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Modal Reasoning = Metric Reasoning, via Lawvere [article]

Ugo Dal Lago, Francesco Gavazzo
<span title="2021-03-05">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In particular, Gavazzo [39] , [53] developed a theory of quantale-based applicative (bi)simulation distances for higher-order languages with algebraic effects [71] - [73] .  ...  Our starting point is the work by Dal Lago, Gavazzo, and collaborators [31] - [33] , [39] , [53] , who defined several coinductively-defined notions of equivalence and prove general congruence theorems  ...  We follow the same structure given by Gavazzo [39] , to which we refer for details.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.03871v1">arXiv:2103.03871v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x6prnsruf5fdtpyrhgv5fzk4vu">fatcat:x6prnsruf5fdtpyrhgv5fzk4vu</a> </span>
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Effectful Normal Form Bisimulation [chapter]

Ugo Dal Lago, Francesco Gavazzo
<span title="">2019</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Normal form bisimulation, also known as open bisimulation, is a coinductive technique for higher-order program equivalence in which programs are compared by looking at their essentially infinitary tree-like normal forms, i.e. at their Böhm or Lévy-Longo trees. The technique has been shown to be useful not only when proving metatheorems about λ-calculi and their semantics, but also when looking at concrete examples of terms. In this paper, we show that there is a way to generalise normal form
more &raquo; ... imulation to calculi with algebraic effects,à la Plotkin and Power. We show that some mild conditions on monads and relators, which have already been shown to guarantee effectful applicative bisimilarity to be a congruence relation, are enough to prove that the obtained notion of bisimilarity, which we call effectful normal form bisimilarity, is a congruence relation, and thus sound for contextual equivalence. Additionally, contrary to applicative bisimilarity, normal form bisimilarity allows for enhancements of the bisimulation proof method, hence proving a powerful reasoning principle for effectful programming languages.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-030-17184-1_10">doi:10.1007/978-3-030-17184-1_10</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/whayrorjs5eqhdryy65k2fhn2e">fatcat:whayrorjs5eqhdryy65k2fhn2e</a> </span>
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A Diagrammatic Calculus for Algebraic Effects [article]

Ugo Dal Lago, Francesco Gavazzo
<span title="2020-01-07">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce a new diagrammatic notation for representing the result of (algebraic) effectful computations. Our notation explicitly separates the effects produced during a computation from the possible values returned, this way simplifying the extension of definitions and results on pure computations to an effectful setting. Additionally, we show a number of algebraic and order-theoretic laws on diagrams, this way laying the foundations for a diagrammatic calculus of algebraic effects. We give
more &raquo; ... formal foundation for such a calculus in terms of Lawvere theories and generic effects.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.01337v2">arXiv:2001.01337v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xql67jnewrdk5pihzbcrx5hgua">fatcat:xql67jnewrdk5pihzbcrx5hgua</a> </span>
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On Reinforcement Learning, Effect Handlers, and the State Monad [article]

Ugo Dal Lago, Francesco Gavazzo, Alexis Ghyselen
<span title="2022-03-29">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study the algebraic effects and handlers as a way to support decision-making abstractions in functional programs, whereas a user can ask a learning algorithm to resolve choices without implementing the underlying selection mechanism, and give a feedback by way of rewards. Differently from some recently proposed approach to the problem based on the selection monad [Abadi and Plotkin, LICS 2021], we express the underlying intelligence as a reinforcement learning algorithm implemented as a set
more &raquo; ... f handlers for some of these algebraic operations, including those for choices and rewards. We show how we can in practice use algebraic operations and handlers -- as available in the programming language EFF -- to clearly separate the learning algorithm from its environment, thus allowing for a good level of modularity. We then show how the host language can be taken as a lambda-calculus with handlers, this way showing what the essential linguistic features are. We conclude by hinting at how type and effect systems could ensure safety properties, at the same time pointing at some directions for further work.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2203.15426v1">arXiv:2203.15426v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iuolqboopvgzvm3guzwst5qkb4">fatcat:iuolqboopvgzvm3guzwst5qkb4</a> </span>
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Effectful Applicative Bisimilarity: Monads, Relators, and Howe's Method (Long Version) [article]

Ugo Dal Lago, Francesco Gavazzo, Paul Blain Levy
<span title="2017-04-15">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study Abramsky's applicative bisimilarity abstractly, in the context of call-by-value λ-calculi with algebraic effects. We first of all endow a computational λ-calculus with a monadic operational semantics. We then show how the theory of relators provides precisely what is needed to generalise applicative bisimilarity to such a calculus, and to single out those monads and relators for which applicative bisimilarity is a congruence, thus a sound methodology for program equivalence. This is done by studying Howe's method in the abstract.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1704.04647v1">arXiv:1704.04647v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bgnjlbhwr5bbhe34xalgmc3xie">fatcat:bgnjlbhwr5bbhe34xalgmc3xie</a> </span>
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Differential Logical Relations, Part I: The Simply-Typed Case (Long Version) [article]

Ugo Dal Lago, Francesco Gavazzo, Akira Yoshimizu
<span title="2019-04-27">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce a new form of logical relation which, in the spirit of metric relations, allows us to assign each pair of programs a quantity measuring their distance, rather than a boolean value standing for their being equivalent. The novelty of differential logical relations consists in measuring the distance between terms not (necessarily) by a numerical value, but by a mathematical object which somehow reflects the interactive complexity, i.e. the type, of the compared terms. We exemplify
more &raquo; ... concept in the simply-typed lambda-calculus, and show a form of soundness theorem. We also see how ordinary logical relations and metric relations can be seen as instances of differential logical relations. Finally, we show that differential logical relations can be organised in a cartesian closed category, contrarily to metric relations, which are well-known not to have such a structure, but only that of a monoidal closed category.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.12137v1">arXiv:1904.12137v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/n7nnwhirfjahtgjzog7yyu2vmq">fatcat:n7nnwhirfjahtgjzog7yyu2vmq</a> </span>
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Resource Transition Systems and Full Abstraction for Linear Higher-Order Effectful Systems [article]

Ugo Dal Lago, Francesco Gavazzo
<span title="2021-06-24">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We investigate program equivalence for linear higher-order(sequential) languages endowed with primitives for computational effects. More specifically, we study operationally-based notions of program equivalence for a linear λ-calculus with explicit copying and algebraic effects à la Plotkin and Power. Such a calculus makes explicit the interaction between copying and linearity, which are intensional aspects of computation, with effects, which are, instead, extensional. We review some of the
more &raquo; ... ons of equivalences for linear calculi proposed in the literature and show their limitations when applied to effectful calculi where copying is a first-class citizen. We then introduce resource transition systems, namely transition systems whose states are built over tuples of programs representing the available resources, as an operational semantics accounting for both intensional and extensional interactive behaviors of programs. Our main result is a sound and complete characterization of contextual equivalence as trace equivalence defined on top of resource transition systems.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.12849v1">arXiv:2106.12849v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r5q7tcdue5awzkqup4r7juihgi">fatcat:r5q7tcdue5awzkqup4r7juihgi</a> </span>
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On the Versatility of Open Logical Relations: Continuity, Automatic Differentiation, and a Containment Theorem [article]

Gilles Barthe, Raphaëlle Crubillé, Ugo Dal Lago, Francesco Gavazzo
<span title="2020-02-19">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Logical relations are one of the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on
more &raquo; ... of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed λ-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any such a collection of functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.08489v1">arXiv:2002.08489v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wayjsmoewfgyrf5n7iokv27wee">fatcat:wayjsmoewfgyrf5n7iokv27wee</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200321154748/https://arxiv.org/pdf/2002.08489v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.08489v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

On the Versatility of Open Logical Relations [chapter]

Gilles Barthe, Raphaëlle Crubillé, Ugo Dal Lago, Francesco Gavazzo
<span title="">2020</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Logical relations are one among the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on
more &raquo; ... rms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed λ-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued firstorder functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-030-44914-8_3">doi:10.1007/978-3-030-44914-8_3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/k6qdai3k5vdkrnhayohebr6ygi">fatcat:k6qdai3k5vdkrnhayohebr6ygi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200510095123/https://link.springer.com/content/pdf/10.1007%2F978-3-030-44914-8_3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/15/8e/158efc0f8de5e36ab5eda9ae7b6c3ea18a287448.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-030-44914-8_3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Quantitative Behavioural Reasoning for Higher-order Effectful Programs

Francesco Gavazzo
<span title="">2018</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cnybbxuptncftdgxtodn5edz7m" style="color: black;">Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS &#39;18</a> </i> &nbsp;
This paper studies quantitative refinements of Abramsky's applicative similarity and bisimilarity in the context of a generalisation of Fuzz, a call-by-value λ-calculus with a linear type system that can express program sensitivity, enriched with algebraic operations à la Plotkin and Power. To do so a general, abstract framework for studying behavioural relations taking values over quantales is introduced according to Lawvere's analysis of generalised metric spaces. Barr's notion of relator (or
more &raquo; ... lax extension) is then extended to quantale-valued relations, adapting and extending results from the field of monoidal topology. Abstract notions of quantale-valued effectful applicative similarity and bisimilarity are then defined and proved to be a compatible generalised metric (in the sense of Lawvere) and pseudometric, respectively, under mild conditions.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3209108.3209149">doi:10.1145/3209108.3209149</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/lics/Gavazzo18.html">dblp:conf/lics/Gavazzo18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2l3donqqgvg3dfxiss3n34aq5y">fatcat:2l3donqqgvg3dfxiss3n34aq5y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200509033657/https://hal.inria.fr/hal-01926069/file/lics2018CameraReady.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4c/4e/4c4e3d2deef9e84415037aa6ef827d2b928527b5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3209108.3209149"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Effectful applicative bisimilarity: Monads, relators, and Howe's method

Ugo Dal Lago, Francesco Gavazzo, Paul Blain Levy
<span title="">2017</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cnybbxuptncftdgxtodn5edz7m" style="color: black;">2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)</a> </i> &nbsp;
We study Abramsky's applicative bisimilarity abstractly, in the context of call-by-value λ-calculi with algebraic effects. We first of all endow a computational λ-calculus with a monadic operational semantics. We then show how the theory of relators provides precisely what is needed to generalise applicative bisimilarity to such a calculus, and to single out those monads and relators for which applicative bisimilarity is a congruence, thus a sound methodology for program equivalence. This is done by studying Howe's method in the abstract.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/lics.2017.8005117">doi:10.1109/lics.2017.8005117</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/lics/LagoGL17.html">dblp:conf/lics/LagoGL17</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vptwgwjebvc53byww7awi32jzy">fatcat:vptwgwjebvc53byww7awi32jzy</a> </span>
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Differential Logical Relations, Part I: The Simply-Typed Case

Ugo Dal Lago, Francesco Gavazzo, Akira Yoshimizu, Michael Wagner
<span title="2019-07-08">2019</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pf6v2q6ji5h7birkbrtpdc2uhy" style="color: black;">International Colloquium on Automata, Languages and Programming</a> </i> &nbsp;
Gavazzo, and A. Yoshimizu 111:9 Let us now prove that (M SIN , f, M ID ) ∈ δ REAL→REAL , where f (x, y) = y + |x − sin x|.  ...  Gavazzo, and A. Yoshimizu 111:7 This is precisely what it is needed to turn τ into a quantale 2 [25] . Proposition 4. For each type τ , τ forms a commutative unital non-idempotent quantale.  ... 
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A Relational Theory of Monadic Rewriting Systems, Part I

Francesco Gavazzo, Claudia Faggian
<span title="2021-06-29">2021</span> <i title="IEEE"> 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) </i> &nbsp;
Motivated by the study of effectful programming languages and computations, we introduce a relational theory of monadic rewriting systems. The latter are rewriting systems whose notion of reduction is effectful, where effects are modelled as monads. Contrary to what happens in the ordinary operational semantics of monadic programming languages, defining meaningful notions of monadic rewriting turns out to problematic for several monads, including the distribution, powerset, reader, and global
more &raquo; ... ate monad. This raises the question of when monadic rewriting is possible. We answer that question by identifying a class of monads, known as weakly cartesian monads, that guarantee monadic rewriting to be well-behaved. In case monads are given as equational theories, as it is the case for algebraic effects, we also show that a sufficient condition to have wellbehaved notion of monadic rewriting is that all equations in the theory are linear. Finally, we apply the abstract theory of monadic rewriting systems to the call-by-value λ-calculus with algebraic effects, this way obtaining effectful (surface) standardisation and confluence theorems.
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220308032044/https://hal.archives-ouvertes.fr/hal-03455778/file/On_Monadic_Rewriting_Systems__Part_I.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/87/bd/87bd7f2900d3cbebe245f87aa45f2e58ddf2fced.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/lics52264.2021.9470633"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version) [article]

Francesco Gavazzo
<span title="2018-02-06">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This paper studies the quantitative refinements of Abramsky's applicative similarity and bisimilarity in the context of a generalisation of Fuzz, a call-by-value λ-calculus with a linear type system that can express programs sensitivity, enriched with algebraic operations à la Plotkin and Power. To do so a general, abstract framework for studying behavioural relations taking values over quantales is defined according to Lawvere's analysis of generalised metric spaces. Barr's notion of relator
more &raquo; ... r lax extension) is then extended to quantale-valued relations adapting and extending results from the field of monoidal topology. Abstract notions of quantale-valued effectful applicative similarity and bisimilarity are then defined and proved to be a compatible generalised metric (in the sense of Lawvere) and pseudometric, respectively, under mild conditions.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1801.09072v3">arXiv:1801.09072v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hvpl33atmjaorlruldhne35ine">fatcat:hvpl33atmjaorlruldhne35ine</a> </span>
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Coinductive Equivalences and Metrics for Higher-order Languages with Algebraic Effects

Francesco Gavazzo
<span title="">2019</span>
This dissertation investigates notions of program equivalence and metric for higher-order sequential languages with algebraic e ects. Computational e ects are those aspects of computation that involve forms of interaction with the environment. Due to such an interactive behaviour, reasoning about e ectful programs is well-known to be hard. This is especially true for higher-order e ectful languages, where programs can be passed as input to, and returned as output by other programs, as well as
more &raquo; ... rform side-e ects. Additionally, when dealing with e ectful languages, program equivalence is oftentimes too coarse, not allowing, for instance, to quantify the observable di erences between programs. A natural way to overcome this problem is to re ne the notion of a program equivalence into the one of a program distance or program metric, this way allowing for a ner, quantitative analysis of program behaviour. A proper account of program distance, however, requires a more sophisticated theory than program equivalence, both conceptually and mathematically. This often makes the study of program distance way more di cult than the corresponding study of program equivalence. Algebraic e ects provide a powerful formalism to structure e ectful higher-order (sequential) computations. Accordingly, e ectful computations are produced by means of e ect-triggering operations which act as sources of the side e ects of interest. Such operations are algebraic, in the sense that their (operational) semantics is independent of their continuation, and thus e ects act independently of the evaluation context in which they are executed. Algebraic e ects can be used to model several computational e ects, proving formal models for higher-order languages with nondeterministic, probabilistic, and imperative features, as well as combinations thereof. In fact, contrary to other theories of computational e ects, algebraic e ects naturally support operations to combine algebraic theories, and thus allow for the combination of e ects with one another. These features make reasoning about program equivalence for languages with algebraic e ects challenging, as the operational behaviour of a program may be determined by complex interactions between the program and the environment. The rst part of this dissertation studies bisimulation-based notions of equivalence and re nement for λ-calculi enriched with algebraic e ects. In particular, notions of e ectful applicative and normal form (bi)similarity are de ned for both call-by-name and call-by-value λ-calculi, as well as a notion of monadic applicative (bi)similarity for call-by-name calculi only. For all these notions, congruence and precongruence theorems are proved, which directly lead to soundness results with respect to an extension of Morris' contextual equivalence to e ectful calculi. In order to design the aforementioned notions of equivalence and re nement, an abstract relational framework is developed, which is based on the notions of a monad and of a relator, the latter being an abstract construction axiomatising relation lifting operations. The second part of this dissertation is devoted to the study of program distances for languages with algebraic e ects. Following Lawvere analysis of metric spaces as enriched categories, the abstract relational framework developed in the rst part of the dissertation, is extended to relations taking values over quantales, the latter being algebraic structures whose elements represent 'abstract quantities'. Using such a framework, the notion of an e ectful applicative bisimulation distance is de ned, and its Broadly speaking, the latter is the notion of equality arising from the β-rule 10 . The major drawback of Church's proposal is that not all λ-terms converge 11 , and thus all diverging terms should be regarded as equal (or, from a linguistic perspective, as meaningless). Unfortunately, the theory induced by such a notion of equality has been proved to be inconsistent (see (Barendregt, 1984) for a nice exposition of the subject). Historically, this problem has been solved by looking at a more informative notion of a value, which can be elegantly described via the notion of a Böhm tree of a λ-term, giving raise to the so-called standard theory (Barendregt, 1984) . Roughly speaking, the Böhm tree of a λ-term e is a kind of in nitary value 12 representing all the stable amount of information obtained by evaluating e. As a consequence, all λ-terms have a Böhm tree, and we can thus regard two λ-terms as equivalent if and only if they have the same Böhm trees. Actually, the Böhm tree of a λ-term e is obtained by evaluating e using neither the call-by-name nor the call-by-value reduction strategy, but the so-called head reduction strategy (Barendregt, 1984) . We will be sloppy on that for the moment, and simply recall that Böhm tree-like structures can be de ned for both the call-by-name (giving the notion of a Lévy-Longo tree (Lévy, 1975; Longo, 1983) ) and the call-by-value (see e.g. (Carraro & Guerrieri, 2014; S. B. Lassen, 2005) ) λ-calculus. At this point of the story it is then natural to ask whether there is a link between the above 'treelike' equivalences, and the interactive approach to program equivalence of previous section. This is indeed the case, as shown in (Sangiorgi, 1992 (Sangiorgi, , 1994)) . Moving from the theory of open bisimulation for the π -calculus (Sangiorgi, 1993) and from encodings of the λ-calculus into the π -calculus (Milner, 1992) , Sangiorgi modi ed Abramsky's testing scenario as follows: instead of testing a value (which, for the sake of the argument, we assume to be a λ-abstraction) λx.e by passing it a value as argument, the environment can now inspect the body of the function, i.e. the term e under the lambda 13 . Compared to Abramsky's idea of testing λ-terms extensionally, open bisimulation can be seen as an intensional notion of equivalence, whereby functions are not tested for their input-output behaviour, but for their intensional (syntactical, to some extent) structure. The tree-like structures associated to λ-terms can be now seen as unfolded representation of this new testing process, and their equality thus coincides with open bisimilarity, the latter being open bisimulation equality. This result had several implications, as it provided (to the best of the author's knowledge) the rst coinductive account to treelike equivalences, whose theory was essentially induction-based at the time (see e.g. (Barendregt, 1984) ). The results proved in (Sangiorgi, 1992 (Sangiorgi, , 1994) ) concerned the call-by-name λ-calculus and the associated notion of Lévy-Longo tree equality. Such results have been then extended to Böhm trees (S. B. Lassen, 1999) and to the call-by-value λ-calculus (S. B. Lassen, 2005) , introducing the general notion of a normal form bisimulation 14 . Open bisimilarity provides a powerful candidate proof technique for contextual equivalence, as λterms are essentially tested in isolation (the environment can interact with them inspecting their intensional structure only, and does not have the power to in uence computations 15 ), meaning that in order to prove two λ-terms to be open bisimilar it is enough to reason about them locally. Open bisimilarity can indeed be proved to be a sound proof technique for contextual equivalence. Unfortunately, due to the limited testing power provided by open bisimulations, open bisimilarity turned out to strictly ner (i.e. strictly included) than contextual equivalence, in general 16 . 10 That is, the congruence relation (inductively) generated by the relation relating terms of the form (λx .e ) with e[x := ]. 11 Think, for instance, to Ω (λx .x x )(λx .x x ). 12 Normal form, actually. We will say more on that later. 13 De ning a λ-term e to be closed if all its variables are bound by a λ (and open otherwise), we see that the environment now tests open terms, from which the name open bisimulation. 14 The expressions normal form bisimulation and open bisimulation are used interchangeably, although the former is arguably the most used one. We will stick with this convention, although we remark that the name open bisimulation seems more appropriate given both the foundational link with the notion of an open bisimulation for the π -calculus, and the relevance of testing open terms (the latter being a central di erence with applicative bisimulation). 15 E.g. by passing values as arguments to the tested terms. 16 That holds for both the call-by-name and the call-by-value λ-calculus, as we will see. However, due to the celebrated Böhm
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