Gradual Effect Handlers

 

As part of the SAGE project funded by Huawei, Li-yao Xia has written in Agda a model of a gradual type system for effects and handlers. It is available on GitHub.

My First Type Theory

Who knew? Add eyeballs and rhymes, and type theory becomes cute! An introductory video by Arved Friedemann.

PhD position in Certifying Compilation of Smart Contracts

Wouter Swierstra at Utrecht and my IOHK colleague Manuel Chakravarty are looking for someone to undertake a PhD to build a certifying compiler for our smart contract language, Plutus.

This project aims to develop a certifying compiler for Plutus Tx, a subset of the purely functional language Haskell that is used to implement smart contracts for the Cardano blockchain. The Plutus smart contract framework is being developed by IOHK for Cardano and the present project is a joint effort of IOHK and Utrecht University. The Plutus Tx compiler is based on the GHC Haskell compiler and adds a translation step from GHC Core to a minimal lambda calculus. Programmes in this lambda calculus are executed during transaction validation in a sandboxed execution environment in a manner that is crucial to the security of the blockchain. The aim of this project is to formalise the semantics of the languages involved in a proof assistant such as Coq, to reason about the transformation and optimisation steps that the compiler performs, and finally, to generate a proof object certifying the correctness of the generated code together with that code.

Every Proof Assistant

Andrej Bauer writes:

For a while now I have been contemplating a series of seminars titled "Every proof assistant" that would be devoted to all the different proof assistants out there. Apart from the established ones (Isabelle/HOL, Coq, Agda, Lean), there are other interesting experimental proof assistants, and some that are still under development, or just proofs of concept. I would like to know more about them, and I suspect I am not the only one. 

Getting the authors of proof assistants to travel to Ljubljana and giving talks at our Foundations of mathematics and theoretical computer science seminar has largely become impossible. But luckily research seminars world-wide are rapidly moving online, and so is our Foundations seminar. I am therefore delighted to announce the first "Every proof assistant" seminar. ... 

I have a couple more in the pipeline, so follow this blog, the Foundations seminar announcements or my Twitter account @andrejbauer.

A Profound Pun

My attempt to explain Propositions as Types to a general audience. Recorded at Bright Club, The Stand, Edinburgh, 23 September 2019.

PLFA in Padova

Next week, I will teach six hours of classes based on PLFA, and deliver a departmental seminar on the same topic. Maybe I will see you there!

Seminar: 16.00-17.00 Tue 28 May (details)

Classes: 12.30-15.30 Wed-Thu 29-30 May (details)

Programming Language Foundations in Agda

Wen Kokke and I are pleased to announce the availability of the textbook:

  Programming Language Foundations in Agda
  plfa.inf.ed.ac.uk
  github.com/plfa/plfa.github.io/

It is written as a literate script in Agda, and available at the above URLs. The books has so far been used for teaching at the Universities of Edinburgh and Vermont, and at Google Seattle. Please send your comments and pull requests!

The book was presented in a paper (of the same title) at the XXI Brazilian Symposium on Formal Methods, 28--30 Nov 2018, and is available here:

  http://homepages.inf.ed.ac.uk/wadler/topics/agda.html#sbmf

The paper won the SBMF 2018 Best Paper Award, 1st Place.

Papers We Love Remote Meetup: John Reynolds, Definitional Interpreters for Higher-Order Languages

I will reprise my June presentation to Papers We Love London at Papers We Love Remote Meetup 2, today at 7pm UK time, with the subject John Reynolds, Definitional Interpreters for Higher-Order Languages. Learn the origins of denotational semantics and continuations. Additional citations here. See you there!

Propositions as Types generalised: The Rosetta Stone

From Physics,Topology, Logic and Computation: A Rosetta Stone by John C. Baez and Mike Stay, courtesy of @CompSciFact, @sigfpe, and @notjfmc.

Papers We Love: John Reynolds, Definitional Interpreters for Higher-Order Programming Languages

I've added online links to the relevant papers (not behind paywalls), copied here.

Papers we love: John Reynolds, Definitional Interpreters for Higher-Order Programming Languages

7 June 2016, Skills Matter, London.
Certain papers change your life. McCarthy's 'Recursive Functions of Symbolic Expressions and their Computation by Machine (Part I)' (1960) changed mine, and so did Landin's 'The Next 700 Programming Languages' (1966). And I remember the moment, halfway through my graduate career, when Guy Steele handed me Reynolds's 'Definitional Interpreters for Higher-Order Programming Languages' (1972).
It is now common to explicate the structure of a programming language by presenting an interpreter for that language. If the language interpreted is the same as the language doing the interpreting, the interpreter is called meta-circular.
Interpreters may be written at differing levels of detail, to explicate different implementation strategies. For instance, the interpreter may be written in a continuation-passing style; or some of the higher-order functions may be represented explicitly using data-structures, via defunctionalisation.
More elaborate interpreters may be derived from simpler versions, thus providing a methodology for discovering an implementation strategy and showing it correct. Each of these techniques has become a mainstay of the study of programming languages, and all of them were introduced in this single paper by Reynolds.

Related material

• John Reynolds, Definitional Interpreters for Higher-Order Programming Languages, 1972.
• John Reynolds, Definitional Interpreters for Higher-Order Programming Languages, 1998.
• John Reynolds, Definitional Interpreters Revisited, 1998.
• John Reynolds, The Discoveries of Continuations, 1993.
• John McCarthy, Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I, 1960.
• John McCarthy, Towards a Mathematical Science of Computation, 1962.
• Peter Landin, The Next 700 Programming Languages, 1966.
• Gordon Plotkin, Call-by-value, Call-by-name, and the Lambda Calculus, 1975.
• Robin Milner, A Theory of Type Polymorphism in Programming, 1978.
• Fermin Reig, ed, Reminiscences of Influential Papers, SIGPLAN Notices, 38(12):9—10, December 2003.

Papers We Love: John Reynolds, Definitional Interpreters for Higher Order Languages

I will be speaking on John Reynolds paper, Definitional Interpreters for Higher Order Languages, at Papers We Love, London, 6:30pm Tuesday 7 June; details here.

A brief bibliography on parametricity

Henry Story asked HoTT Cafe about parametricity, prompting creation of a handy, short bibliography including Hermida, Reddy, and Robinson; Ahmed; Dreyer; Milewski (artist of the elegant diagram above); and Shulman.

A complement to blame

A complement to blame
Philip Wadler.
SNAPL, May 2015.

Contracts, gradual typing, and hybrid typing all permit less-precisely typed and more-precisely typed code to interact. Blame calculus encompasses these, and guarantees blame safety: blame for type errors always lays with less-precisely typed code. This paper serves as a complement to the literature on blame calculus: it elaborates on motivation, comments on the reception of the work, critiques some work for not properly attending to blame, and looks forward to applications. No knowledge of contracts, gradual typing, hybrid typing, or blame calculus is assumed.

Blame and Coercion:Together Again for the First Time

Blame and Coercion:Together Again for the First Time
Jeremy Siek, Peter Thiemann, Philip Wadler.
PLDI, June 2015.

C#, Dart, Pyret, Racket, TypeScript, VB: many recent languages integrate dynamic and static types via gradual typing. We systematically develop three calculi for gradual typing and the relations between them, building on and strengthening previous work. The calculi are: λB, based on the blame calculus of Wadler and Findler (2009); λC, inspired by the coercion calculus of Henglein (1994); λS inspired by the space-efficient calculus of Herman, Tomb, and Flanagan (2006) and the threesome calculus of Siek and Wadler (2010). While λB is little changed from previous work, λC and λS are new. Together, λB, λC, and λS provide a coherent foundation for design, implementation, and optimisation of gradual types.

We define translations from λB to λC and from λC to λS. Much previous work lacked proofs of correctness or had weak correctness criteria; here we demonstrate the strongest correctness criterion one could hope for, that each of the translations is fully abstract. Each of the calculi reinforces the design of the others: λC has a particularly simple definition, and the subtle definition of blame safety for λB is justified by the simple definition of blame safety for λC. Our calculus λS is implementation-ready: the first space-efficient calculus that is both straightforward to implement and easy to understand. We give two applications: first, using full abstraction from λC to λS to validate the challenging part of full abstraction between λB and λC; and, second, using full abstraction from λB to λS to easily establish the Fundamental Property of Casts, which required a custom bisimulation and six lemmas in earlier work.

Propositions as Types, with Howard on Curry-Howard

Propositions as Types has been updated, again. Thanks again to all the readers and reviewers who helped me improve the paper. This version includes an appendix, Howard on Curry-Howard, including additional correspondence with Howard not previously published.

Propositions as Types
Philip Wadler
Draft, 29 November 2014

The principle of Propositions as Types links logic to computation. At first sight it appears to be a simple coincidence---almost a pun---but it turns out to be remarkably robust, inspiring the design of theorem provers and programming languages, and continuing to influence the forefronts of computing. Propositions as Types has many names and many origins, and is a notion with depth, breadth, and mystery.

Comments still solicited!

Howard on Curry-Howard

When writing Propositions as Types, I realised I was unclear on parts of the history. Below is a letter I wrote to William Howard and his response (with corrections he provided after I asked to publish it). I believe it is a useful historical document, and am grateful to Howard for his permission to publish.

Update. References to Figures 5 and 6 in the following are to Propositions as Types. Thanks to Ezra Cooper for pointing out this was unclear.

Here is my original request.

Subject: The Formulae-as-Types Notion of Construction

Dear Prof Howard,

My research has been greatly influenced by your own, particularly the paper cited in my subject. I am now writing a paper on the field of work that grew out of that paper, which was solicited for publications by the Communications of the ACM (the flagship of the professional organisation for computer scientists). A draft of the paper is attached.

I would like to portray the history of the subject accurately. I have read your interview with Shell-Gallasch, but a few questions remain, which I hope you will be kind enough to answer.

Your paper breaks into two halves. The first describes the correspondence between propositional logic and simple types, the second introduces the correspondence between predicate logic and dependent types. Did you consider the first half to be new material or merely a reprise of what was known? To what extent do you consider your work draws on or was anticipated by the work of Heyting and Kolmogorov, and Kleene's realisability? To what extent did your work influence the subsequent work of de Bruijn and Martin Lof? What was the history of your mimeograph on the subject, and why was it not published until the Curry Festschrift in 1980?

Many thanks for your consideration, not to mention for founding my field! Yours, -- P

And here is his response: 

Dear Prof. Wadler,

As mentioned in the interview with Shell-Gellasch, my work on propositions as types (p-a-t) originated from my correspondence with Kreisel, who was very interested in getting a mathematical notion (i.e., in ordinary mathematics) for Brouwer's idea of a construction (as explained by Heyting). I was not familiar with the work of Brouwer or Heyting, let alone Kolmogorov, but, from what Kreisel had to say, the idea was clear enough: a construction of  alpha - > beta was to be a construction F which, acting on a construction A of alpha, gives a construction B of beta. So we have constructions acting on constructions, rather like functionals acting on functionals. So, as an approximation,

(1)   let's take "construction" to mean "functional".

But what kind of functionals? In constructive mathematics, a functional is not given as a set of ordered pairs. Rather,

(2)   to give a functional is to give not only the action or process it performs but also to give its type (domain and counterdomain).

Clearly, the type structure is going to be complicated. I set myself the project of finding a suitable notation for the type symbols. So one needs a suitable type symbol for the functional F, above. Well, just take it to be alpha itself (at this point, I was thinking of propositional logic). Suddenly I remembered something that Curry had talked about in the logic seminar during my time at Penn State. If we consider typed combinators, and look at the structure of the type symbols of the basic combinators (e.g., S, K, I), we see that each of the type symbols corresponds to (is isomorphic to) one of the axioms of pure implicative logic. Well! This was just what I needed!

How do we formulate the following notion?

(3)   F is a construction of phi.

Consider the case in which phi has the form alpha - > beta. The temptation is to define "F is a construction of alpha - > beta" to mean "for all A: if A is a construction of alpha, then FA is a construction of beta". Well, that is circular, because we have used if ... then ... to define implication. This is what you call "Zeno’s paradox of logic". I avoided this circularity by taking (3) to mean:

(4)   F is assigned the type phi according to the way F is built up; i.e., the way in which F is constructed.

Thus F is a construction of phi {\em by construction}. Your figure 6 illustrates precisely what I meant by this. (I did not have that beautiful notation at the time but it conveys what I meant.)

To summarize: My basic insight consisted simultaneously of the thoughts (2) and (4) plus the thought that Curry's observation provided the means to implement (2), (4). Let me say this in a different way. The thought (2) was not new. I had had the thought (2) for many years, ever since I had begun to study primitive recursive functionals of finite type. What was new was the thought (4) plus the recognition that Curry's idea provided the way to implement (4). I got this basic insight in the summer of 1966. Once I saw how to do it with combinators, I wondered what it would look like from the vewpoint of the lambda calculus, and saw, to my delight, that this corresponded to the intuitionistic version of Gentzen's sequent calculus.

Incidentally, Curry's observation concerning the types of the basic combinators is presented in his book with Feys (Curry-Feys), but I was unaware of this, though I had owned a copy for several years (since 1959, when I was hired at Penn State). After working out the details of p-a-t over a period of several months, I began to think about writing it up, so I thought I had better see if it is in the book. Well, it is easy enough to find if you know what you are looking for. On looking at it, I got a shock: not only had they extended the ideas to Gentzen's sequent calculus, but they had the connection between elimination of cuts from a derivation and normalization of the corresponding lambda term. But, on a closer look, I concluded that they had {\em a} connection but not {\em the} connection. It turns out that I was not quite right about that either. See my remark about their Theorem 5, below. Not that it would have mattered much for anything I might have published: even if they had the connection between Gentzen's sequent calculus and the lambda calculus, I had a far-reaching generalization (i.e., to Heyting arithmetic).

The above is more detailed than would be required to answer your questions, but I needed to write this out to clarify my thoughts about the matter; so I may as well include the above, since I think it will interest you. It answers one of your questions, "To what extent do you consider your work draws on or was anticipated by the work of Heyting and Kolmogorov, and Kleene's realisability?" Namely, my work draws on the work of Heyting and Brouwer, via Kreisel's explanation of that work to me. None of it was anticipated by the work of Heyting, Kolmogorov or Kleene: they were not thinking of functionals of finite type. Though I was familiar with Kleene's recursive realizability, I was not thinking about it at the time. Admittedly, it touches on ideas about Brouwer's constructions but is far from capturing the notion of a construction (actually, Kleene once made remarks to this effect, I forget where). Because of the relation between constructions and Kleene's recursive realizability, there could have been some unconscious influence; but, in any case, not a significant influence.

"did your work influence the subsequent work of de Bruijn and Martin Lof? " As far as I know, my work had no influence on the work of de Bruijn. His work appears to be completely independent of mine. I do recall that he once sent me a package of Automath material. The project of a computer program for checking existing proofs did not appear very interesting and I did not reply. What I would have been interested in is a program for finding proofs of results that had not yet been proved! Even a proof-assistant would have been fine. Why did he send me the Automath material? I don't recall what year it was. Sometime in the 1970s. Whatever the accompanying letter, it was not informative; merely something like: "Dear Professor Howard, you may be interested in the following material ...". Since that time, I have seen two or three articles by him, and I have a more favorable impression. It is good, solid work. Obviously original. He discovered the idea of derivations as terms, and the accompanying idea of formulae-as-types, on his own. He uses lambda terms but, I think, only for purposes of description. In other words, I don't think that he has the connection between normalization and cut-elimination, but I have not made an extensive examination of his work. In fact, does he use a Gentzen system at all? I just don't know. The latter two questions would easily be answered by anyone familiar with his work. In any case, give him credit where credit is due. There are enough goodies for everyone!

My influence on Martin-Löf? No problem there. I met him at the Buffalo 1968 conference and I told him my ideas. His instant reaction was: "Now, why didn't I think of that?" He had a visiting appointment at UIC for the academic year 1968-1969, so we had lot's of opportunity to talk, and he started developing his own approach to the ideas. In Jan. 1969, mainly to make sure that we were both clear on who had discovered what, I wrote up my own ideas in the form of handwritten notes. By then, Xerox machines were prevalent, so I sent a copy to Kreisel, and he gave copies to various people, including Girard. At least, I think that is how Girard got a copy, or maybe Martin-Löf gave him one. I like Martin-Löf's work. I could say more about this, but the short answer to your question is: Martin-Löf's work originated from mine. He has always given me credit and we are good friends.

On further thought, I need to mention that, in that first conversation, Martin-Löf suggested that the derivations-as-terms idea would work particularly well in connection with Prawitz's theory of natural deduction. I thought: okay, but no big deal. Actually, at that time, I was not familiar with Prawitz's results (or, if at all, then only vaguely). But it was a bigger deal than I had thought, because Prawitz's reductions steps for a deduction correspond direcly to reduction steps for the associated lambda term! Actually, for most purposes, I like the sequent formulation of natural deduction as given in pages 33 and 88 of Sorensen and Urzyczyn (2006). In fact, if we add left-implication-introduction to this (let's confine ourselves to pure implicative logic), the resulting system P# is pretty interesting. All occurrences of modus ponens can be eliminated, not just those which are preceded by left-implication-introduction. This is what I am up to in my JSL 1980 paper, "Ordinal analysis of terms of finite type." Also, the cut rule is easy to derive in P# (just consider, for typed lambda terms: a well-formed term substituted into a well-formed term results in a well-formed term); hence P# is is a conservative extension of the system P* in Part I of my little paper in the Curry Festschrift.

The phrase formulae-as-types was coined by Kreisel in order that we would have a name for the subject matter in our correspondence back and forth. I would assume that the phrase "propositions as types" was coined by Martin-Löf; at least, during our first conversation at the Buffalo 1968 meeting, he suggested that one could think of a type as a proposition, according to the idea that, in intuitionistic mathematics, the meaning of a proposition phi is given by the species of "all" proofs of phi. I use quotes here because we are not talking about a set-theoretic, completed infinity.

"the second [part] intrudes the correspondence between predicate logic and dependent types." I was not thinking about it in that way at all. I wanted to provided an interpretation of the notion of construction to some nontrivial part of intuitionistic mathematics (Heyting arithmetic). Part I of the paper was just the preliminaries for this. Actually, what you say in the pdf is consistent with this. No need for change here.

"Did you consider the first half to be new material or merely a reprise of what was known?" New. But in January of last year I had occasion to take a really hard look at the material in Curry-Feys, pp. 313-314; and I now see that there is a much closer relationship between my Theorem 2 in Part I and their Theorem 5, page 326, than I had thought. The issues here are quite interesting. I can provide a discussion if you want.

In the introduction to my little paper, I mention that Tait had influenced me. Let me say a few words about that. In the summer of 1963 we had conversations in which he explained to me that he had developed a theory of infinite terms in analogy to Schütte's theory of infinite proofs, where normalization (via lambda reductions) of an infinite terms corresponds to cut elimination of the corresponding proof. He did not know what to make of it. He thought of his theory of infinite terms as a sort of pun on Schütte's theory of infinite proofs. But we both agreed that  there must be a deep connection between normalization of lambda terms and Gentzen's cut elimination. We puzzled over this during two or three of our conversations but could not come up with an answer.

As explained in the first paragraph of this e-mail, my work originated with a problem posed by Kreisel; so, at the start of this work, certainly I was not thinking of those conversations with Tait. But, as mentioned above, as soon as I got the basic insight about the relevance of Curry's combinators, I considered how it would work for lambda terms. At that point, I remembered my conversations with Tait. In other words, when I verified that

(5)   cut elimination for a derivation corresponds to normalization for the term,

the conversations with Tait were very much on my mind. Most likely I would have noticed (5) without having had the conversations with Tait. But who knows? In any case, he deserves credit for having noticed the correspondence between derivations and terms. What he did not have was the associated correspondence between propositions and types. In fact, he was not using a general enough notion of type for this. By hindsight we can see that in his system there is a homomorphism, not an isomorphism, from propositions to types.

I need to say a bit more about Tait and types. Since Schütte had extended his system of proofs to transfinite orders, Tait extended his system of terms to transfinite type levels. I already had my own system of primitive recursive functionals of transfinite type. In our very first conversation, we compared out ideas on this topic. This topic requires that one think very hard about the notion of type. Certainly, I had already thought extensively about the notion of type (because of (2), above) before I ever met Tait, but my conversations with him reinforced this tendency. Thoughts about types were very much on my mind when I began to consider (1), (2), above.

As already mentioned, the notes were handwritten and xeroxed; no mimeographs. "why [were they] not published until the Curry Festschrift in 1980?" First let me mention why they got published in the Curry Festschrift. Selden was bringing out the Festschrift for Curry's 80th birthday. He asked me to contribute the notes. I said: "Sure. I'll write up an improved version. I can now do much better." He replied: "No, I want the original notes. It is a historical document." In other words, by that time various copies had been passed around and there were a number of references to them in the literature. So I had them typed up and I sent them in.

Why didn't I publish them before that? Simply because they did not solve the original problem. That was Kreisel's and Gödel’s verdict (Kreisel had shown or described the work to Gödel). In fact, even before communicating the work to Kreisel, I knew that I had gotten only an approximation to the notion of construction, and that more work had to be done. Essentially, the criticism is as follows. In my little paper, I do not provide axioms and rules of inference for proving statements of the form

(3)   F is a construction of phi.

Remember, we have to avoid "Zeno’s paradox of logic"! The answer is that the proofs will look like what you have in Figure 6. In other words, Figure 6 is not only a program; it is also a proof (or: it can be reinterpreted as a proof). But Figure 6 can also be interpreted as an explanation of how a construction (blue) is to be built up in order to have a given type (red). In other words, figures such as Figure 6 implements the idea (4) mentioned near the beginning of this e-mail; i.e., F is assigned the type phi according to the way F is built up.

I hope this tickles you; it certainly tickles me. Of course, the rules of inference are as in Figure 5. So these simple ideas provide the missing theory of constructions; or, at the very least, provide a significant step in that direction.

In Jan. 2013, I exchanged a few e-mails with Tait and Constable on the history of p-a-t. This caused me to take a really careful look at the Curry-Feys book. Here is something I found that really made me laugh: the required theory, whose inferences are of the form given in Figure 5 is already in Curry-Feys. Admittedly, to see this you first have to erase all the turnstyles ( |-- ); Curry seems to have some kind of obsession with them. In particular, erase the turnstiles from the proof tree on page 281. The result is exactly a proof tree of the general form given in your Figure 6. (Hint: (...)X is to be read "X has type (...)". In other words, rewrite (...)X as X : (...).) What does Fbc mean, where F is boldface? Just rewrite Fbc as b -> c. You see? I am an expert. I could probably make money writing a translation manual. In summary, the required theory is essentially just Curry's theory of functionality (more precisely, the appropriate variant of Curry's theory). So, did I miss the boat here? Might I have seen all this in 1969 if only I had had the determination to take a hard look at Curry-Feys? I don't know. It may require the clarity of mind represented by the notation of Figure 5. Do you have any idea when and where this notation came into use?

One more remark concerning my reason for not publishing. Didn't I feel that I had made an important breakthrough, in spite of Kreisel's and Gödel’s criticisms? On the one hand, yes. On the other hand, I had reservations. Except for Martin-Löf, Prawitz, Tait and Girard, very few people showed an interest in the ideas. But maybe Martin-Löf, Prawitz, Tait and Girard should have been enough. You say: "Certainly Howard was proud of the connection he drew, citing it as one of the two great achievements of his career [43]." Should we let that passage stand? Sure. The interview occurred in the spring of 2000. By that time, I was getting lots of praise from the computer science community. So, pride is a peculiar thing. Let me end this on a positive note. In 1969 Prawitz was in the US and came to UIC to give a talk. When he entered the room, he made a beeline for me, looked me in the eye and shook my hand. The message was: Well done! THAT made me proud.

There is more to say; but this answers your questions, I think; so I'll send it to avoid further delay. 

Your pdf, Propositions as Types, is very readable.

Bill

Propositions as Types, updated

Propositions as Types has been updated. Thanks to all the readers and reviewers who helped me improve the paper.

Propositions as Types
Philip Wadler
Draft, 26 June 2014

The principle of Propositions as Types links logic to computation. At first sight it appears to be a simple coincidence---almost a pun---but it turns out to be remarkably robust, inspiring the design of theorem provers and programming languages, and continuing to influence the forefronts of computing. Propositions as Types has many names and many origins, and is a notion with depth, breadth, and mystery.

Comments still solicited!

Microsoft Research PhD Scholarship: TypeScript, The Next Generation

We are recruiting one PhD student to work on design and implementation of programming languages. The post is on the project TypeScript, The Next Generation, and is funded by a Microsoft Research PhD Scholarship.

There is increasing interest in integrating dynamically and statically typed programming languages, as witnessed in industry by the development of the languages TypeScript and Dart, and in academia by the development of the theories of gradual types, hybrid types, and the blame calculus. The purpose of our project is to bring the academic and industrial developments together, applying theory to improve practice.

Our project focusses on JavaScript, an ECMA standard, and its typed variant TypeScript, an open-source project sponsored by Microsoft. JavaScript plays a central role in web-based applications and the new Windows 8 framework, and TypeScript is seeing rapid takeup, with over 150 JavaScript libraries now provided with TypeScript declarations. Our project has two parts, one aimed at immediate short-term application, and one aimed at fundamental long-term research.

• TypeScript TNG The short-term goal is to build a tool, TypeScript: The Next Generation (or TypeScript TNG for short), that generates wrapper code from TypeScript import declarations to detect and pinpoint type errors. A wrapper will accept any JavaScript value as input, and either raise an error or return a value guaranteed to satisfy the invariant associated with the corresponding type. In particular, wrappers for generic types will assure surprisingly strong guarantees, known as “theorems for free”. Our hypothesis is that TypeScript TNG will aid debugging and increase reliablility of TypeScript and JavaScript code.
  
• A wide-spectrum type system. The long-term goal is to extend the foundations of the blame calculus to support a wide-spectrum of type systems, ranging from dynamic types (as in JavaScript or Racket) through generic types (as in F# or Haskell) to dependent types (as in F* or Coq). Our hypothesis is that a wide-spectrum type system will increase the utility of dependent types, by allowing dynamic checks to be used as a fallback when static validation is problematic.
  

The workplan is likely to be too ambitious for a single PhD studentship. Which aspects are carried out will depend on which seem the most promising as our work develops, and on the abilities and desires of the student.

The successful candidate will join the ABCD team, carrying out a research programme investigating sesion types and web programming. The project is jointly supervised by Andrew Gordon of Microsoft Research Cambridge and the University of Edinburgh.

You should possess an undergraduate degree in a relevant area, or being nearing completion of same, or have comparable experience. You should have evidence of ability to undertake research and communicate well. You should have a background in programming languages, including type systems, and programming and software engineering skills.

It is desirable for candidates to also have one or more of the following: a combination of theoretical and practical skills; experience of JavaScript or web programming; knowledge of dependent type theory; or training in empirical measurement of programming tasks. We especially welcome applications from women and minorities.

We seek applicants at an international level of excellence. The School of Informatics at Edinburgh is among the strongest in the world, and Edinburgh is known as a cultural centre providing a high quality of life.

The successful candidate will receive a studentship covering tuition and subsistence. Students from the UK or EU are preferred. Consult the University of Edinburgh website for details of how to apply.

If you are interested, please send an outline of your qualifications to: Prof. Philip Wadler (wadler@inf.ed.ac.uk).

Propositions as Types

Propositions as Types
Philip Wadler
Draft, March 2014

The principle of Propositions as Types links logic to computation. At first sight it appears to be a simple coincidence---almost a pun---but it turns out to be remarkably robust, inspiring the design of theorem provers and programming languages, and continuing to influence the forefronts of computing. Propositions as Types has many names and many origins, and is a notion with depth, breadth, and mystery.

Comments solicited!

Blame, coercions, and threesomes, precisely

Blame, coercions, and threesomes, precisely
Jeremy Siek, Peter Thiemann, and Philip Wadler
Draft, March 2014

We systematically present four calculi for gradual typing: the blame calculus of Wadler and Findler (2009); a novel calculus that pinpoints blame precisely; the coercion calculus of Henglein (1994); and the threesome calculus of Siek and Wadler (2010). Threesomes are given a syntax that directly exposes their origin as coercions in normal form, a more transparent presentation than that found in Siek and Wadler (2010) or Garcia (2013).

Comments welcome!