The Experts below are selected from a list of 573 Experts worldwide ranked by ideXlab platform

Andre Platzer - One of the best experts on this subject based on the ideXlab platform.

  • uniform substitution for differential game logic
    International Joint Conference on Automated Reasoning, 2018
    Co-Authors: Andre Platzer
    Abstract:

    This paper presents a uniform substitution calculus for differential game logic ( Open image in new window ). Church’s uniform substitutions substitute a term or formula for a function or predicate symbol everywhere. After generalizing them to differential game logic and allowing for the substitution of hybrid games for game symbols, uniform substitutions make it possible to only use Axioms instead of Axiom Schemata, thereby substantially simplifying implementations. Instead of subtle Schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many Axiom Schema instances to sound ones, the resulting Axiomatization adopts only a finite number of ordinary Open image in new window formulas as Axioms, which uniform substitutions instantiate soundly. This paper proves soundness and completeness of uniform substitutions for the monotone modal logic Open image in new window . The resulting Axiomatization admits a straightforward modular implementation of Open image in new window in theorem provers.

  • a complete uniform substitution calculus for differential dynamic logic
    Journal of Automated Reasoning, 2017
    Co-Authors: Andre Platzer
    Abstract:

    This article introduces a relatively complete proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to use Axioms instead of Axiom Schemata, thereby substantially simplifying implementations. Instead of subtle Schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many Axiom Schema instances to sound ones, the resulting calculus adopts only a finite number of ordinary dLformulas as Axioms, which uniform substitutions instantiate soundly. The static semantics of differential dynamic logic and the soundness-critical restrictions it imposes on proof steps is captured exclusively in uniform substitutions and variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this article introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivatives as first-class Axioms to reason about differential equations Axiomatically. The resulting Axiomatization of differential dynamic logic is proved to be sound and relatively complete.

Platzer André - One of the best experts on this subject based on the ideXlab platform.

  • Uniform Substitution for Differential Game Logic
    'Springer Science and Business Media LLC', 2018
    Co-Authors: Platzer André
    Abstract:

    This paper presents a uniform substitution calculus for differential game logic (dGL). Church's uniform substitutions substitute a term or formula for a function or predicate symbol everywhere. After generalizing them to differential game logic and allowing for the substitution of hybrid games for game symbols, uniform substitutions make it possible to only use Axioms instead of Axiom Schemata, thereby substantially simplifying implementations. Instead of subtle Schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many Axiom Schema instances to sound ones, the resulting Axiomatization adopts only a finite number of ordinary dGL formulas as Axioms, which uniform substitutions instantiate soundly. This paper proves soundness and completeness of uniform substitutions for the monotone modal logic dGL. The resulting Axiomatization admits a straightforward modular implementation of dGL in theorem provers

  • A Complete Uniform Substitution Calculus for Differential Dynamic Logic
    'Springer Science and Business Media LLC', 2016
    Co-Authors: Platzer André
    Abstract:

    This article introduces a relatively complete proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to use Axioms instead of Axiom Schemata, thereby substantially simplifying implementations. Instead of subtle Schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many Axiom Schema instances to sound ones, the resulting calculus adopts only a finite number of ordinary dL formulas as Axioms, which uniform substitutions instantiate soundly. The static semantics of differential dynamic logic and the soundness-critical restrictions it imposes on proof steps is captured exclusively in uniform substitutions and variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this article introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivatives as first-class Axioms to reason about differential equations Axiomatically. The resulting Axiomatization of differential dynamic logic is proved to be sound and relatively complete.Comment: Long article extending the conference version that appeared at CADE 2015, arXiv:1503.0198

Wensheng Yu - One of the best experts on this subject based on the ideXlab platform.

  • a formal system of Axiomatic set theory in coq
    IEEE Access, 2020
    Co-Authors: Wensheng Yu
    Abstract:

    Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal system of Axiomatic set theory based on the Coq proof assistant. The Axiomatic system used in the formal system refers to Morse-Kelley set theory which is a relatively complete and concise Axiomatic set theory. In this formal system, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano’s postulates are proved as theorems. According to the Axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schroeder-Bernstein theorem. The whole formalization of the system includes eight Axioms, one Axiom Schema, 62 definitions, and 148 corollaries or theorems. The “Axiomatic set theory” formal system is free from the more apparent paradoxes, and a complete Axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

Wang Quanlong - One of the best experts on this subject based on the ideXlab platform.

  • ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics
    HAL CCSD, 2017
    Co-Authors: Jeandel Emmanuel, Perdrix Simon, Vilmart Renaud, Wang Quanlong
    Abstract:

    International audienceThe ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e.~the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new Axioms for diagrammatic quantum reasoning.We exhibit another new simple Axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new Axiomatisation of an extended ZX-Calculus, including an Axiom Schema for the cyclotomic supplementarity

  • ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics
    2017
    Co-Authors: Jeandel Emmanuel, Perdrix Simon, Vilmart Renaud, Wang Quanlong
    Abstract:

    The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new Axioms for diagrammatic quantum reasoning.We exhibit another new simple Axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new Axiomatisation of an extended ZX-Calculus, including an Axiom Schema for the cyclotomic supplementarity.Comment: Mathematical Foundations of Computer Science, Aug 2017, Aalborg, Denmar

O'donovan Richard - One of the best experts on this subject based on the ideXlab platform.

  • Une théorie des objets et de leur visibilité. Un lien entre l'analyse relative et la théorie alternative des ensembles
    2011
    Co-Authors: O'donovan Richard
    Abstract:

    La théorie présentée ici est issue d'années d'enseignement de l'analyse au niveau pré-universitaire en utilisant d'abord le concept d'infiniment petit, tel que défini dans l'analyse nonstandard de Robinson, puis ensuite d'ultrapetit, tel que défini dans notre travail en collaboration avec Hrbacek et Lessmann et présenté en annexe. A la suite de ces recherches, s'est posée la question : Si l'on a à disposition des quantités finies mais ultragrandes, est-il possible de se passer de quantités dites infinies ? La théorie alternative des ensembles de Vopěnka est une théorie avec des ensembles finis et des classes qui, elles, peuvent être infinies. La théorie des objets est le résultat d'un mélange de certains Axiomes de Vopěnka avec des Axiomes déterminant des niveaux de visibilité tels que dans l'analyse relative. On s'est donné comme premier principe : x⊆y⇒x⊑y qui spécifie que si l'objet x est inclus dans l'objet y, alors x "paraît" au niveau de y. Cette affirmation serait fausse avec des quantités infinies ; elle est néanmoins une caractérisation des ensembles finis : cela est bien connu en analyse nonstandard. L'introduction de ce principe comme point de départ est donc une affirmation forte que les objets devront être finis au sens habituel de ce terme. L'autre Axiome fondateur ici est le schéma d'Axiomes d'induction de Gordon et Andreev : Si Φ est une formule, et si Φ(∅) est vrai et que Φ(x) et Φ(y) impliquent Φ(x∪{y}), alors Φ(x) est vrai pour tout x. Un accent particulier est mis sur le concept de formules dites contextuelles. Ce concept est une de nos contributions à l'analyse relative de Hrbacek et détermine les formules bien formées. On montre que le système qui en résulte est relativement cohérent avec la théorie FRIST de Hrbacek et la théorie RIST de Péraire qui sont elles-mêmes des extensions conservatives de ZFC. La théorie des objets est une extension de la théorie des ensembles de Zermelo et Fraenkel sans Axiome du choix et négation de l'Axiome de l'infini. Les nombres entiers et rationnels sont définis et ces derniers sont munis de relations d'ultraproximité. Une ébauche d'une construction de "grains numériques" est présentée : ces nombres pourraient avoir des propriétés suffisamment semblables aux nombres réels pour permettre de faire de l'analyse.The theory presented here stemmed from years of teaching analysis at pre-university level first using the concept of infinitesimal as defined in nonstandard analysis by Robinson, then the concept of ultrasmall as defined in our joint work with Hrbacek and Lessmann presented in the appendix. This research led to the question : If one has finite yet ultralarge quantities, is it possible to avoid infinite quantities ? The alternative set theory of Vopěnka is a theory of finite sets including classes that can be infinite. The theory of objects is a merger of certain Axioms of Vopěnka with Axioms that determine levels of visibility as in relative analysis. We took as first principle : x⊆y⇒x⊑y, which specifies that if object x is included in object y, then x "appears" at the level of y. This statement would be false with infinite quantities and is in fact a characterisation of finite sets : this is a well-known theorem of nonstandard analysis. The introduction of this principle as starting point is making a strong point that all objects will be finite - in the usual sense of the word. The other founding Axiom is Gordon and Andreev's Axiom Schema : If Φ is a formula, and if Φ(∅) is true and that Φ(x) and Φ(y) imply Φ(x∪{y}), then Φ(x) is true for all x. An emphasis is made on the concept of contextual formulae. This concept is one of our contributions to relative analysis of Hrbacek and determines an equivalence to well-formed formulae. We show that the resulting system is relatively consistent with Hrbacek's FRIST and Péraire's RIST which are conservative extensions of ZFC. The theory of objects extends set theory of Zermelo and Fraenkel without choice and with negation of the infinity Axiom. Integers and rationals are defined and endowed with an ultraproximity relation. A draft of a construction of "numeric grains" is presented : these numbers could prove to have properties sufficiently similar to real numbers to allow to perform analysis

  • A theory of objects and visibility. A link between relative analysis and alternative set theory
    2011
    Co-Authors: O'donovan Richard, Peraire Yves
    Abstract:

    La théorie présentée ici est issue d'années d'enseignement de l'analyse au niveau pré-universitaire en utilisant d'abord le concept d'infiniment petit, tel que défini dans l'analyse nonstandard de Robinson, puis ensuite d'ultrapetit, tel que défini dans notre travail en collaboration avec Hrbacek et Lessmann et présenté en annexe. A la suite de ces recherches, s'est posée la question : Si l'on a à disposition des quantités finies mais ultragrandes, est-il possible de se passer de quantités dites infinies ? La théorie alternative des ensembles de Vopěnka est une théorie avec des ensembles finis et des classes qui, elles, peuvent être infinies. La théorie des objets est le résultat d'un mélange de certains Axiomes de Vopěnka avec des Axiomes déterminant des niveaux de visibilité tels que dans l'analyse relative. On s'est donné comme premier principe : $x\subseteq y\Rightarrow x\sqsubseteq y$ qui spécifie que si l'objet $x$ est inclus dans l'objet $y$, alors $x$ "paraît" au niveau de $y$. Cette affirmation serait fausse avec des quantités infinies ; elle est néanmoins une caractérisation des ensembles finis : cela est bien connu en analyse nonstandard. L'introduction de ce principe comme point de départ est donc une affirmation forte que les objets devront être finis au sens habituel de ce terme. L'autre Axiome fondateur ici est le schéma d'Axiomes d'induction de Gordon et Andreev : Si $\Phi$ est une formule, et si $\Phi(\emptyset)$ est vrai et que $\Phi(x)$ et $\Phi(y)$ impliquent $\Phi(x\cup\{y\})$, alors $\Phi(x)$ est vrai pour tout $x$. Un accent particulier est mis sur le concept de formules dites contextuelles. Ce concept est une de nos contributions à l'analyse relative de Hrbacek et détermine les formules bien formées. On montre que le système qui en résulte est relativement cohérent avec la théorie FRIST de Hrbacek et la théorie RIST de Péraire qui sont elles-mêmes des extensions conservatives de ZFC. La théorie des objets est une extension de la théorie des ensembles de Zermelo et Fraenkel sans Axiome du choix et négation de l'Axiome de l'infini. Les nombres entiers et rationnels sont définis et ces derniers sont munis de relations d'ultraproximité. Une ébauche d'une construction de "grains numériques" est présentée : ces nombres pourraient avoir des propriétés suffisamment semblables aux nombres réels pour permettre de faire de l'analyse.The theory presented here stemmed from years of teaching analysis at pre-university level first using the concept of infinitesimal as defined in nonstandard analysis by Robinson, then the concept of ultrasmall as defined in our joint work with Hrbacek and Lessmann presented in the appendix. This research led to the question : If one has finite yet ultralarge quantities, is it possible to avoid infinite quantities ? The alternative set theory of Vopěnka is a theory of finite sets including classes that can be infinite. The theory of objects is a merger of certain Axioms of Vopěnka with Axioms that determine levels of visibility as in relative analysis. We took as first principle : $x\subseteq y\Rightarrow x\sqsubseteq y$, which specifies that if object $x$ is included in object $y$, then $x$ "appears" at the level of $y$. This statement would be false with infinite quantities and is in fact a characterisation of finite sets : this is a well-known theorem of nonstandard analysis. The introduction of this principle as starting point is making a strong point that all objects will be finite - in the usual sense of the word. The other founding Axiom is Gordon and Andreev's Axiom Schema : If $\Phi$ is a formula, and if $\Phi(\emptyset)$ is true and that $\Phi(x)$ and $\Phi(y)$ imply $\Phi(x\cup\{y\})$, then $\Phi(x)$ is true for all $x$. An emphasis is made on the concept of contextual formulae. This concept is one of our contributions to relative analysis of Hrbacek and determines an equivalence to well-formed formulae. We show that the resulting system is relatively consistent with Hrbacek's FRIST and Péraire's RIST which are conservative extensions of ZFC. The theory of objects extends set theory of Zermelo and Fraenkel without choice and with negation of the infinity Axiom. Integers and rationals are defined and endowed with an ultraproximity relation. A draft of a construction of "numeric grains" is presented : these numbers could prove to have properties sufficiently similar to real numbers to allow to perform analysis.CLERMONT FD-Bib.électronique (631139902) / SudocSudocFranceF

  • Une théorie des objets et de leur visibilité. Un lien entre l'analyse relative et la théorie alternative des ensembles
    HAL CCSD, 2011
    Co-Authors: O'donovan Richard
    Abstract:

    The theory presented here stemmed from years of teaching analysis at pre-university level first using the concept of infinitesimal as defined in nonstandard analysis by Robinson, then the concept of ultrasmall as defined in our joint work with Hrbacek and Lessmann presented in the appendix. This research led to the question : If one has finite yet ultralarge quantities, is it possible to avoid infinite quantities ? The alternative set theory of Vopěnka is a theory of finite sets including classes that can be infinite. The theory of objects is a merger of certain Axioms of Vopěnka with Axioms that determine levels of visibility as in relative analysis. We took as first principle : $x\subseteq y\Rightarrow x\sqsubseteq y$, which specifies that if object $x$ is included in object $y$, then $x$ "appears" at the level of $y$. This statement would be false with infinite quantities and is in fact a characterisation of finite sets : this is a well-known theorem of nonstandard analysis. The introduction of this principle as starting point is making a strong point that all objects will be finite - in the usual sense of the word. The other founding Axiom is Gordon and Andreev's Axiom Schema : If $\Phi$ is a formula, and if $\Phi(\emptyset)$ is true and that $\Phi(x)$ and $\Phi(y)$ imply $\Phi(x\cup\{y\})$, then $\Phi(x)$ is true for all $x$. An emphasis is made on the concept of contextual formulae. This concept is one of our contributions to relative analysis of Hrbacek and determines an equivalence to well-formed formulae. We show that the resulting system is relatively consistent with Hrbacek's FRIST and Péraire's RIST which are conservative extensions of ZFC. The theory of objects extends set theory of Zermelo and Fraenkel without choice and with negation of the infinity Axiom. Integers and rationals are defined and endowed with an ultraproximity relation. A draft of a construction of "numeric grains" is presented : these numbers could prove to have properties sufficiently similar to real numbers to allow to perform analysis.La théorie présentée ici est issue d'années d'enseignement de l'analyse au niveau pré-universitaire en utilisant d'abord le concept d'infiniment petit, tel que défini dans l'analyse nonstandard de Robinson, puis ensuite d'ultrapetit, tel que défini dans notre travail en collaboration avec Hrbacek et Lessmann et présenté en annexe. A la suite de ces recherches, s'est posée la question : Si l'on a à disposition des quantités finies mais ultragrandes, est-il possible de se passer de quantités dites infinies ? La théorie alternative des ensembles de Vopěnka est une théorie avec des ensembles finis et des classes qui, elles, peuvent être infinies. La théorie des objets est le résultat d'un mélange de certains Axiomes de Vopěnka avec des Axiomes déterminant des niveaux de visibilité tels que dans l'analyse relative. On s'est donné comme premier principe : $x\subseteq y\Rightarrow x\sqsubseteq y$ qui spécifie que si l'objet $x$ est inclus dans l'objet $y$, alors $x$ "paraît" au niveau de $y$. Cette affirmation serait fausse avec des quantités infinies ; elle est néanmoins une caractérisation des ensembles finis : cela est bien connu en analyse nonstandard. L'introduction de ce principe comme point de départ est donc une affirmation forte que les objets devront être finis au sens habituel de ce terme. L'autre Axiome fondateur ici est le schéma d'Axiomes d'induction de Gordon et Andreev : Si $\Phi$ est une formule, et si $\Phi(\emptyset)$ est vrai et que $\Phi(x)$ et $\Phi(y)$ impliquent $\Phi(x\cup\{y\})$, alors $\Phi(x)$ est vrai pour tout $x$. Un accent particulier est mis sur le concept de formules dites contextuelles. Ce concept est une de nos contributions à l'analyse relative de Hrbacek et détermine les formules bien formées. On montre que le système qui en résulte est relativement cohérent avec la théorie FRIST de Hrbacek et la théorie RIST de Péraire qui sont elles-mêmes des extensions conservatives de ZFC. La théorie des objets est une extension de la théorie des ensembles de Zermelo et Fraenkel sans Axiome du choix et négation de l'Axiome de l'infini. Les nombres entiers et rationnels sont définis et ces derniers sont munis de relations d'ultraproximité. Une ébauche d'une construction de "grains numériques" est présentée : ces nombres pourraient avoir des propriétés suffisamment semblables aux nombres réels pour permettre de faire de l'analyse