Email: jgross@mit.edu
Office: 32G822
Eventually, my projects will be integrated into this page. In the meantime, please visit my GitHub account, or look at my résumé (general, for computer science, or for mathematics) or curriculum vitæ.
These days, I'm working on program synthesis and category theory on top of homotopy type theory in Coq. I also occasionally commit to the BarnOwl project.
What I do, with only the tenhundred most used words (checked by The UpGoer Five Words TypingBox): It would be nice if we could tell computers what should be done in only a few simple words (but in words that can only mean one thing), and the computers would just know how to do it, and never be slow and never be wrong. I'm working on making this dream come true.
[1] 
Jason Gross, Jack Gallagher, and Benya Fallenstein.
Löb's theorem: A functional pearl of dependently typed quining,
March 2016.
Submitted to ICFP
2016.
[ bib 
project () 
artifact (.zip) 
code (.agda) 
code (.html) 
bibliography 
.pdf ]
Löb's theorem states that to prove that a proposition is provable, it is sufficient to prove the proposition under the assumption that it is provable. The CurryHoward isomorphism identifies formal proofs with abstract syntax trees of programs; Löb's theorem thus implies, for total languages which validate it, that selfinterpreters are impossible. We formalize a few variations of Löb's theorem in Agda using an inductiveinductive encoding of terms indexed over types. We verify the consistency of our formalizations relative to Agda by giving them semantics via interpretation functions.

[2] 
Jason Gross.
An extensible framework for synthesizing efficient, verified parsers.
Master's thesis, Massachusetts Institute of Technology, September
2015.
[ bib 
.pdf ]
Parsers have a long history in computer science. This thesis proposes a novel approach to synthesizing efficient, verified parsers by refinement, and presents a demonstration of this approach in the Fiat framework by synthesizing a parser for arithmetic expressions. The benefits of this framework may include more flexibility in the parsers that can be described, more control over the lowlevel details when necessary for performance, and automatic or mostly automatic correctness proofs.

[3] 
Jason Gross and Adam Chlipala.
Parsing parses: A pearl of (dependently typed) programming and proof,
August 2015.
Submitted to ICFP
2015.
[ bib 
.pdf ]
We present a functional parser for arbitrary contextfree grammars, together with soundness and completeness proofs, all inside Coq. By exposing the parser in the right way with parametric polymorphism and dependent types, we are able to use the parser to prove its own soundness, and, with a little help from relational parametricity, prove its own completeness, too. Of particular interest is one strange instantiation of the type and value parameters: by parsing parse trees instead of strings, we convince the parser to generate its own completeness proof. We conclude with highlights of our experiences iterating through several versions of the Coq development, and some general lessons about dependently typed programming.

[4] 
Clément PitClaudel, Peng Wang, Jason Gross, Ben Delaware, and Adam Chlipala.
Correctbyconstruction program derivation from specifications to
assembly language, June 2015.
Submitted to PLDI 2015.
[ bib 
.pdf ]
We present a Coqbased system to certify the entire process of implementing declarative mathematical specifications with efficient assembly code. That is, we produce formal assemblycode libraries with proofs, in the style of Hoare logic, demonstrating compatibility with relational specifications in higherorder logic. Most codegeneration paths from highlevel languages involve the introduction of garbage collection and other runtime support for sourcelevel abstractions, but we generate code suitable for resourceconstrained embedded systems, using manual memory management and inplace updating of heapallocated data structures. We start from very highlevel source code, applying the Fiat framework to refine settheory expressions into functional programs; then we further apply Fiat's refinement tools to translate functional programs into Facade, a simple imperative language without a heap or aliasing; and finally we plug into the assemblygeneration features of the Bedrock framework, where we link with handwritten datastructure implementations and their associated proofs. Each program refinement leads to a proved Hoarelogic specification for an assembly function, with no trust dependencies on any aspect of our synthesis process, which is highly automated.

[5] 
Ben Delaware, Clément PitClaudel, Jason Gross, and Adam Chlipala.
Fiat: Deductive synthesis of abstract data types in a proof
assistant.
In Proceedings of the 42nd
ACM SIGPLANSIGACT Symposium on Principles of Programming Languages
(POPL'15), January 2015.
[ bib 
.pdf ]
We present Fiat, a library for the Coq proof assistant supporting refinement of declarative specifications into efficient functional programs with a high degree of automation. Each refinement process leaves a proof trail, checkable by the normal Coq kernel, justifying its soundness. We focus on the synthesis of abstract data types that package methods with private data. We demonstrate the utility of our framework by applying it to the synthesis of query structures  abstract data types with SQLlike query and insert operations. Fiat includes a library for writing specifications of query structures in SQLinspired notation, expressing operations over relations (tables) in terms of mathematical sets. This library includes a set of tactics for automating the refinement of these specifications into efficient, correctbyconstruction OCaml code. Using these tactics, a programmer can generate such an implementation completely automatically by only specifying the equivalent of SQL indexes, data structures capturing useful views of the abstract data. We conclude by speculating on the new programming modularity possibilities enabled by an automated refinement system with provedcorrect rules.

[6] 
Jason Gross.
Coq bug minimizer, January 2015.
Accepted for presentation at CoqPL 2015.
[ bib 
reviews 
.pdf ]
Are bugs the bane of your existence? Do you dread Coq upgrades, because they mean you'll have to spend days tracking down subtle failures deep in your developments? Have you ever hit an anomaly that just wouldn't go away, and wished you understood what triggered it? Have you ever been tormented by two blocks of code that looked identical, but behaved differently? Do you wish you submit more helpful error reports, but don't want to put in the time to construct minimal examples? If you answered “yes” to any of these questions, then the Coq Bug Minimizer is for you! Clone your own copy at https://github.com/JasonGross/coqbugfinder.

[7] 
Tobias Tebbi and Jason Gross.
A profiler for Ltac, January 2015.
Accepted for presentation at CoqPL 2015.
[ bib 
.pdf ]
We present a simple profiler for the Ltac tactic language of the Coq Proof Assistent. It measures the time spent in invocations of primitive tactics as well as tactics defined in Ltac and their inner invocations. The profiler is controlled using Vernacular commands and prints an aggregated view that differentiates between tactic invocations depending on their call tree location.

[8] 
Jason Gross.
Presentation: Input, output, and automation in x86 proved, August
2014.
Presented at Microsoft Research, Cambridge, UK.
[ bib 
.pptx 
.pdf ]
The x86proved project can now verify assembly programs with input and output! The codereasoning throughout the project is now mostly automatic. Although not yet pushbutton verification (specificationlevel reasoning, in particular, leaves a lot to be desired) these tactics make a significant step towards that goal. This presentation will cover:

[9] 
Jason Gross, Adam Chlipala, and David I. Spivak.
Experience implementing a performant categorytheory library in
Coq.
In Proceedings of the
5th International Conference on
Interactive Theorem Proving (ITP'14), July 2014.
[ bib 
arXiv 
Springer publication 
presentation (.pdf) 
presentation (.pptx, annotated with notes) 
original conference submission (.pdf) 
full bibliography 
reviews 
.pdf ]
We describe our experience implementing a broad categorytheory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows categorytheoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.

[10] 
Jason Gross.
Presentation proposal for of three neat tricks in Coq 8.5, April
2014.
Accepted to the 6th Coq Workshop.
[ bib 
code (.html) 
code (.v) 
reviews 
.pdf ]
Coq 8.5 has a number of new features. It has more powerful universe polymorphism support. It allows tactics to be run at interpretation to construct other terms. The ability to switch from Gallina to Ltac in arbitrary locations nicely complements the constr: notation permitting the switch from Ltac to Gallina in tactics, and opens up many new possibilities. I propose to present three tricks involving these new features: tactics in terms allows the construction of tactics that recurse under binders; tactics in terms together with typeclasses allows overloading notations based on the type of their arguments; and there is a way to talk about universe levels explicitly, helped along by tactics in terms.

[11]  Jason Gross. Jason Gross' wishlist for Coq, January 2014. [ bib  .pdf ] 
[12]  Jason Gross. POPL: Minute madness: Category theory in Coq, and program synthesis, January 2014. [ bib  .pdf ] 
[13]  Jason Gross. CSAIL student workshop 2013: Computational higher inductive types: Computing with custom equalities, October 2013. [ bib  .pdf ] 
[14]  Jason Gross. Building database management on top of category theory in Coq, January 2013. [ bib  .pdf ] 
[15]  Jason Gross. POPL: Minute madness: Database management on top of category theory in Coq: Category of relational schemas = category of categories, January 2013. [ bib  .pdf ] 
[16] 
Brenden M. Lake, Ruslan Salakhutdinov, Jason Gross, and Joshua B. Tenenbaum.
One shot learning of simple visual concepts.
In Proceedings of the 33rd Annual Conference of the Cognitive
Science Society, 2011.
[ bib 
videos 
.pdf ]
People can learn visual concepts from just one example, but it remains a mystery how this is accomplished. Many authors have proposed that transferred knowledge from more familiar concepts is a route to one shot learning, but what is the form of this abstract knowledge? One hypothesis is that the sharing of parts is core to one shot learning, and we evaluate this idea in the domain of handwritten characters, using a massive new dataset. These simple visual concepts have a rich internal part structure, yet they are particularly tractable for computational models. We introduce a generative model of how characters are composed from strokes, where knowledge from previous characters helps to infer the latent strokes in novel characters. The stroke model outperforms a competing stateoftheart character model on a challenging one shot learning task, and it provides a good fit to human perceptual data.

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