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Example-Driven Construction of Functional Programs

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Example-Driven Construction of Functional Programs Inductive Programming – Example-driven construction of functional programs Ute Schmid, Martin Hofmann, Emanuel Kitzelmann We developed an efficient, analytical approach for learning recursive functional programs from examples. Igor2 is a realization of this approach as constructor term rewriting system. We can show that our approach compares very well to other systems of inductive programming and we will discuss applications in the domains of cognitive modelling, end-user programming and automated software engineering. 1 Introduction programs: In a first step, input/output examples were rewritten into traces, in a second step recurrent patterns were searched-for Inductive programming research addresses the problem of learn- in the traces and the found regularities were generalised to a re- ing computer programs from incomplete specifications, typically cursive function. Thesys was restricted to structural problems samples of the desired input/output behaviour and possibly addi- on lists such as unpacking elements or reverting. Due to that tional constraints (Biermann, Guiho, & Kodratoff, 1984; Flener restriction rewriting of input/output examples could be realized & Schmid, to appear). Induced programs are typically in a with a deterministic algorithm where inputs where characterised declarative – functional or logic – programming language. As by the list structure using empty as only predicate. Folding of a special application of machine learning (Mitchell, 1997), in- traces was restricted to linear recursion based on Summers’ syn- ductive programming creates program hypotheses, that is, gen- thesis theorem which stated that a Kleene sequence of traces eralised, typically recursive, programs. In contrast to classifica- can be interpreted as being generated by the kth sequence of tion learning, program hypotheses must cover all given exam- unfoldings of the searched-for linear recursive program. ples correctly since for programs it is expected that a desired For Thesys and all later analytical approaches it is enough input/output relation holds for all possible inputs. In contrast to present a small set of only positive input/output examples. to deductive approaches to automated program construction, These examples must be the first representants of the underlying there is no guarantee that the induced program meets the in- data-type of the input parameter. In contrast, in search-based tension of the user. Inductive programming approaches can be approaches, an arbitrary set of positive examples can be pre- characterised by learning biases, especially by a language bias sented. Typically, more examples than for analytical approaches which restricts the syntactic form of programs which can be are necessary. In addition negative examples can be used to characterised by a program scheme and by a search bias which eliminate unsuitable hypotheses. determines the sequence in which hypotheses are constructed. An extension of Thesys to a larger class of recursive pro- In contrast to deductive approaches, the barrier for application is grams is Igor1 (Schmid & Wysotzki, 1998; Schmid, 2003; lower since the system user is not required to become an expert Kitzelmann & Schmid, 2006). Rewriting of input/output ex- in the domain of formal specifications. Therefore, inductive pro- amples allowed nested expressions and folding was guided by a gramming approaches are good candidates for the development more general program scheme allowing for linear and tree recur- of programming assistants (Flener & Partridge, 2001). sions and especially for inducing sets instead of single recursive There are two general approaches to inductive programming: functions. Another analytical system in the context of ILP is analytical and search-based methods. Search-based methods Dialogs (Flener, 1996). Here synthesis relies on interaction enumerate syntactically correct programs and test these pro- with a user who selects a suitable program scheme for the given grams against the given examples guided by some search strat- problem. The most recent analytical approach to inductive pro- egy. The inductive logic programming (ILP) system Foil con- gramming is Igor2 which will be presented in the following. structs sets of Horn clauses by sequential covering, that is by a first construct then test approach. This approach was also ap- plied to learning recursive rule sets (Quinlan & Cameron-Jones, 2 IGOR2 1995). The most successful search-based system is Adate (Ols- son, 1995) which constructs ML programs using evolutionary Igor2 (Kitzelmann & Schmid, 2007; Kitzelmann, 2008; Kitzel- principles. The system MagicHaskeller (Katayama, 2005) mann & Hofmann, 2008; Kitzelmann, 2009) is a new analytical enumerates Haskell programs with higher-order functions. approach to inductive programming which overcomes inherent Analytical methods are guided by the structure underlying limitations of systems in the Thesys tradition, includes pow- the given input/output examples. Beginning of research on in- erful concepts introduced in inductive logic programming, and ductive programming in the nineteen-seventies was concerned relates to the state of art in functional programming languages. with such analytical strategies for learning Lisp programs from The two step approach of Thesys results in an inherent bottle- small sets of positive input/output examples (Biermann et al., neck to program synthesis due to the first step – rewriting in- 1984). The most influential of the early systems is Thesys put/output examples to traces. A rewriting algorithm relies on a (Summers, 1977). It realized a two step approach to synthesise set of predefined predicates to discern different inputs and on a Page 1 set of predefined operators such as list constructors and decon- (* data type specification omitted *) eq Reverse([]) = [] structors to express the construction of outputs from the inputs. eq Reverse(cons(X, [])) = cons(X,[]) Only if both sets are kept very small and if discerning of in- eq Reverse(cons(X, cons(Y, []))) = cons(Y, cons(X, [])) puts is possible without nested conditions an efficient algorithm eq Reverse(cons(X, cons(Y, cons(Z, [])))) = with unique results can be defined. In consequence, folding of cons(Z, cons(Y, cons(X, []))) traces into recursive programs is restricted to such simple trace eq Reverse(cons(X, cons(Y, cons(Z, cons(V, []))))) = structures also. To allow the generation of more complex traces, cons(V, cons(Z, cons(Y, cons(X, [])))) the rewriting algorithm needs to employ more complex pattern Reverse([]) = [] recognition abilities and partially has to rely on the same infer- Reverse(cons(X0,X1)) = cons(Sub1(cons(X0,X1)), ence mechanisms which are used in the folding step of synthesis. Reverse(Sub2(cons(X0,X1)))) Igor2 therefore relies on a single-step algorithm to generalise input/output examples to recursive programs. (* last *) The idea of allowing the use of background knowledge in Sub1(cons(X0, [])) = X0 Sub1(cons(X0, cons(X1, X2))) = Sub1(cons(X1, X2)) learning as it was introduced in ILP (Muggleton & De Raedt, 1994) was also successfully applied in ILP approaches to induc- (* init *) tive programming (Muggleton & Feng, 1990). More complex Sub2(cons(X0,[])) = [] programs can be learned by providing information of functions Sub2(cons(X0, cons(X1, X2))) = cons(X0, Sub2(cons(X1, X2))) which can be used in program construction. For example, quick- sort can be learned by providing not only examples for sorting Figure 1: Induction of reverse by Igor2 but also examples for partitioning lists and for appending lists. One drawback of search-based approaches is, however, that the background knowledge needs to be restricted to such functions of the example outputs with according parameter substitutions which are necessary to induce the target program because oth- and empty condition. If the resulting rule bodies contain un- erwise there is combinatorial explosion. Igor2 allows the use of bound variables, successor hypotheses are computed applying background knowledge but the introduction of such additional the following three methods: (1) Partitioning of the inputs by functions is restricted by matching against the given examples replacing one pattern by a set of disjoint more specific patterns due to the analytical strategy used. or by adding a predicate to the condition. (2) Replacing the Contemporary functional languages such as ML or Haskell body by a (recursive) call of a defined function, where finding allow to define functions using patterns for case distinction and the argument of the function call is treated as a new induc- rely on the definition of data types. These characteristics are re- tion problem. (3) Replacing the sub-terms in which unbound flected in constructor term rewriting systems (Baader & Nipkow, variables occur by a call to new sub-programs. In cases (2) 1998). Igor2 is defined in this framework and implemented in and (3) auxiliary functions are invented, abducing input/output Maude (Clavel et al., 2003). Input for Igor2 is examples for them. The algorithm is described in more detail • a set of non-recursive equations specifying the input/output in Kitzelmann (2009) the system can be downloaded from the examples of the target function, project website. • definitions of the data types for the target program (which can be very general, e.g. just sort, or more specific) by means of constructors, 3 Comparison of IP Systems • optional definitions of background knowledge, also in form Igor2 of non-recursive equations. To get closer insights in the strengths and weaknesses of The
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