## Cost semantics for functional languages

There is an ongoing discussion in LtU (there, and there) on whether RAM and other machine models are inherently a better basis to reason about (time and) memory usage than lambda-calculus and functional languages. Guy Blelloch and his colleagues have been doing very important work on this question that seems to have escaped LtU's notice so far.

A portion of the functional programming community has long been of the opinion that we *do not* need to refer to machines of the Turing tradition to reason about execution of functional programs. Dynamic semantics (which are often perceived as more abstract and elegant) are adequate, self-contained descriptions of computational behavior, which we can elevate to the status of (functional) machine model -- just like "abstract machines" can be seen as just machines.

This opinion has been made scientifically precise by various brands of work, including for example implicit (computational) complexity, resource analysis and cost semantics for functional languages. Guy Blelloch developed a family of *cost semantics*, which correspond to annotations of operational semantics of functional languages with new information that captures more intentional behavior of the computation: not only the result, but also running time, memory usage, degree of parallelism and, more recently, interaction with a memory cache. Cost semantics are self-contained way to think of the efficiency of functional programs; they can of course be put in correspondence with existing machine models, and Blelloch and his colleagues have proved a vast amount of two-way correspondences, with the occasional extra logarithmic overhead -- or, from another point of view, provided probably cost-effective implementations of functional languages in imperative languages and conversely.

This topic has been discussed by Robert Harper in two blog posts, Language and Machines which develops the general argument, and a second post on recent joint work by Guy and him on integrating cache-efficiency into the model. Harper also presents various cost semantics (called "cost dynamics") in his book "Practical Foundations for Programming Languages".

In chronological order, three papers that are representative of the evolution of this work are the following.

Parallelism In Sequential Functional Languages

Guy E. Blelloch and John Greiner, 1995.

This paper is focused on parallelism, but is also one of the earliest work carefully relating a lambda-calculus cost semantics with several machine models.

This paper formally studies the question of how much parallelism is available in call-by-value functional languages with no parallel extensions (i.e., the functional subsets of ML or Scheme). In particular we are interested in placing bounds on how much parallelism is available for various problems. To do this we introduce a complexity model, the PAL, based on the call-by-value lambda-calculus. The model is defined in terms of a profiling semantics and measures complexity in terms of the total work and the parallel depth of a computation. We describe a simulation of the A-PAL (the PAL extended with arithmetic operations) on various parallel machine models, including the butterfly, hypercube, and PRAM models and prove simulation bounds. In particular the simulations are work-efficient (the processor-time product on the machines is within a constant factor of the work on the A-PAL), and for P processors the slowdown (time on the machines divided by depth on the A-PAL) is proportional to at most O(log P). We also prove bounds for simulating the PRAM on the A-PAL.

Space Profiling for Functional Programs

Daniel Spoonhower, Guy E. Blelloch, Robert Harper, and Phillip B. Gibbons, 2011 (conference version 2008)

This paper clearly defines a notion of ideal memory usage (the set of store locations that are referenced by a value or an ongoing computation) that is highly reminiscent of garbage collection specifications, but without making any reference to an actual garbage collection implementation.

We present a semantic space profiler for parallel functional programs. Building on previous work in sequential profiling, our tools help programmers to relate runtime resource use back to program source code. Unlike many profiling tools, our profiler is based on a cost semantics. This provides a means to reason about performance without requiring a detailed understanding of the compiler or runtime system. It also provides a specification for language implementers. This is critical in that it enables us to separate cleanly the performance of the application from that of the language implementation. Some aspects of the implementation can have significant effects on performance. Our cost semantics enables programmers to understand the impact of different scheduling policies while hiding many of the details of their implementations. We show applications where the choice of scheduling policy has asymptotic effects on space use. We explain these use patterns through a demonstration of our tools. We also validate our methodology by observing similar performance in our implementation of a parallel extension of Standard ML

Cache and I/O efficient functional algorithms

Guy E. Blelloch, Robert Harper, 2013 (see also the shorter CACM version)

The cost semantics in this last work incorporates more notions from garbage collection, to reason about cache-efficient allocation of values -- in that it relies on work on formalizing garbage collection that has been mentioned on LtU before.

The widely studied I/O and ideal-cache models were developed to account for the large difference in costs to access memory at different levels of the memory hierarchy. Both models are based on a two level memory hierarchy with a fixed size primary memory (cache) of size \(M\), an unbounded secondary memory, and assume unit cost for transferring blocks of size \(B\) between the two. Many algorithms have been analyzed in these models and indeed these models predict the relative performance of algorithms much more accurately than the standard RAM model. The models, however, require specifying algorithms at a very low level requiring the user to carefully lay out their data in arrays in memory and manage their own memory allocation.

In this paper we present a cost model for analyzing the memory efficiency of algorithms expressed in a simple functional language. We show how many algorithms written in standard forms using just lists and trees (no arrays) and requiring no explicit memory layout or memory management are efficient in the model. We then describe an implementation of the language and show provable bounds for mapping the cost in our model to the cost in the ideal-cache model. These bound imply that purely functional programs based on lists and trees with no special attention to any details of memory layout can be as asymptotically as efficient as the carefully designed imperative I/O efficient algorithms. For example we describe an \(O(\frac{n}{B} \log_{M/B} \frac{n}{B})\) cost sorting algorithm, which is optimal in the ideal cache and I/O models.