The expression problem describes a fundamental trade-off in program design: Should a program's primary decomposition be determined by the way its domain objects are constructed ("functional" decomposition), or by the way they are destructed ("object-oriented" decomposition)? We argue that programming languages should not force one of these decompositions on the programmer; rather, a programming language should support both ways of decomposing a program in a symmetric way, with an easy translation between these decompositions. However, current programming languages are usually not symmetric and hence make it unnecessarily hard to switch the decomposition. We propose a language that is symmetric in this regard and allows a fully automatic translation between "functional" and "object-oriented" decomposition. We present a language with algebraic data types and pattern matching for "functional" decomposition and codata types and copattern matching for "object-oriented" decomposition, together with a bijective translation that turns a data type into a codata type ("destructorization") or vice versa ("constructorization"). We present the first symmetric programming language with support for local (co)pattern matching, which includes local anonymous function or object definitions, that allows an automatic translation as described above. We also present the first mechanical formalization of such a language and prove i) that the type system is sound, that the translations between data and codata types are ii) type-preserving, iii) behavior-preserving and iv) inverses of each other. We also extract a mechanically verified implementation from our formalization and have implemented an IDE with direct support for these translations.
No abstract
Functional programming language design has been shaped by the framework of natural deduction, in which language constructs are divided into introduction and elimination rules for producers of values. In sequent calculus-based languages, left introduction rules replace (right) elimination rules and provide a dedicated sublanguage for consumers of values. In this paper, we present and analyze a wider design space of programming languages which encompasses four kinds of rules: Introduction and elimination, both left and right. We analyze the influence of rule choice on program structure and argue that having all kinds of rules enriches a programmer’s modularity arsenal. In particular, we identify four ways of adhering to the principle that ”the structure of the program follows the structure of the data“ and show that they correspond to the four possible choices of rules. We also propose the principle of bi-expressibility to guide and validate the design of rules for a connective. Finally, we deepen the well-known dualities between different connectives by means of the proof/refutation duality.
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