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Invited Talk

**Published in:** LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)

The computational view of intuitionistic dependent type theory is as an intrinsic logic of (functional) programs in which types are viewed as specifications of their behavior. Equational reasoning is particularly relevant in the functional case, where correctness can be formulated as equality between two implementations of the same behavior. Besides behavior, it is also important to specify and verify the cost of programs, measured in terms of their resource usage, with respect to both sequential and parallel evaluation. Although program cost can - and has been - verified in type theory using an extrinsic formulation of programs as data objects, what we seek here is, instead, an intrinsic account within type theory itself.
In this talk we discuss Calf, the Cost-Aware Logical Framework, which is an extension of dependent call-by-push-value type theory that provides an intrinsic account of both parallel and sequential resource usage for a variety of problem-specific measures of cost. Thus, for example, it is possible to prove that insertion sort and merge sort are equal as regards behavior, but differ in terms of the number of comparisons required to achieve the same results. But how can equal functions have different cost? To provide an intrinsic account of both intensional and extensional properties of programs, we make use of Sterling’s notion of Synthetic Tait Computability, a generalization of Tait’s method originally developed for the study of higher type theory.
In STC the concept of a "phase" plays a central role: originally as the distinction between the syntactic and semantic aspects of a computability structure, but more recently applied to the formulation of type theories for program modules and for information flow properties of programs. In Calf we distinguish two phases, the intensional and extensional, which differ as regards the significance of cost accounting - extensionally it is neglected, intensionally it is of paramount importance. Thus, in the extensional phase insertion sort and merge sort are equal, but in the intensional phase they are distinct, and indeed one is proved to have optimal behavior as regards comparisons, and the other not. Importantly, both phases are needed in a cost verification - the proof of the complexity of an algorithm usually relies on aspects of its correctness.
We will provide an overview of Calf itself, and of its application in the verification of the cost and behavior of a variety of programs. So far we have been able to verify cost bounds on Euclid’s Algorithm, amortized bounds on batched queues, parallel cost bounds on a joinable form of red-black trees, and the equivalence and cost of the aforementioned sorting methods. In a companion paper at this meeting Grodin and I develop an account of amortization that relates the standard inductive view of instruction seequences with the coinductive view of data structures characterized by the same operations. In ongoing work we are extending the base of verified deterministic algorithms to those taught in the undergraduate parallel algorithms course at Carnegie Mellon, and are extending Calf itself to account for probabilistic methods, which are also used in that course.
(This talk represents joint work with Yue Niu, Harrison Grodin, and Jon Sterling.)

Robert Harper. Integrating Cost and Behavior in Type Theory (Invited Talk). In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 1:1-1:2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{harper:LIPIcs.CALCO.2023.1, author = {Harper, Robert}, title = {{Integrating Cost and Behavior in Type Theory}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {1:1--1:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-287-7}, ISSN = {1868-8969}, year = {2023}, volume = {270}, editor = {Baldan, Paolo and de Paiva, Valeria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.1}, URN = {urn:nbn:de:0030-drops-187980}, doi = {10.4230/LIPIcs.CALCO.2023.1}, annote = {Keywords: type theory, analysis of algorithms, program verification} }

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Early Ideas

**Published in:** LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)

Amortized analysis is a program cost analysis technique for data structures in which the cost of operations is specified in aggregate, under the assumption of continued sequential use. Typically, amortized analyses are presented inductively, in terms of finite sequences of operations. We give an alternative coinductive formulation and prove that it is equivalent to the standard inductive definition. We describe a classic amortized data structure, the batched queue, and outline a coinductive proof of its amortized efficiency in calf, a dependent type theory for cost analysis.

Harrison Grodin and Robert Harper. Amortized Analysis via Coinduction (Early Ideas). In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 23:1-23:6, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{grodin_et_al:LIPIcs.CALCO.2023.23, author = {Grodin, Harrison and Harper, Robert}, title = {{Amortized Analysis via Coinduction}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {23:1--23:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-287-7}, ISSN = {1868-8969}, year = {2023}, volume = {270}, editor = {Baldan, Paolo and de Paiva, Valeria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.23}, URN = {urn:nbn:de:0030-drops-188201}, doi = {10.4230/LIPIcs.CALCO.2023.23}, annote = {Keywords: amortized analysis, coinduction, data structure, mechanized proof} }

Document

**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

We propose a new sheaf semantics for secure information flow over a space of abstract behaviors, based on synthetic domain theory: security classes are open/closed partitions, types are sheaves, and redaction of sensitive information corresponds to restricting a sheaf to a closed subspace. Our security-aware computational model satisfies termination-insensitive noninterference automatically, and therefore constitutes an intrinsic alternative to state of the art extrinsic/relational models of noninterference. Our semantics is the latest application of Sterling and Harper’s recent re-interpretation of phase distinctions and noninterference in programming languages in terms of Artin gluing and topos-theoretic open/closed modalities. Prior applications include parametricity for ML modules, the proof of normalization for cubical type theory by Sterling and Angiuli, and the cost-aware logical framework of Niu et al. In this paper we employ the phase distinction perspective twice: first to reconstruct the syntax and semantics of secure information flow as a lattice of phase distinctions between "higher" and "lower" security, and second to verify the computational adequacy of our sheaf semantics with respect to a version of Abadi et al.’s dependency core calculus to which we have added a construct for declassifying termination channels.

Jonathan Sterling and Robert Harper. Sheaf Semantics of Termination-Insensitive Noninterference. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 5:1-5:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{sterling_et_al:LIPIcs.FSCD.2022.5, author = {Sterling, Jonathan and Harper, Robert}, title = {{Sheaf Semantics of Termination-Insensitive Noninterference}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {5:1--5:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-233-4}, ISSN = {1868-8969}, year = {2022}, volume = {228}, editor = {Felty, Amy P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.5}, URN = {urn:nbn:de:0030-drops-162869}, doi = {10.4230/LIPIcs.FSCD.2022.5}, annote = {Keywords: information flow, noninterference, denotational semantics, phase distinction, Artin gluing, modal type theory, topos theory, synthetic domain theory, synthetic Tait computability} }

Document

**Published in:** LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity.

Evan Cavallo and Robert Harper. Internal Parametricity for Cubical Type Theory. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 13:1-13:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cavallo_et_al:LIPIcs.CSL.2020.13, author = {Cavallo, Evan and Harper, Robert}, title = {{Internal Parametricity for Cubical Type Theory}}, booktitle = {28th EACSL Annual Conference on Computer Science Logic (CSL 2020)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-132-0}, ISSN = {1868-8969}, year = {2020}, volume = {152}, editor = {Fern\'{a}ndez, Maribel and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.13}, URN = {urn:nbn:de:0030-drops-116564}, doi = {10.4230/LIPIcs.CSL.2020.13}, annote = {Keywords: parametricity, cubical type theory, higher inductive types} }

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**Published in:** LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)

Broadly speaking, algebraic topology consists of associating algebraic structures to topological spaces that give information about their structure. An elementary, but fundamental, example is provided by the theory of covering spaces, which associate groups to covering spaces in such a way that the universal cover corresponds to the fundamental group of the space. One natural question to ask is whether these connections can be stated in homotopy type theory, a new area linking type theory to homotopy theory. In this paper, we give an affirmative answer with a surprisingly concise definition of covering spaces in type theory; we are able to prove various expected properties about the newly defined covering spaces, including the connections with fundamental groups. An additional merit is that our work has been fully mechanized in the proof assistant Agda.

Kuen-Bang Hou (Favonia) and Robert Harper. Covering Spaces in Homotopy Type Theory. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 11:1-11:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hou(favonia)_et_al:LIPIcs.TYPES.2016.11, author = {Hou (Favonia), Kuen-Bang and Harper, Robert}, title = {{Covering Spaces in Homotopy Type Theory}}, booktitle = {22nd International Conference on Types for Proofs and Programs (TYPES 2016)}, pages = {11:1--11:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-065-1}, ISSN = {1868-8969}, year = {2018}, volume = {97}, editor = {Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.11}, URN = {urn:nbn:de:0030-drops-98512}, doi = {10.4230/LIPIcs.TYPES.2016.11}, annote = {Keywords: homotopy type theory, covering space, fundamental group, mechanized reasoning} }

Document

**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky's univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.

Carlo Angiuli, Kuen-Bang Hou (Favonia), and Robert Harper. Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 6:1-6:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{angiuli_et_al:LIPIcs.CSL.2018.6, author = {Angiuli, Carlo and Hou (Favonia), Kuen-Bang and Harper, Robert}, title = {{Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {6:1--6:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.6}, URN = {urn:nbn:de:0030-drops-96734}, doi = {10.4230/LIPIcs.CSL.2018.6}, annote = {Keywords: Homotopy Type Theory, Two-Level Type Theory, Computational Type Theory, Cubical Sets} }

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