Lambda Calculus
SelfRepresentation in Girard’s System U, by Matt Brown and Jens Palsberg:
In 1991, Pfenning and Lee studied whether System F could support a typed selfinterpreter. They concluded that typed selfrepresentation for System F “seems to be impossible”, but were able to represent System F in Fω. Further, they found that the representation of Fω requires kind polymorphism, which is outside Fω. In 2009, Rendel, Ostermann and Hofer conjectured that the representation of kindpolymorphic terms would require another, higher form of polymorphism. Is this a case of infinite regress?
We show that it is not and present a typed selfrepresentation for Girard’s System U, the first for a λcalculus with decidable type checking. System U extends System Fω with kind polymorphic terms and types. We show that kind polymorphic types (i.e. types that depend on kinds) are sufficient to “tie the knot” – they enable representations of kind polymorphic terms without introducing another form of polymorphism. Our selfrepresentation supports operations that iterate over a term, each of which can be applied to a representation of itself. We present three typed selfapplicable operations: a selfinterpreter that recovers a term from its representation, a predicate that tests the intensional structure of a term, and a typed continuationpassingstyle (CPS) transformation – the first typed selfapplicable CPS transformation. Our techniques could have applications from verifiably typepreserving metaprograms, to growable typed languages, to more efficient selfinterpreters.
Typed selfrepresentation has come up here on LtU in the past. I believe the best selfinterpreter available prior to this work was a variant of Barry Jay's SFcalculus, covered in the paper Typed SelfInterpretation by Pattern Matching (and more fully developed in Structural Types for the Factorisation Calculus). These covered statically typed selfinterpreters without resorting to undecidable type:type rules.
However, being combinator calculi, they're not very similar to most of our programming languages, and so selfinterpretation was still an active problem. Enter Girard's System U, which features a more familiar type system with only kind * and kindpolymorphic types. However, System U is not strongly normalizing and is inconsistent as a logic. Whether selfinterpretation can be achieved in a strongly normalizing language with decidable type checking is still an open problem.
The project Incremental λCalculus is just starting (compared to more mature approaches like selfadjusting computation), with a first publication last year.
A theory of changes for higherorder languages — incrementalizing λcalculi by static differentiation
Paolo Giarusso, Yufei Cai, Tillmann Rendel, and Klaus Ostermann. 2014
If the result of an expensive computation is invalidated by a small change to the input, the old result should be updated incrementally instead of reexecuting the whole computation. We incrementalize programs through their derivative. A derivative maps changes in the program’s input directly to changes in the program’s output, without reexecuting the original program. We present a program transformation taking programs to their derivatives, which is fully static and automatic, supports firstclass functions, and produces derivatives amenable to standard optimization.
We prove the program transformation correct in Agda for a family of simplytyped λcalculi, parameterized by base types and primitives. A precise interface specifies what is required to incrementalize the chosen primitives.
We investigate performance by a case study: We implement in Scala the program transformation, a plugin and improve performance of a nontrivial program by orders of magnitude.
I like the nice dependent types: a key idea of this work is that the "diffs" possible from a value v do not live in some common type diff(T) , but rather in a valuedependent type diff(v) . Intuitively, the empty list and a nonempty list have fairly different types of possible changes. This makes changemerging and changeproducing operations total, and allow to give them a nice operational theory. Good design, through types.
(The program transformation seems related to the programlevel parametricity transformation. Parametricity abstract over equality justifications, differentiation on small differences.)
In this year's POPL, Bob Atkey made a splash by showing how to get from parametricity to conservation laws, via Noether's theorem:
Invariance is of paramount importance in programming languages and in physics. In programming languages, John Reynolds’ theory of relational parametricity demonstrates that parametric polymorphic programs are invariant under change of data representation, a property that yields “free” theorems about programs just from their types. In physics, Emmy Noether showed that if the action of a physical system is invariant under change of coordinates, then the physical system has a conserved quantity: a quantity that remains constant for all time. Knowledge of conserved quantities can reveal deep properties of physical systems. For example, the conservation of energy, which by Noether’s theorem is a consequence of a system’s invariance under timeshifting.
In this paper, we link Reynolds’ relational parametricity with Noether’s theorem for deriving conserved quantities. We propose an extension of System Fω with new kinds, types and term constants for writing programs that describe classical mechanical systems in terms of their Lagrangians. We show, by constructing a relationally parametric model of our extension of Fω, that relational parametricity is enough to satisfy the hypotheses of Noether’s theorem, and so to derive conserved quantities for free, directly from the polymorphic types of Lagrangians expressed in our system.
In a recent LtU discussion, naasking comments that "I always thought languages that don't specify evaluation order should classify possibly effectful expressions that assume an evaluation order to be errors". Recent work on the C language has provided reasonable formal tools to reason about evaluation order for C, which has very complex evaluationorder rules.
An operational and axiomatic semantics for nondeterminism and sequence points in C
Robbert Krebbers
2014
The C11 standard of the C programming language does not specify
the execution order of expressions. Besides, to make more effective
optimizations possible (e.g. delaying of sideeffects and interleav
ing), it gives compilers in certain cases the freedom to use even
more behaviors than just those of all execution orders.
Widely used C compilers actually exploit this freedom given by
the C standard for optimizations, so it should be taken seriously
in formal verification. This paper presents an operational and ax
iomatic semantics (based on separation logic) for nondeterminism
and sequence points in C. We prove soundness of our axiomatic se
mantics with respect to our operational semantics. This proof has
been fully formalized using the Coq proof assistant.
One aspect of this work that I find particularly interesting is that it provides a program (separation) logic: there is a set of inference rules for a judgment of the form \(\Delta; J; R \vdash \{P\} s \{Q\}\), where \(s\) is a C statement and \(P, Q\) are logical pre,postconditions such that if it holds, then the statement \(s\) has no undefined behavior related to expression evaluation order. This opens the door to practical verification that existing C program are safe in a very strong way (this is all validated in the Coq theorem prover).
Earlier this week Microsoft Research Cambridge organised a Festschrift for Luca Cardelli. The preface from the book:
Luca Cardelli has made exceptional contributions to the world of programming
languages and beyond. Throughout his career, he has reinvented himself every
decade or so, while continuing to make true innovations. His achievements span
many areas: software; language design, including experimental languages;
programming language foundations; and the interaction of programming languages
and biology. These achievements form the basis of his lasting scientific leadership
and his wide impact.
...
Luca is always asking "what is new", and is always looking to
the future. Therefore, we have asked authors to produce short pieces that would
indicate where they are today and where they are going. Some of the resulting
pieces are short scientific papers, or abridged versions of longer papers; others are
less technical, with thoughts on the past and ideas for the future. We hope that
they will all interest Luca.
Hopefully the videos will be posted soon.
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 lambdacalculus 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, selfcontained 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 selfcontained 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 twoway correspondences, with the occasional extra logarithmic overhead  or, from another point of view, provided probably costeffective 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 cacheefficiency 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 lambdacalculus cost semantics with several machine models.
This paper formally studies the question of how much parallelism is available in callbyvalue 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 callbyvalue lambdacalculus. 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 APAL (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 workefficient (the processortime product on the machines is within a constant factor of the work on the APAL), and for P processors the slowdown (time on the machines divided by depth on the APAL) is proportional to at most O(log P). We also prove bounds for simulating the PRAM on the APAL.
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 cacheefficient 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 idealcache 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 idealcache 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.
Pure Subtype Systems, by DeLesley S. Hutchins:
This paper introduces a new approach to type theory called pure subtype systems. Pure subtype systems differ from traditional approaches to type theory (such as pure type systems) because the theory is based on subtyping, rather than typing. Proper types and typing are completely absent from the theory; the subtype relation is defined directly over objects. The traditional typing relation is shown to be a special case of subtyping, so the loss of types comes without any loss of generality.
Pure subtype systems provide a uniform framework which seamlessly integrates subtyping with dependent and singleton types. The framework was designed as a theoretical foundation for several problems of practical interest, including mixin modules, virtual classes, and featureoriented programming.
The cost of using pure subtype systems is the complexity of the metatheory. We formulate the subtype relation as an abstract reduction system, and show that the theory is sound if the underlying reductions commute. We are able to show that the reductions commute locally, but have thus far been unable to show that they commute globally. Although the proof is incomplete, it is â€œclose enoughâ€ to rule out obvious counterexamples. We present it as an open problem in type theory.
A thoughtprovoking take on type theory using subtyping as the foundation for all relations. He collapses the type hierarchy and unifies types and terms via the subtyping relation. This also has the sideeffect of combining type checking and partial evaluation. Functions can accept "types" and can also return "types".
Of course, it's not all sunshine and roses. As the abstract explains, the metatheory is quite complicated and soundness is still an open question. Not too surprising considering type checking Type:Type is undecidable.
Hutchins' thesis is also available for a more thorough treatment. This work is all in pursuit of Hitchens' goal of featureoriented programming.
Conor McBride gave an 8lecture summer course on Dependently typed metaprogramming (in Agda) at the Cambridge University Computer Laboratory:
Dependently typed functional programming languages such as Agda are capable of expressing very precise types for data. When those data themselves encode types, we gain a powerful mechanism for abstracting generic operations over carefully circumscribed universes. This course will begin with a rapid depedentlytyped programming primer in Agda, then explore techniques for and consequences of universe constructions. Of central importance are the â€œpattern functorsâ€ which determine the node structure of inductive and coinductive datatypes. We shall consider syntactic presentations of these functors (allowing operations as useful as symbolic differentiation), and relate them to the more uniform abstract notion of â€œcontainerâ€. We shall expose the doublelife containers lead as â€œinteraction structuresâ€ describing systems of effects. Later, we step up to functors over universes, acquiring the power of inductiverecursive definitions, and we use that power to build universes of dependent types.
The lecture notes, code, and video captures are available online.
As with his previous course, the notes contain many(!) mind expanding exploratory exercises, some of which quite challenging.
Mechanized Î»_{JS}
The Brown PLT Blog, 20120604
In an earlier post, we introduced Î»_{JS}, our operational semantics for JavaScript. Unlike many other operational semantics, Î»_{JS} is no toy, but strives to correctly model JavaScript's messy details. To validate these claims, we test Î»_{JS} with randomly generated tests and with portions of the Mozilla JavaScript test suite.
Testing is not enough. Despite our work, other researchers found a missing case in Î»_{JS}. Today, we're introducing Mechanized Î»_{JS}, which comes with a machinechecked proof of correctness, using the Coq proof assistant.
More work on mechanizing the actual, implemented semantics of a real language, rather than a toy.
Milawa
Aug 2010  May 2011. Magnus Myreen has developed a verified Lisp system, named Jitawa, which can run Milawa. Our paper about this project was accepted to ITP 2011.
This is pretty interesting: Milawa was already "selfverifying," in the sense explained on the page. More recently, it's been made to run on a verified Lisp runtime, so that means the entire stack down to the X86_64 machine code is verified. Milawa itself is "ACL2like," so it's not as interesting logically as, say, Isabelle or Coq, but it's far from a toy. Also, the Jitawa formalization apparently took place in HOL4, so you need to trust HOL4. Since HOL4 is an "LCFlike" system, you can do that to the extent that you trust the LCF process, but it doesn't satisfy the de Bruijn criterion in the same way Milawa or Coq do. Nevertheless, this seems like an important step toward the ultimate goal of having a stack that is verified "all the way down," as it were.

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