The complexity of abstract machines

I previously wrote about a brand of research by Guy Blelloch on the Cost semantics for functional languages, which let us make precise claim about the complexity of functional programs without leaving their usual and comfortable programming models (beta-reduction).

While the complexity behavior of weak reduction strategies, such as call-by-value and call-by-name, is by now relatively well-understood, the lambda-calculus has a much richer range of reduction strategies, in particular those that can reduce under lambda-abstractions, whose complexity behavior is sensibly more subtle and was, until recently, not very well understood. (This has become a practical concern since the rise in usage of proof assistants that must implement reduction under binders and are very concerned about the complexity of their reduction strategy, which consumes a lot of time during type/proof-checking.)

Beniamino Accatoli, who has been co-authoring a lot of work in that area, recently published on arXiv a new paper that has survey quality, and is a good introduction to this area of work and other pointers from the literature.

The Complexity of Abstract Machines

Beniamino Accatoli, 2017

The lambda-calculus is a peculiar computational model whose definition does not come with a notion of machine. Unsurprisingly, implementations of the lambda-calculus have been studied for decades. Abstract machines are implementations schema for fixed evaluation strategies that are a compromise between theory and practice: they are concrete enough to provide a notion of machine and abstract enough to avoid the many intricacies of actual implementations. There is an extensive literature about abstract machines for the lambda-calculus, and yet -- quite mysteriously -- the efficiency of these machines with respect to the strategy that they implement has almost never been studied.

This paper provides an unusual introduction to abstract machines, based on the complexity of their overhead with respect to the length of the implemented strategies. It is conceived to be a tutorial, focusing on the case study of implementing the weak head (call-by-name) strategy, and yet it is an original re-elaboration of known results. Moreover, some of the observation contained here never appeared in print before.

Philip Wadler: Category Theory for the Working Hacker

Nothing you don't already know, if you are inteo this sort of thing (and many if not most LtU-ers are), but a quick way to get the basic idea if you are not. Wadler has papers that explain Curry-Howard better, and the category theory content here is very basic -- but it's an easy listen that will give you the fundamental points if you still wonder what this category thing is all about.

To make this a bit more fun for those already in the know: what is totally missing from the talk (understandable given time constraints) is why this should interest the "working hacker". So how about pointing out a few cool uses/ideas that discerning hackers will appreciate? Go for it!

Fully Abstract Compilation via Universal Embedding

Fully Abstract Compilation via Universal Embedding by Max S. New, William J. Bowman, and Amal Ahmed:

A fully abstract compiler guarantees that two source components are observationally equivalent in the source language if and only if their translations are observationally equivalent in the target. Full abstraction implies the translation is secure: target-language attackers can make no more observations of a compiled component than a source-language attacker interacting with the original source component. Proving full abstraction for realistic compilers is challenging because realistic target languages contain features (such as control effects) unavailable in the source, while proofs of full abstraction require showing that every target context to which a compiled component may be linked can be back-translated to a behaviorally equivalent source context.

We prove the first full abstraction result for a translation whose target language contains exceptions, but the source does not. Our translation—specifically, closure conversion of simply typed λ-calculus with recursive types—uses types at the target level to ensure that a compiled component is never linked with attackers that have more distinguishing power than source-level attackers. We present a new back-translation technique based on a deep embedding of the target language into the source language at a dynamic type. Then boundaries are inserted that mediate terms between the untyped embedding and the strongly-typed source. This technique allows back-translating non-terminating programs, target features that are untypeable in the source, and well-bracketed effects.

Potentially a promising step forward to secure multilanguage runtimes. We've previously discussed security vulnerabilities caused by full abstraction failures here and here. The paper also provides a comprehensive review of associated literature, like various means of protection, back translations, embeddings, etc.

Among many interesting works, the POPL 2016 papers have a bunch of nice articles on Gradual Typing.

by Matteo Cimini, Jeremy Siek
2016

Many languages are beginning to integrate dynamic and static typing. Siek and Taha offered gradual typing as an approach to this integration that provides the benefits of a coherent and full-span migration between the two disciplines. However, the literature lacks a general methodology for designing gradually typed languages. Our first contribution is to provide such a methodology insofar as the static aspects of gradual typing are concerned: deriving the gradual type system and the compilation to the cast calculus.

Based on this methodology, we present the Gradualizer, an algorithm that generates a gradual type system from a normal type system (expressed as a logic program) and generates a compiler to the cast calculus. Our algorithm handles a large class of type systems and generates systems that are correct with respect to the formal criteria of gradual typing. We also report on an implementation of the Gradualizer that takes type systems expressed in lambda-prolog and outputs their gradually typed version (and compiler to the cast calculus) in lambda-prolog.

One can think of the Gradualizer as a kind of meta-programming algorithm that takes a type system in input, and returns a gradual version of this type system as output. I find it interesting that these type systems are encoded as lambda-prolog programs (a notable use-case for functional logic programming). This is a very nice way to bridge the gap between describing a transformation that is "in principle" mechanizable and a running implementation.

An interesting phenomenon happening once you want to implement these ideas in practice is that it forced the authors to define precisely many intuitions everyone has when reading the description of a type system as a system of inference rules. These intuitions are, broadly, about the relation between the static and the dynamic semantics of a system, the flow of typing information, and the flow of values; two occurrences of the same type in a typing rule may play very different roles, some of which are discussed in this article.

by Asumu Takikawa, Daniel Feltey, Ben Greenman, Max New, Jan Vitek, Matthias Felleisen
2016

Programmers have come to embrace dynamically typed languages for prototyping and delivering large and complex systems. When it comes to maintaining and evolving these systems, the lack of explicit static typing becomes a bottleneck. In response, researchers have explored the idea of gradually typed programming languages which allow the post-hoc addition of type annotations to software written in one of these “untyped” languages. Some of these new hybrid languages insert run-time checks at the boundary between typed and untyped code to establish type soundness for the overall system. With sound gradual typing programmers can rely on the language implementation to provide meaningful error messages when “untyped” code misbehaves.

While most research on sound gradual typing has remained theoretical, the few emerging implementations incur performance overheads due to these checks. Indeed, none of the publications on this topic come with a comprehensive performance evaluation; a few report disastrous numbers on toy benchmarks. In response, this paper proposes a methodology for evaluating the performance of gradually typed programming languages. The key is to explore the performance impact of adding type annotations to different parts of a software system. The methodology takes takes the idea of a gradual conversion from untyped to typed seriously and calls for measuring the performance of all possible conversions of a given untyped benchmark. Finally the paper validates the proposed methodology using Typed Racket, a mature implementation of sound gradual typing, and a suite of real-world programs of various sizes and complexities. Based on the results obtained in this study, the paper concludes that, given the state of current implementation technologies, sound gradual typing is dead. Conversely, it raises the question of how implementations could reduce the overheads associated with ensuring soundness and how tools could be used to steer programmers clear from pathological cases.

In a fully dynamic system, typing checks are often superficial (only the existence of a particular field is tested) and done lazily (the check is made when the field is accessed). Gradual typing changes this, as typing assumptions can be made earlier than the value is used, and range over parts of the program that are not exercised in all execution branches. This has the potentially counter-intuitive consequence that the overhead of runtime checks may be sensibly larger than for fully-dynamic systems. This paper presents a methodology to evaluate the "annotation space" of a Typed Racket program, studying how the possible choices of which parts to annotate affect overall performance.

Many would find this article surprisingly grounded in reality for a POPL paper. It puts the spotlight on a question that is too rarely discussed, and could be presented as a strong illustration of why it matters to be serious about implementing our research.

by Ronald Garcia, Alison M. Clark, Éric Tanter
2016

Language researchers and designers have extended a wide variety of type systems to support gradual typing, which enables languages to seamlessly combine dynamic and static checking. These efforts consistently demonstrate that designing a satisfactory gradual counterpart to a static type system is challenging, and this challenge only increases with the sophistication of the type system. Gradual type system designers need more formal tools to help them conceptualize, structure, and evaluate their designs.

In this paper, we propose a new formal foundation for gradual typing, drawing on principles from abstract interpretation to give gradual types a semantics in terms of pre-existing static types. Abstracting Gradual Typing (AGT for short) yields a formal account of consistency—one of the cornerstones of the gradual typing approach—that subsumes existing notions of consistency, which were developed through intuition and ad hoc reasoning.

Given a syntax-directed static typing judgment, the AGT approach induces a corresponding gradual typing judgment. Then the subject-reduction proof for the underlying static discipline induces a dynamic semantics for gradual programs defined over source-language typing derivations. The AGT approach does not recourse to an externally justified cast calculus: instead, run-time checks naturally arise by deducing evidence for consistent judgments during proof-reduction.

To illustrate our approach, we develop novel gradually-typed counterparts for two languages: one with record subtyping and one with information-flow security types. Gradual languages designed with the AGT approach satisfy, by construction, the refined criteria for gradual typing set forth by Siek and colleagues.

At first sight this description seems to overlap with the Gradualizer work cited above, but in fact the two approaches are highly complementary. The Abstract Gradual Typing effort seems mechanizable, but it is far from being implementable in practice as done in the Gradualizer work. It remains a translation to be done on paper by skilled expert, although, as standard in abstract interpretation works, many aspects are deeply computational -- computing the best abstractions. On the other hand, it is extremely powerful to guide system design, as it provides not only a static semantics for a gradual system, but also a model dynamic semantics.

The central idea of the paper is to think of a missing type annotation not as "a special Dyn type that can contain anything" but "a specific static type, but I don't know which one it is". A problem is then to be understood as a family of potential programs, one for each possible static choice that could have been put there. Not all choices are consistent (type soundness imposes constraints on different missing annotations), so we can study the space of possible interpretations -- using only the original, non-gradually-typed system to make those deductions.

An obvious consequence is that a static type error occurs exactly when we can prove that there is no possible consistent typing. A much less obvious contribution is that, when there is a consistent set of types, we can consider this set as "evidence" that the program may be correct, and transport evidence along values while running the program. This gives a runtime semantics for the gradual system that automatically does what it should -- but it, of course, would fare terribly in the performance harness described above.

Some context

The Abstract Gradual Typing work feels like a real breakthrough, and it is interesting to idly wonder about which previous works in particular enabled this advance. I would make two guesses.

First, there was a very nice conceptualization work in 2015, drawing general principles from existing gradual typing system, and highlighting in particular a specific difficulty in designing dynamic semantics for gradual systems (removing annotations must not make program fail more).

by Jeremy Siek, Michael Vitousek, Matteo Cimini, and John Tang Boyland
2015

Second, the marriage of gradual typing and abstract interpretation was already consumed in previous work (2014), studying the gradual classification of effects rather than types.

A Theory of Gradual Effect Systems
2014

Effect systems have the potential to help software developers, but their practical adoption has been very limited. We conjecture that this limited adoption is due in part to the difficulty of transitioning from a system where effects are implicit and unrestricted to a system with a static effect discipline, which must settle for conservative checking in order to be decidable. To address this hindrance, we develop a theory of gradual effect checking, which makes it possible to incrementally annotate and statically check effects, while still rejecting statically inconsistent programs. We extend the generic type-and-effect framework of Marino and Millstein with a notion of unknown effects, which turns out to be significantly more subtle than unknown types in traditional gradual typing. We appeal to abstract interpretation to develop and validate the concepts of gradual effect checking. We also demonstrate how an effect system formulated in Marino and Millstein’s framework can be automatically extended to support gradual checking.

Difficulty rewards: gradual effects are more difficult than gradual simply-typed systems, so you get strong and powerful ideas when you study them. The choice of working on effect systems is also useful in practice, as nicely said by Philip Wadler in the conclusion of his 2015 article A Complement to Blame:

I [Philip Wadler] always assumed gradual types were to help those poor schmucks using untyped languages to migrate to typed languages. I now realize that I am one of the poor schmucks. My recent research involves session types, a linear type system that declares protocols for sending messages along channels. Sending messages along channels is an example of an effect. Haskell uses monads to track effects (Wadler, 1992), and a few experimental languages such as Links (Cooper et al., 2007), Eff (Bauer and Pretnar, 2014), and Koka (Leijen, 2014) support effect typing. But, by and large, every programming language is untyped when it comes to effects. To encourage migration from legacy code to code with effect types, such as session types, some form of gradual typing may be essential.

Self-Representation in Girard’s System U

Self-Representation in Girard’s System U, by Matt Brown and Jens Palsberg:

In 1991, Pfenning and Lee studied whether System F could support a typed self-interpreter. They concluded that typed self-representation 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 kind-polymorphic 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 self-representation for Girard’s System U, the first for a λ-calculus 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 self-representation supports operations that iterate over a term, each of which can be applied to a representation of itself. We present three typed self-applicable operations: a self-interpreter that recovers a term from its representation, a predicate that tests the intensional structure of a term, and a typed continuation-passing-style (CPS) transformation – the first typed self-applicable CPS transformation. Our techniques could have applications from verifiably type-preserving metaprograms, to growable typed languages, to more efficient self-interpreters.

Typed self-representation has come up here on LtU in the past. I believe the best self-interpreter available prior to this work was a variant of Barry Jay's SF-calculus, covered in the paper Typed Self-Interpretation by Pattern Matching (and more fully developed in Structural Types for the Factorisation Calculus). These covered statically typed self-interpreters without resorting to undecidable type:type rules.

However, being combinator calculi, they're not very similar to most of our programming languages, and so self-interpretation was still an active problem. Enter Girard's System U, which features a more familiar type system with only kind * and kind-polymorphic types. However, System U is not strongly normalizing and is inconsistent as a logic. Whether self-interpretation can be achieved in a strongly normalizing language with decidable type checking is still an open problem.

A theory of changes for higher-order languages — incrementalizing λ-calculi by static differentiation

The project Incremental λ-Calculus is just starting (compared to more mature approaches like self-adjusting computation), with a first publication last year.

A theory of changes for higher-order languages — incrementalizing λ-calculi 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 first-class functions, and produces derivatives amenable to standard optimization.

We prove the program transformation correct in Agda for a family of simply-typed λ-calculi, 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 value-dependent type diff(v). Intuitively, the empty list and a non-empty list have fairly different types of possible changes. This makes change-merging and change-producing operations total, and allow to give them a nice operational theory. Good design, through types.

(The program transformation seems related to the program-level 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 time-shifting.

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.

An operational and axiomatic semantics for non-determinism and sequence points in C

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 evaluation-order rules.

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 side-effects 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 non-determinism 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,post-conditions 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).

Luca Cardelli Festschrift

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 re-invented 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.

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.