Preliminary Proceedings of the 7 th International Workshop on Expressiveness in Concurrency

Compare rst-order functional programs with higher-order programs allowing functions asfunction parameters. Can the the rst program class solve fewer problems than the second?The answer is no: both classes are Turing complete, meaning that they can compute allpartial recursive functions. In particular, higher-order values may be rst-order simulatedby use of the list constructor \cons" to build function closures.This paper uses complexity theory to prove some expressivity results about small pro-gramming languages that are less than Turing complete. Complexity classes of decisionproblems are used to characterize the expressive power of functional programming lan-guage features. An example: second-order programs are more powerful than rst-order,since a function f of type [Bool]->Bool is computable by a cons-free rst-order functionalprogram if and only if f is in ptime, whereas f is computable by a cons-free second-orderprogram if and only if f is in exptime.Exact characterizations are given for those problems of type [Bool]->Bool solvableby programs with several combinations of operations on data: presence or absence ofconstructors; the order of data values: 0, 1, or higher; and program control structures:general recursion, tail recursion, primitive recursion. 1 On expressivity in programming languages Does the programming style we use a ect the problems we can solve, or the e ciencyof the programs we can write to solve a given problem? Some especially relevantquestions: 1. Does the use of functions as data values give a greater problem-solving ability?2. Is recursion more powerful than iteration?3. Would the use of a strongly normalizing programming language, as proposedby several researchers, including (Turner, 1996) and (Lloyd, 1999), imposeany limitations on the problems that can be solved? This paper gives some answers to these questions, when restricted to simply typedfunctional programs without \cons". In particular, the answer to question 1 is This paper is an extract from a forthcoming article of the same title in the Journal of FunctionalProgramming. This research was partially supported by the Danish Natural Science ResearchCouncil (DART project).