, contrary to previous approaches, the program schemes are neither fixed, nor selected by an expert user: The applicability of those recursion schemes can be automatically detected in the given IO examples of a target function by checking the universal properties of the corresponding type morphisms. Applying this to the analytical system Igor2, both the capabilities and the expressiveness can be extended without a decrease in efficiency. An extension of the analytical functional inductive programming system Igor2 is proposed and its algorithms described. An empirical evaluation demonstrates the improvements with respect to efficiency and effectiveness that can be achieved by the use of type morphisms for Igor2 due to a reduction in search space complexity.
Inductive programming is concerned with the automated construction of declarative, often functional, recursive programs from incomplete specifications such as input/output examples. The inferred program must be correct with respect to the provided examples in a generalising sense: it should be neither equivalent to them, nor inconsistent. Inductive programming algorithms are guided explicitly or implicitly by a language bias (the class of programs that can be induced) and a search bias (determining which generalised program is constructed first). Induction strategies are either generate-and-test or example-driven. In generate-and-test approaches, hypotheses about candidate programs are generated independently from the given specifications. Program candidates are tested against the given specification and one or more of the best evaluated candidates are developed further. In analytical approaches, candidate programs are constructed in an example-driven way. While generate-and-test approaches can -- in principle -- construct any kind of program, analytical approaches have a more limited scope. On the other hand, efficiency of induction is much higher in analytical approaches. Inductive programming is still mainly a topic of basic research, exploring how the intellectual ability of humans to infer generalised recursive procedures from incomplete evidence can be captured in the form of synthesis methods. Intended applications are mainly in the domain of programming assistance -- either to relieve professional programmers from routine tasks or to enable non-programmers to some limited form of end-user programming. Furthermore, in the future, inductive programming techniques might be applied to further areas such as supporting the inference of lemmata in theorem proving or learning grammar rules. Inductive automated program construction has been originally addressed by researchers in artificial intelligence and machine learning. During the last years, some work on exploiting induction techniques has been started also in the functional programming community. Therefore, the third workshop on |Approaches and Applications of Inductive Programming| took place for the first time in conjunction with the ACM SIGPLAN International Conference on Functional Programming (ICFP 2009). The first and second workshop were associated with the International Conference on Machine Learning (ICML 2005) and the European Conference on Machine Learning (ECML 2007). AAIP´09 aimed to bring together researchers from the functional programming and the artificial intelligence communities, working in the field of inductive functional programming, and advance fruitful interactions between these communities with respect to programming techniques for inductive programming algorithms, the identification of challenge problems and potential applications. For everybody interested in inductive programming we recommend to visit the website: www.inductive-programming.org.