Pub Date : 2017-09-03DOI: 10.4230/LIPIcs.FSCD.2017.31
Lutz Straßburger
This paper introduces combinatorial flows that generalize combinatorial proofs such that they also include cut and substitution as methods of proof compression. We show a normalization procedure for combinatorial flows, and how syntactic proofs are translated into combinatorial flows and vice versa.
{"title":"Combinatorial Flows and Their Normalisation","authors":"Lutz Straßburger","doi":"10.4230/LIPIcs.FSCD.2017.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.31","url":null,"abstract":"This paper introduces combinatorial flows that generalize combinatorial proofs such that they also include cut and substitution as methods of proof compression. We show a normalization procedure for combinatorial flows, and how syntactic proofs are translated into combinatorial flows and vice versa.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115089823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-03DOI: 10.4230/LIPIcs.FSCD.2017.18
Daniel Hillerström, S. Lindley, R. Atkey, K. Sivaramakrishnan
We present Continuation Passing Style (CPS) translations for Plotkin and Pretnar's effect handlers with Hillerstrom and Lindley's row-typed fine-grain call-by-value calculus of effect handlers as the source language. CPS translations of handlers are interesting theoretically, to explain the semantics of handlers, and also offer a practical implementation technique that does not require special support in the target language's runtime. We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation first by uncurrying it to yield a properly tail-recursive translation and second by making it higher-order in order to contract administrative redexes at translation time. We prove that the higher-order CPS translation simulates effect handler reduction. We have implemented the higher-order CPS translation as a JavaScript backend for the Links programming language.
{"title":"Continuation Passing Style for Effect Handlers","authors":"Daniel Hillerström, S. Lindley, R. Atkey, K. Sivaramakrishnan","doi":"10.4230/LIPIcs.FSCD.2017.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.18","url":null,"abstract":"We present Continuation Passing Style (CPS) translations for Plotkin and Pretnar's effect handlers with Hillerstrom and Lindley's row-typed fine-grain call-by-value calculus of effect handlers as the source language. CPS translations of handlers are interesting theoretically, to explain the semantics of handlers, and also offer a practical implementation technique that does not require special support in the target language's runtime. \u0000 \u0000We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation first by uncurrying it to yield a properly tail-recursive translation and second by making it higher-order in order to contract administrative redexes at translation time. We prove that the higher-order CPS translation simulates effect handler reduction. We have implemented the higher-order CPS translation as a JavaScript backend for the Links programming language.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117337887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-01DOI: 10.4230/LIPIcs.FSCD.2017.11
J. Blanchette, M. Fleury, Dmitriy Traytel
We present a collection of formalized results about finite nested multisets, developed using the Isabelle/HOL proof assistant. The nested multiset order is a generalization of the multiset order that can be used to prove termination of processes. Hereditary multisets, a variant of nested multisets, offer a convenient representation of ordinals below 0. In Isabelle/HOL, both nested and hereditary multisets can be comfortably defined as inductive datatypes. Our formal library also provides, somewhat nonstandardly, multisets with negative multiplicities and syntactic or-dinals with negative coefficients. We present applications of the library to formalizations of Goodstein's theorem and the decidability of unary PCF (programming computable functions). 1 Introduction In their seminal article on proving termination using multisets [15], Dershowitz and Manna introduced two orders of increasing strength. The multiset order lifts a base partial order on a set A to finite multisets over A. It forms the basis of the multiset path order, which has many applications in term rewriting [41] and automatic theorem proving [1]. The nested multiset order is a generalization of the multiset order that operates on multisets that can be nested in arbitrary ways. Nesting can increase the order's strength: If (A, <) has ordinal type α < 0 , the associated multiset order has ordinal type ω α , whereas the nested order has ordinal type 0 = ω ω ω. .. . In this paper, we present formal proofs of the main properties of the nested multiset order that are useful in applications: preservation of well-foundedness and preservation of totality (linearity). The proofs are developed in the Isabelle/HOL proof assistant [27]. To our knowledge, this is the first development of its kind in any proof assistant. Our starting point is the following inductive datatype of nested finite multisets over a type a (Section 4): datatype a nmultiset = Elem a | MSet ((a nmultiset) multiset) The above Isabelle/HOL command introduces a (unary postfix) type constructor, nmultiset, equipped with two constructors, Elem : a → a nmultiset and MSet : (a nmultiset)multiset → a nmultiset, where a is a type variable and multiset is the type constructor of (finite) multisets.
我们提出了关于有限嵌套多集的形式化结果的集合，使用Isabelle/HOL证明助手开发。嵌套多集顺序是多集顺序的推广，可以用来证明过程的终止。遗传多集是嵌套多集的一种变体，它提供了0以下序数的方便表示。在Isabelle/HOL中，嵌套多集和遗传多集都可以轻松地定义为归纳数据类型。我们的正式库还提供了具有负多重性的多集和具有负系数的语法或dinals(有点不标准)。给出了该库在Goodstein定理形式化和一元可编程可计算函数(PCF)可判决性方面的应用。在他们关于使用多集[15]证明终止的开创性文章中，Dershowitz和Manna引入了两个增加强度的阶数。多集阶将集合a上的基偏阶提升到a上的有限多集，它构成了多集路径阶的基础，在项重写[41]和自动定理证明[1]中有许多应用。嵌套多集顺序是多集顺序的泛化，它作用于可以以任意方式嵌套的多集。嵌套可以增加顺序的强度:如果(A， <)具有序数类型α < 0，则关联的多集顺序具有序数类型ω α，而嵌套顺序具有序数类型0 = ω ω ω. .. .在本文中，我们给出了嵌套多集序在应用中有用的主要性质的形式化证明:保持良基性和保持整体(线性)。证明是在Isabelle/HOL证明助手[27]中开发的。据我们所知，这是其在任何证明助理的第一次发展。我们的起点是a类型上嵌套有限多集的以下归纳数据类型(第4节):datatype a nmultiset = Elem a | MSet ((a nmultiset)multiset)上面的Isabelle/HOL命令引入了一个(一元后置)类型构造函数nmultiset，它带有两个构造函数Elem: a→一个nmultiset和MSet: (a nmultiset)multiset→一个nmultiset，其中a是类型变量，multiset是(有限)多集的类型构造函数。
{"title":"Nested Multisets, Hereditary Multisets, and Syntactic Ordinals in Isabelle/HOL","authors":"J. Blanchette, M. Fleury, Dmitriy Traytel","doi":"10.4230/LIPIcs.FSCD.2017.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.11","url":null,"abstract":"We present a collection of formalized results about finite nested multisets, developed using the Isabelle/HOL proof assistant. The nested multiset order is a generalization of the multiset order that can be used to prove termination of processes. Hereditary multisets, a variant of nested multisets, offer a convenient representation of ordinals below 0. In Isabelle/HOL, both nested and hereditary multisets can be comfortably defined as inductive datatypes. Our formal library also provides, somewhat nonstandardly, multisets with negative multiplicities and syntactic or-dinals with negative coefficients. We present applications of the library to formalizations of Goodstein's theorem and the decidability of unary PCF (programming computable functions). 1 Introduction In their seminal article on proving termination using multisets [15], Dershowitz and Manna introduced two orders of increasing strength. The multiset order lifts a base partial order on a set A to finite multisets over A. It forms the basis of the multiset path order, which has many applications in term rewriting [41] and automatic theorem proving [1]. The nested multiset order is a generalization of the multiset order that operates on multisets that can be nested in arbitrary ways. Nesting can increase the order's strength: If (A, <) has ordinal type α < 0 , the associated multiset order has ordinal type ω α , whereas the nested order has ordinal type 0 = ω ω ω. .. . In this paper, we present formal proofs of the main properties of the nested multiset order that are useful in applications: preservation of well-foundedness and preservation of totality (linearity). The proofs are developed in the Isabelle/HOL proof assistant [27]. To our knowledge, this is the first development of its kind in any proof assistant. Our starting point is the following inductive datatype of nested finite multisets over a type a (Section 4): datatype a nmultiset = Elem a | MSet ((a nmultiset) multiset) The above Isabelle/HOL command introduces a (unary postfix) type constructor, nmultiset, equipped with two constructors, Elem : a → a nmultiset and MSet : (a nmultiset)multiset → a nmultiset, where a is a type variable and multiset is the type constructor of (finite) multisets.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130126717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-09-01DOI: 10.4230/LIPIcs.FSCD.2017.16
M. Fiore, P. Saville
We introduce and study the notion of list object with algebraic structure. The first key aspect of our development is that the notion of list object is considered in the context of monoidal structure; the second key aspect is that we further equip list objects with algebraic structure in this setting. Within our framework, we observe that list objects give rise to free monoids and moreover show that this remains so in the presence of algebraic structure. In addition, we provide a basic theory explicitly describing as an inductively defined object such free monoids with suitably compatible algebraic structure in common practical situations. This theory is accompanied by the study of two technical themes that, besides being of interest in their own right, are important for establishing applications. These themes are: parametrised initiality, central to the universal property defining list objects; and approaches to algebraic structure, in particular in the context of monoidal theories. The latter leads naturally to a notion of nsr (or near semiring) category of independent interest. With the theoretical development in place, we touch upon a variety of applications, considering Natural Numbers Objects in domain theory, giving a universal property for the monadic list transformer, providing free instances of algebraic extensions of the Haskell Monad type class, elucidating the algebraic character of the construction of opetopes in higher-dimensional algebra, and considering free models of second-order algebraic theories.
{"title":"List Objects with Algebraic Structure","authors":"M. Fiore, P. Saville","doi":"10.4230/LIPIcs.FSCD.2017.16","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.16","url":null,"abstract":"We introduce and study the notion of list object with algebraic structure. The first key aspect of our development is that the notion of list object is considered in the context of monoidal structure; the second key aspect is that we further equip list objects with algebraic structure in this setting. Within our framework, we observe that list objects give rise to free monoids and moreover show that this remains so in the presence of algebraic structure. In addition, we provide a basic theory explicitly describing as an inductively defined object such free monoids with suitably compatible algebraic structure in common practical situations. This theory is accompanied by the study of two technical themes that, besides being of interest in their own right, are important for establishing applications. These themes are: parametrised initiality, central to the universal property defining list objects; and approaches to algebraic structure, in particular in the context of monoidal theories. The latter leads naturally to a notion of nsr (or near semiring) category of independent interest. With the theoretical development in place, we touch upon a variety of applications, considering Natural Numbers Objects in domain theory, giving a universal property for the monadic list transformer, providing free instances of algebraic extensions of the Haskell Monad type class, elucidating the algebraic character of the construction of opetopes in higher-dimensional algebra, and considering free models of second-order algebraic theories.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122645198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-08-21DOI: 10.4230/LIPIcs.FSCD.2017.25
Daniel R. Licata, Michael Shulman, Mitchell Riley
We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics / type theories. The framework is a sequent calculus / normal-form type theory parametrized by a mode theory, which is used to describe the structure of contexts and the structural properties they obey. In this sequent calculus, the context itself obeys standard structural properties, while a term, drawn from the mode theory, constrains how the context can be used. Product types, implications, and modalities are defined as instances of two general connectives, one positive and one negative, that manipulate these terms. Specific mode theories can express a range of substructural and modal connectives, including non-associative, ordered, linear, affine, relevant, and cartesian products and implications; monoidal and non-monoidal functors, (co)monads and adjunctions; n-linear variables; and bunched implications. We prove cut (and identity) admissibility independently of the mode theory, obtaining it for many different logics at once. Further, we give a general equational theory on derivations / terms that, in addition to the usual beta/eta-rules, characterizes when two derivations differ only by the placement of structural rules. Additionally, we give an equivalent semantic presentation of these ideas, in which a mode theory corresponds to a 2-dimensional cartesian multicategory, the framework corresponds to another such multicategory with a functor to the mode theory, and the logical connectives make this into a bifibration. Finally, we show how the framework can be used both to encode existing existing logics / type theories and to design new ones.
{"title":"A Fibrational Framework for Substructural and Modal Logics","authors":"Daniel R. Licata, Michael Shulman, Mitchell Riley","doi":"10.4230/LIPIcs.FSCD.2017.25","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.25","url":null,"abstract":"We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics / type theories. The framework is a sequent calculus / normal-form type theory parametrized by a mode theory, which is used to describe the structure of contexts and the structural properties they obey. In this sequent calculus, the context itself obeys standard structural properties, while a term, drawn from the mode theory, constrains how the context can be used. Product types, implications, and modalities are defined as instances of two general connectives, one positive and one negative, that manipulate these terms. Specific mode theories can express a range of substructural and modal connectives, including non-associative, ordered, linear, affine, relevant, and cartesian products and implications; monoidal and non-monoidal functors, (co)monads and adjunctions; n-linear variables; and bunched implications. We prove cut (and identity) admissibility independently of the mode theory, obtaining it for many different logics at once. Further, we give a general equational theory on derivations / terms that, in addition to the usual beta/eta-rules, characterizes when two derivations differ only by the placement of structural rules. Additionally, we give an equivalent semantic presentation of these ideas, in which a mode theory corresponds to a 2-dimensional cartesian multicategory, the framework corresponds to another such multicategory with a functor to the mode theory, and the logical connectives make this into a bifibration. Finally, we show how the framework can be used both to encode existing existing logics / type theories and to design new ones.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"50 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120839421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-07-07DOI: 10.4230/LIPIcs.FSCD.2017.17
S. Guerrini, M. Solieri
Sharing graphs are a local and asynchronous implementation of lambda-calculus beta-reduction (or linear logic proof-net cut-elimination) that avoids useless duplications. Empirical benchmarks suggest that they are one of the most efficient machineries, when one wants to fully exploit the higher-order features of lambda-calculus. However, we still lack confirming grounds with theoretical solidity to dispel uncertainties about the adoption of sharing graphs. Aiming at analysing in detail the worst-case overhead cost of sharing operators, we restrict to the case of elementary and light linear logic, two subsystems with bounded computational complexity of multiplicative exponential linear logic. In these two cases, the bookkeeping component is unnecessary, and sharing graphs are simplified to the so-called "abstract algorithm". By a modular cost comparison over a syntactical simulation, we prove that the overhead of shared reductions is quadratically bounded to cost of the naive implementation, i.e. proof-net reduction. This result generalises and strengthens a previous complexity result, and implies that the price of sharing is negligible, if compared to the obtainable benefits on reductions requiring a large amount of duplication.
{"title":"Is the Optimal Implementation Inefficient? Elementarily Not","authors":"S. Guerrini, M. Solieri","doi":"10.4230/LIPIcs.FSCD.2017.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.17","url":null,"abstract":"Sharing graphs are a local and asynchronous implementation of lambda-calculus beta-reduction (or linear logic proof-net cut-elimination) that avoids useless duplications. Empirical benchmarks suggest that they are one of the most efficient machineries, when one wants to fully exploit the higher-order features of lambda-calculus. However, we still lack confirming grounds with theoretical solidity to dispel uncertainties about the adoption of sharing graphs. \u0000 \u0000Aiming at analysing in detail the worst-case overhead cost of sharing operators, we restrict to the case of elementary and light linear logic, two subsystems with bounded computational complexity of multiplicative exponential linear logic. In these two cases, the bookkeeping component is unnecessary, and sharing graphs are simplified to the so-called \"abstract algorithm\". By a modular cost comparison over a syntactical simulation, we prove that the overhead of shared reductions is quadratically bounded to cost of the naive implementation, i.e. proof-net reduction. This result generalises and strengthens a previous complexity result, and implies that the price of sharing is negligible, if compared to the obtainable benefits on reductions requiring a large amount of duplication.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"100 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115544068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-04-05DOI: 10.4230/LIPIcs.FSCD.2017.23
A. Kawamura, Florian Steinberg
This paper introduces a more restrictive notion of feasibility of functionals on Baire space than the established one from second-order complexity theory. Thereby making it possible to consider functions on the natural numbers as running times of oracle Turing machines and avoiding second-order polynomials, which are notoriously difficult to handle. Furthermore, all machines that witness this stronger kind of feasibility can be clocked and the different traditions of treating partial functionals from computable analysis and second-order complexity theory are equated in a precise sense. The new notion is named "strong polynomial-time computability", and proven to be a strictly stronger requirement than polynomial-time computability. It is proven that within the framework for complexity of operators from analysis introduced by Kawamura and Cook the classes of strongly polynomial-time computable functionals and polynomial-time computable functionals coincide.
{"title":"Polynomial Running Times for Polynomial-Time Oracle Machines","authors":"A. Kawamura, Florian Steinberg","doi":"10.4230/LIPIcs.FSCD.2017.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2017.23","url":null,"abstract":"This paper introduces a more restrictive notion of feasibility of functionals on Baire space than the established one from second-order complexity theory. Thereby making it possible to consider functions on the natural numbers as running times of oracle Turing machines and avoiding second-order polynomials, which are notoriously difficult to handle. Furthermore, all machines that witness this stronger kind of feasibility can be clocked and the different traditions of treating partial functionals from computable analysis and second-order complexity theory are equated in a precise sense. The new notion is named \"strong polynomial-time computability\", and proven to be a strictly stronger requirement than polynomial-time computability. It is proven that within the framework for complexity of operators from analysis introduced by Kawamura and Cook the classes of strongly polynomial-time computable functionals and polynomial-time computable functionals coincide.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"194 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124290438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-11-17DOI: 10.4230/LIPIcs.FSCD.2016.19
Andrej Dudenhefner, M. Martens, J. Rehof
The intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the unification problem is decidable. We give the first nontrivial lower bound for the problem by showing (our main result) that it is exponential time hard. Furthermore, we show that this holds even under rank 1 solutions (substitutions whose codomains are restricted to contain rank 1 types). In addition, we provide a fixed-parameter intractability result for intersection type matching (one-sided unification), which is known to be NP-complete. We place the intersection type unification problem in the context of unification theory. The equational theory of intersection types can be presented as an algebraic theory with an ACI (associative, commutative, and idempotent) operator (intersection type) combined with distributivity properties with respect to a second operator (function type). Although the problem is algebraically natural and interesting, it appears to occupy a hitherto unstudied place in the theory of unification, and our investigation of the problem suggests that new methods are required to understand the problem. Thus, for the lower bound proof, we were not able to reduce from known results in ACI-unification theory and use game-theoretic methods for two-player tiling games.
{"title":"The Intersection Type Unification Problem","authors":"Andrej Dudenhefner, M. Martens, J. Rehof","doi":"10.4230/LIPIcs.FSCD.2016.19","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2016.19","url":null,"abstract":"The intersection type unification problem is an important component in \u0000proof search related to several natural decision problems in \u0000intersection type systems. It is unknown and remains open whether the \u0000unification problem is decidable. We give the first nontrivial lower \u0000bound for the problem by showing (our main result) that it is \u0000exponential time hard. Furthermore, we show that this holds even under \u0000rank 1 solutions (substitutions whose codomains are restricted to \u0000contain rank 1 types). In addition, we provide a fixed-parameter \u0000intractability result for intersection type matching (one-sided \u0000unification), which is known to be NP-complete. \u0000 \u0000We place the intersection type unification problem in the context of \u0000unification theory. The equational theory of intersection types can \u0000be presented as an algebraic theory with an ACI (associative, \u0000commutative, and idempotent) operator (intersection type) combined \u0000with distributivity properties with respect to a second operator \u0000(function type). Although the problem is algebraically natural and \u0000interesting, it appears to occupy a hitherto unstudied place in the \u0000theory of unification, and our investigation of the problem suggests \u0000that new methods are required to understand the problem. Thus, for the \u0000lower bound proof, we were not able to reduce from known results in \u0000ACI-unification theory and use game-theoretic methods for two-player \u0000tiling games.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127350415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-06-26DOI: 10.4230/LIPIcs.FSCD.2016.6
Thorsten Altenkirch, A. Kaposi
We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated using internal type theory using quotient inductive types. We use a typed presentation hence there are no preterms or realizers in our construction. NBE for simple types is using a logical relation between the syntax and the presheaf interpretation. In our construction, we merge the presheaf interpretation and the logical relation into a proof-relevant logical predicate. We have formalized parts of the construction in Agda.
{"title":"Normalisation by Evaluation for Dependent Types","authors":"Thorsten Altenkirch, A. Kaposi","doi":"10.4230/LIPIcs.FSCD.2016.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2016.6","url":null,"abstract":"We develop normalisation by evaluation (NBE) for dependent types based \u0000on presheaf categories. Our construction is formulated using internal \u0000type theory using quotient inductive types. We use a typed \u0000presentation hence there are no preterms or realizers in our \u0000construction. NBE for simple types is using a logical relation between \u0000the syntax and the presheaf interpretation. In our construction, we \u0000merge the presheaf interpretation and the logical relation into a \u0000proof-relevant logical predicate. We have formalized parts of the \u0000construction in Agda.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133044663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-06-22DOI: 10.4230/LIPIcs.FSCD.2016.18
F. D. Vries
We distinguish between undefined terms as used in lambda definability of partial recursive functions and meaningless terms as used in infinite lambda calculus for the infinitary terms models that generalise the Bohm model. While there are uncountable many known sets of meaningless terms, there are four known sets of undefined terms. Two of these four are sets of meaningless terms. In this paper we first present set of sufficient conditions for a set of lambda terms to serve as set of undefined terms in lambda definability of partial functions. The four known sets of undefined terms satisfy these conditions. Next we locate the smallest set of meaningless terms satisfying these conditions. This set sits very low in the lattice of all sets of meaningless terms. Any larger set of meaningless terms than this smallest set is a set of undefined terms. Thus we find uncountably many new sets of undefined terms. As an unexpected bonus of our careful analysis of lambda definability we obtain a natural modification, strict lambda-definability, which allows for a Barendregt style of proof in which the representation of composition is truly the composition of representations.
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