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Thierry Coquand - One of the best experts on this subject based on the ideXlab platform.
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a normalizing Computation Rule for propositional extensionality in higher order minimal logic
Types for Proofs and Programs, 2018Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. How- ever, if we simply add the univalence axiom to type theory, then we lose the property of canonicity — that every closed term computes to a canonical form. A Computation becomes ‘stuck’ when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed us- ing a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Ω. There are proofs which inhabit propositions, which are the terms of type Ω. The canonical propositions are those constructed from ⊥ by implication ⊃. Thirdly, there are paths which inhabit equations M = A N , where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality — logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
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notions of anonymous existence in martin lof type theory
Logical Methods in Computer Science, 2017Co-Authors: Nicolai Kraus, Thierry Coquand, Martin Hotzel Escardo, Thorsten AltenkirchAbstract:As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental Computation Rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.
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a normalizing Computation Rule for propositional extensionality in higher order minimal logic
2016Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A Computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
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a strongly normalizing Computation Rule for univalence in higher order minimal logic
arXiv: Logic in Computer Science, 2016Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A Computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
Robin Adams - One of the best experts on this subject based on the ideXlab platform.
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a normalizing Computation Rule for propositional extensionality in higher order minimal logic
Types for Proofs and Programs, 2018Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. How- ever, if we simply add the univalence axiom to type theory, then we lose the property of canonicity — that every closed term computes to a canonical form. A Computation becomes ‘stuck’ when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed us- ing a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Ω. There are proofs which inhabit propositions, which are the terms of type Ω. The canonical propositions are those constructed from ⊥ by implication ⊃. Thirdly, there are paths which inhabit equations M = A N , where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality — logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
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a normalizing Computation Rule for propositional extensionality in higher order minimal logic
2016Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A Computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
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a strongly normalizing Computation Rule for univalence in higher order minimal logic
arXiv: Logic in Computer Science, 2016Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A Computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
Marc Bezem - One of the best experts on this subject based on the ideXlab platform.
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a normalizing Computation Rule for propositional extensionality in higher order minimal logic
Types for Proofs and Programs, 2018Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. How- ever, if we simply add the univalence axiom to type theory, then we lose the property of canonicity — that every closed term computes to a canonical form. A Computation becomes ‘stuck’ when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed us- ing a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Ω. There are proofs which inhabit propositions, which are the terms of type Ω. The canonical propositions are those constructed from ⊥ by implication ⊃. Thirdly, there are paths which inhabit equations M = A N , where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality — logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
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a normalizing Computation Rule for propositional extensionality in higher order minimal logic
2016Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A Computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
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a strongly normalizing Computation Rule for univalence in higher order minimal logic
arXiv: Logic in Computer Science, 2016Co-Authors: Robin Adams, Marc Bezem, Thierry CoquandAbstract:The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A Computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the Computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.
David S. Warren - One of the best experts on this subject based on the ideXlab platform.
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PODS - Query evaluation under the well-founded semantics
Proceedings of the twelfth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems - PODS '93, 1993Co-Authors: Weidong Chen, David S. WarrenAbstract:SLD resolution with negation as finite failure (or SLDNF) reflects the procedural interpretation of Horn-clause predicate logic as a programming language and forms the Computational basis for prolog systems. Despite its advantages in memory management, SLDNF is often not appropriate for query evaluation for three reasons; a) it may not terminate due to infinite positive recursion; b) it may not terminate due to infinite recursion through negation; and c) it may repeatedly evaluate the same clause body literal, leading to unacceptable performance. We address all three problems for goal-oriented query evaluation of arbitrary programs by presenting an extension of SLDNF, called SLG resolution , with the following distinctive features: (i) SLG resolution is a partial deduction procedure, consisting of several transformations. Each query is transformed step by step into a set of answer clauses; (ii) SLG resolution is sound and ideally complete for all non-floundering queries with respect to all three-valued stable models (including the well founded partial model); (iii) SLG resolution allows an arbitrary Computation Rule and an arbitrary control strategy for selecting transformations to apply; (iv) SLG resolution avoids both positive and negative loops and always terminates for programs with the bounded-term-size property; (v) SLG resolution has a polynomial time data complexity for well founded negation. Restricted forms of SLG resolution are identified for definite, locally stratified, and modularly stratified programs, thereby shedding light on the role each transformation plays. To provide answers to a query under different three-valued stable models, SLG resolution can be enhanced by further processing of the derived set of answer clauses. SLG resolution makes many more clausal specifications into effective programs. With simple (user or computer generated) annotations, SLDNF resolution and SLG resolution can be fully integrated. Thus a system including SLG resolution can be fully integrated. Thus a system including SLG resolution is naturally upward compatible with Prolog. For all these reasons we believe that SLG resolution will provide the Computational basis for the next generation of logic programming systems.
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PODS - Query evaluation under the well-founded semantics
Proceedings of the twelfth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems - PODS '93, 1993Co-Authors: Weidong Chen, David S. WarrenAbstract:SLD resolution with negation as finite failure (or SLDNF) reflects the procedural interpretation of Horn-clause predicate logic as a programming language and forms the Computational basis for prolog systems. Despite its advantages in memory management, SLDNF is often not appropriate for query evaluation for three reasons; a) it may not terminate due to infinite positive recursion; b) it may not terminate due to infinite recursion through negation; and c) it may repeatedly evaluate the same clause body literal, leading to unacceptable performance. We address all three problems for goal-oriented query evaluation of arbitrary programs by presenting an extension of SLDNF, called SLG resolution , with the following distinctive features: (i) SLG resolution is a partial deduction procedure, consisting of several transformations. Each query is transformed step by step into a set of answer clauses; (ii) SLG resolution is sound and ideally complete for all non-floundering queries with respect to all three-valued stable models (including the well founded partial model); (iii) SLG resolution allows an arbitrary Computation Rule and an arbitrary control strategy for selecting transformations to apply; (iv) SLG resolution avoids both positive and negative loops and always terminates for programs with the bounded-term-size property; (v) SLG resolution has a polynomial time data complexity for well founded negation. Restricted forms of SLG resolution are identified for definite, locally stratified, and modularly stratified programs, thereby shedding light on the role each transformation plays. To provide answers to a query under different three-valued stable models, SLG resolution can be enhanced by further processing of the derived set of answer clauses. SLG resolution makes many more clausal specifications into effective programs. With simple (user or computer generated) annotations, SLDNF resolution and SLG resolution can be fully integrated. Thus a system including SLG resolution can be fully integrated. Thus a system including SLG resolution is naturally upward compatible with Prolog. For all these reasons we believe that SLG resolution will provide the Computational basis for the next generation of logic programming systems.
Thorsten Altenkirch - One of the best experts on this subject based on the ideXlab platform.
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Setoid type theory - a syntactic translation
2019Co-Authors: Thorsten Altenkirch, Simon Boulier, Ambrus Kaposi, Nicolas TabareauAbstract:We introduce setoid type theory, an intensional type theory with a proof-irrelevant universe of propositions and an equality type satisfying function extensionality, propositional extensionality and a definitional Computation Rule for transport. We justify the Rules of setoid type theory by a syntactic translation into a pure type theory with a universe of propositions. We conjecture that our syntax is complete with regards to this translation.
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notions of anonymous existence in martin lof type theory
Logical Methods in Computer Science, 2017Co-Authors: Nicolai Kraus, Thierry Coquand, Martin Hotzel Escardo, Thorsten AltenkirchAbstract:As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental Computation Rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.