8 Search Results for "Zeilberger, Noam"

Document
DOI: 10.4230/LIPIcs.ECOOP.2024.33
Compiling with Arrays

Authors: David Richter, Timon Böhler, Pascal Weisenburger, and Mira Mezini

Published in: LIPIcs, Volume 313, 38th European Conference on Object-Oriented Programming (ECOOP 2024)

Abstract
Linear algebra computations are foundational for neural networks and machine learning, often handled through arrays. While many functional programming languages feature lists and recursion, arrays in linear algebra demand constant-time access and bulk operations. To bridge this gap, some languages represent arrays as (eager) functions instead of lists. In this paper, we connect this idea to a formal logical foundation by interpreting functions as the usual negative types from polarized type theory, and arrays as the corresponding dual positive version of the function type. Positive types are defined to have a single elimination form whose computational interpretation is pattern matching. Just like (positive) product types bind two variables during pattern matching, (positive) array types bind variables with multiplicity during pattern matching. We follow a similar approach for Booleans by introducing conditionally-defined variables. The positive formulation for the array type enables us to combine typed partial evaluation and common subexpression elimination into an elegant algorithm whose result enjoys a property we call maximal fission, which we argue can be beneficial for further optimizations. For this purpose, we present the novel intermediate representation indexed administrative normal form (A_{i}NF), which relies on the formal logical foundation of the positive formulation for the array type to facilitate maximal loop fission and subsequent optimizations. A_{i}NF is normal with regard to commuting conversion for both let-bindings and for-loops, leading to flat and maximally fissioned terms. We mechanize the translation and normalization from a simple surface language to A_{i}NF, establishing that the process terminates, preserves types, and produces maximally fissioned terms.
Cite as

David Richter, Timon Böhler, Pascal Weisenburger, and Mira Mezini. Compiling with Arrays. In 38th European Conference on Object-Oriented Programming (ECOOP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 313, pp. 33:1-33:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{richter_et_al:LIPIcs.ECOOP.2024.33,
  author =	{Richter, David and B\"{o}hler, Timon and Weisenburger, Pascal and Mezini, Mira},
  title =	{{Compiling with Arrays}},
  booktitle =	{38th European Conference on Object-Oriented Programming (ECOOP 2024)},
  pages =	{33:1--33:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-341-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{313},
  editor =	{Aldrich, Jonathan and Salvaneschi, Guido},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2024.33},
  URN =		{urn:nbn:de:0030-drops-208823},
  doi =		{10.4230/LIPIcs.ECOOP.2024.33},
  annote =	{Keywords: array languages, functional programming, domain-specific languages, normalization by evaluation, common subexpression elimination, polarity, positive function type, intrinsic types}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.30
When Lawvere Meets Peirce: An Equational Presentation of Boolean Hyperdoctrines

Authors: Filippo Bonchi, Alessandro Di Giorgio, and Davide Trotta

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Fo-bicategories are a categorification of Peirce’s calculus of relations. Notably, their laws provide a proof system for first-order logic that is both purely equational and complete. This paper illustrates a correspondence between fo-bicategories and Lawvere’s hyperdoctrines. To streamline our proof, we introduce peircean bicategories, which offer a more succinct characterization of fo-bicategories.
Cite as

Filippo Bonchi, Alessandro Di Giorgio, and Davide Trotta. When Lawvere Meets Peirce: An Equational Presentation of Boolean Hyperdoctrines. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bonchi_et_al:LIPIcs.MFCS.2024.30,
  author =	{Bonchi, Filippo and Di Giorgio, Alessandro and Trotta, Davide},
  title =	{{When Lawvere Meets Peirce: An Equational Presentation of Boolean Hyperdoctrines}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.30},
  URN =		{urn:nbn:de:0030-drops-205867},
  doi =		{10.4230/LIPIcs.MFCS.2024.30},
  annote =	{Keywords: relational algebra, hyperdoctrines, cartesian bicategories, string diagrams}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.3
Categorical Models of Subtyping

Authors: Greta Coraglia and Jacopo Emmenegger

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context - and for each type a set of terms of said type. This is the case for categories with families, categories with attributes, and natural models; in particular, all of them can be traced back to certain discrete Grothendieck fibrations. We extend this intuition to the case of general, not necessarily discrete, fibrations, so that over a given context one has not only a set but a category of types. We argue that the added structure can be attributed to a notion of subtyping that shares many features with that of coercive subtyping, in the sense that it is the product of thinking about subtyping as an abbreviation mechanism: we say that a given type A' is a subtype of A if there is a unique coercion from A' to A. Whenever we need a term of type A, then, it suffices to have a term of type A', which we can "plug-in" into A. For this version of subtyping we provide rules, coherences, and explicit models, and we compare and contrast it to coercive subtyping as introduced by Z. Luo and others. We conclude by suggesting how the tools we present can be employed in finding appropriate rules relating subtyping and certain type constructors.
Cite as

Greta Coraglia and Jacopo Emmenegger. Categorical Models of Subtyping. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{coraglia_et_al:LIPIcs.TYPES.2023.3,
  author =	{Coraglia, Greta and Emmenegger, Jacopo},
  title =	{{Categorical Models of Subtyping}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{3:1--3:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.3},
  URN =		{urn:nbn:de:0030-drops-204811},
  doi =		{10.4230/LIPIcs.TYPES.2023.3},
  annote =	{Keywords: dependent types, subtyping, coercive subtyping, categorical semantics, categories with families, monad}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2024.5
The Flower Calculus

Authors: Pablo Donato

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract
We introduce the flower calculus, a deep inference proof system for intuitionistic first-order logic inspired by Peirce’s existential graphs. It works as a rewriting system over inductive objects called "flowers", that enjoy both a graphical interpretation as topological diagrams, and a textual presentation as nested sequents akin to coherent formulas. Importantly, the calculus dispenses completely with the traditional notion of symbolic connective, operating solely on nested flowers containing atomic predicates. We prove both the soundness of the full calculus and the completeness of an analytic fragment with respect to Kripke semantics. This provides to our knowledge the first analyticity result for a proof system based on existential graphs, adapting semantic cut-elimination techniques to a deep inference setting. Furthermore, the kernel of rules targetted by completeness is fully invertible, a desirable property for both automated and interactive proof search.
Cite as

Pablo Donato. The Flower Calculus. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{donato:LIPIcs.FSCD.2024.5,
  author =	{Donato, Pablo},
  title =	{{The Flower Calculus}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{5:1--5:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.5},
  URN =		{urn:nbn:de:0030-drops-203343},
  doi =		{10.4230/LIPIcs.FSCD.2024.5},
  annote =	{Keywords: deep inference, graphical calculi, existential graphs, intuitionistic logic, Kripke semantics, cut-elimination}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2023.17
Convolution Products on Double Categories and Categorification of Rule Algebras

Authors: Nicolas Behr, Paul-André Melliès, and Noam Zeilberger

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

Abstract
Motivated by compositional categorical rewriting theory, we introduce a convolution product over presheaves of double categories which generalizes the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, we identify several classes of double categories for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product.
Cite as

Nicolas Behr, Paul-André Melliès, and Noam Zeilberger. Convolution Products on Double Categories and Categorification of Rule Algebras. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{behr_et_al:LIPIcs.FSCD.2023.17,
  author =	{Behr, Nicolas and Melli\`{e}s, Paul-Andr\'{e} and Zeilberger, Noam},
  title =	{{Convolution Products on Double Categories and Categorification of Rule Algebras}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{17:1--17:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.17},
  URN =		{urn:nbn:de:0030-drops-180017},
  doi =		{10.4230/LIPIcs.FSCD.2023.17},
  annote =	{Keywords: Categorical rewriting, double pushout, sesqui-pushout, double categories, convolution product, presheaf categories, framed bicategories, opfibrations, rule algebra}
}
Document
DOI: 10.4230/LIPIcs.ECOOP.2022.5
How to Take the Inverse of a Type

Authors: Danielle Marshall and Dominic Orchard

Published in: LIPIcs, Volume 222, 36th European Conference on Object-Oriented Programming (ECOOP 2022)

Abstract
In functional programming, regular types are a subset of algebraic data types formed from products and sums with their respective units. One can view regular types as forming a commutative semiring but where the usual axioms are isomorphisms rather than equalities. In this pearl, we show that regular types in a linear setting permit a useful notion of multiplicative inverse, allowing us to "divide" one type by another. Our adventure begins with an exploration of the properties and applications of this construction, visiting various topics from the literature including program calculation, Laurent polynomials, and derivatives of data types. Examples are given throughout using Haskell’s linear types extension to demonstrate the ideas. We then step through the looking glass to discover what might be possible in richer settings; the functional language Granule offers linear functions that incorporate local side effects, which allow us to demonstrate further algebraic structure. Lastly, we discuss whether dualities in linear logic might permit the related notion of an additive inverse.
Cite as

Danielle Marshall and Dominic Orchard. How to Take the Inverse of a Type. In 36th European Conference on Object-Oriented Programming (ECOOP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 222, pp. 5:1-5:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{marshall_et_al:LIPIcs.ECOOP.2022.5,
  author =	{Marshall, Danielle and Orchard, Dominic},
  title =	{{How to Take the Inverse of a Type}},
  booktitle =	{36th European Conference on Object-Oriented Programming (ECOOP 2022)},
  pages =	{5:1--5:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-225-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{222},
  editor =	{Ali, Karim and Vitek, Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2022.5},
  URN =		{urn:nbn:de:0030-drops-162339},
  doi =		{10.4230/LIPIcs.ECOOP.2022.5},
  annote =	{Keywords: linear types, regular types, algebra of programming, derivatives}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2020.135
Implicit Automata in Typed λ-Calculi I: Aperiodicity in a Non-Commutative Logic

Authors: Lê Thành Dũng Nguyễn and Pierre Pradic

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
We give a characterization of star-free languages in a λ-calculus with support for non-commutative affine types (in the sense of linear logic), via the algebraic characterization of the former using aperiodic monoids. When the type system is made commutative, we show that we get regular languages instead. A key ingredient in our approach – that it shares with higher-order model checking – is the use of Church encodings for inputs and outputs. Our result is, to our knowledge, the first use of non-commutativity in implicit computational complexity.
Cite as

Lê Thành Dũng Nguyễn and Pierre Pradic. Implicit Automata in Typed λ-Calculi I: Aperiodicity in a Non-Commutative Logic. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 135:1-135:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{nguyen_et_al:LIPIcs.ICALP.2020.135,
  author =	{Nguy\~{ê}n, L\^{e} Th\`{a}nh D\~{u}ng and Pradic, Pierre},
  title =	{{Implicit Automata in Typed \lambda-Calculi I: Aperiodicity in a Non-Commutative Logic}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{135:1--135:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.135},
  URN =		{urn:nbn:de:0030-drops-125426},
  doi =		{10.4230/LIPIcs.ICALP.2020.135},
  annote =	{Keywords: Church encodings, ordered linear types, star-free languages}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2017.33
A Sequent Calculus for a Semi-Associative Law

Authors: Noam Zeilberger

Published in: LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)

Abstract
We introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Y_n. Elsewhere, we have also used the sequent calculus and the coherence theorem to build a surprising bijection between intervals of the Tamari order and a natural fragment of lambda calculus, consisting of the beta-normal planar lambda terms with no closed proper subterms.
Cite as

Noam Zeilberger. A Sequent Calculus for a Semi-Associative Law. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{zeilberger:LIPIcs.FSCD.2017.33,
  author =	{Zeilberger, Noam},
  title =	{{A Sequent Calculus for a Semi-Associative Law}},
  booktitle =	{2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)},
  pages =	{33:1--33:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-047-7},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{84},
  editor =	{Miller, Dale},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.33},
  URN =		{urn:nbn:de:0030-drops-77179},
  doi =		{10.4230/LIPIcs.FSCD.2017.33},
  annote =	{Keywords: proof theory, combinatorics, coherence theorem, substructural logic, associativity}
}
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