70 research outputs found

    On Notions of Provability

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    In this thesis, we study notions of provability, i.e. formulas B(x,y) such that a formula ϕ is provable in T if, and only if, there is m ∈ N such that T ⊢ B(⌜ϕ⌝,m) (m plays the role of a parameter); the usual notion of provability, k-step provability (also known as k-provability), s-symbols provability are examples of notions of provability. We develop general results concerning notions of provability, but we also study in detail concrete notions. We present partial results concerning the decidability of kprovability for Peano Arithmetic (PA), and we study important problems concerning k-provability, such as Kreisel’s Conjecture and Montagna’s Problem: (∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Kreisel’s Conjecture] and Does PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ imply PA ⊢k steps ϕ? [Montagna’s Problem] Incompleteness, Undefinability of Truth, and Recursion are different entities that share important features; we study this in detail and we trace these entities to common results. We present numeral forms of completeness and consistency, numeral completeness and numeral consistency, respectively; numeral completeness guarantees that, whenever a Σb 1(S12 )-formula ϕ(⃗x ) is such that ⃗Q ⃗x .ϕ(⃗x ) is true (where ⃗Q is any array of quantifiers), then this very fact can be proved inside S12 , more precisely S12 ⊢ ⃗Q ⃗x .Prτ (⌜ϕ( •⃗ x )⌝). We examine these two results from a mathematical point of view by presenting the minimal conditions to state them and by finding consequences of them, and from a philosophical point of view by relating them to Hilbert’s Program. The derivability condition “provability implies provable provability” is one of the main derivability conditions used to derive the Second Incompleteness Theorem and is known to be very sensitive to the underlying theory one has at hand. We create a weak theory G2 to study this condition; this is a theory for the complexity class FLINSPACE. We also relate properties of G2 to equality between computational classes.O tema desta tese são noções de demonstração; estas últimas são fórmulas B(x,y) tais que uma fórmula ϕ é demonstrável em T se, e só se, existe m ∈ N tal que T ⊢ B(⌜ϕ⌝,m) (m desempenha o papel de um parâmetro). A noção usual de demonstração, demonstração em k-linhas (demonstração-k), demonstração em s-símbolos são exemplos de noções de demonstração. Desenvolvemos resultados gerais sobre noções de demonstração, mas também estudamos exemplos concretos. Damos a conhecer resultados parciais sobre a decidibilidade da demonstração-k para a Aritmética de Peano (PA), e estudamos dois problemas conhecidos desta área, a Conjectura de Kreisel e o Problema de Montagna: (∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Conjectura de Kreisel] e PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ implica PA ⊢k steps ϕ? [Problema de Montagna] A Incompletude, a Incapacidade de Definir Verdade, e Recursão são entidades que têm em comum características relevantes; nós estudamos estas entidades em detalhe e apresentamos resultados que são simultaneamente responsáveis pelas mesmas. Além disso, apresentamos formas numerais de completude e consistência, a completude numeral e a consistência numeral, respectivamente; a completude numeral assegura que, quando uma fórmula-Σb 1(S12) ϕ(⃗x ) é tal que ⃗Q ⃗x .ϕ(⃗x ) é verdadeira, então este facto pode ser verificado dentro de S12, mais precisamente S12 ⊢ ⃗Q ⃗x .Prτ (⌜ϕ( •⃗ x )⌝). Este dois resultados são analisados de um ponto de vista matemático onde apresentamos as condições mínimas para os demonstrar e apresentamos consequências dos mesmos, e de um ponto de vista filosófico, onde relacionamos os mesmos com o Programa de Hilbert. A condição de derivabilidade “demonstração implica demonstrabilidade da demonstração” é uma das condições usadas para derivar o Segundo Teorema da Incompletude e sabemos ser muito sensível à teoria de base escolhida. Nós criámos uma teoria fraca G2 para estudar esta condição; esta é uma teoria para a classe de complexidade FLINSPACE. Também relacionámos propriedades de G2 com igualdades entre classes de complexidade computacional

    On Notions of Provability

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    In this thesis, we study notions of provability, i.e. formulas B(x,y) such that a formula ' is provable in T if, and only if, there is m 2 N such that T ` B(p'q,m) (m plays the role of a parameter); the usual notion of provability, k-step provability (also known as k-provability), s-symbols provability are examples of notions of provability. We develop general results concerning notions of provability, but we also study in detail concrete notions. We present partial results concerning the decidability of k- provability for Peano Arithmetic (PA), and we study important problems concerning k-provability, such as Kreisel’s Conjecture and Montagna’s Problem: (8n 2 N.T `k steps '(n)) =) T ` 8x.'(x), [Kreisel’s Conjecture] Does PA `k steps PrPA(p'q) ! ' imply PA `k steps '? [Montagna’s Problem] Incompleteness, Undefinability of Truth, and Recursion are di↵erent entities that share important features; we study this in detail and we trace these entities to common results. We present numeral forms of completeness and consistency, numeral completeness and numeral consistency, respectively; numeral completeness guarantees that, whenever a⌃b1(S12)-formula'(x~)issuchthatQ~x~.'(x~)istrue(whereQ~ isanyarrayofquantifiers), then this very fact can be proved inside S12, more precisely S12 ` Q~ x~.Pr⌧(p'(x~• )q). We examine these two results from a mathematical point of view by presenting the minimal conditions to state them and by finding consequences of them, and from a philosophical point of view by relating them to Hilbert’s Program. The derivability condition “provability implies provable provability” is one of the main derivability conditions used to derive the Second Incompleteness Theorem and is known to be very sensitive to the underlying theory one has at hand. We create a weak theory G2 to study this condition; this is a theory for the complexity class FLINSPACE. We also relate properties of G2 to equality between computational classes

    A Computational Cantor-Bernstein and Myhill's Isomorphism Theorem in Constructive Type Theory: Proof Pearl

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    International audienceThe Cantor-Bernstein theorem (CB) from set theory, stating that two sets which can be injectively embedded into each other are in bijection, is inherently classical in its full generality, i.e. implies the law of excluded middle, a result due to Pradic and Brown. Recently, Escardó has provided a proof of CB in univalent type theory, assuming the law of excluded middle. It is a natural question to ask which restrictions of CB can be proved without axiomatic assumptions. We give a partial answer to this question contributing an assumptionfree proof of CB restricted to enumerable discrete types, i.e. types which can be computationally treated. In fact, we construct several bijections from injections: The first is by translating a proof of the Myhill isomorphism theorem from computability theory-stating that 1-equivalent predicates are recursively isomorphic-to constructive type theory, where the bijection is constructed in stages and an algorithm with an intricate termination argument is used to extend the bijection in every step. The second is also constructed in stages, but with a simpler extension algorithm sufficient for CB. The third is constructed directly in such a way that it only relies on the given enumerations of the types, not on the given injections. We aim at keeping the explanations simple, accessible, and concise in the style of a "proof pearl". All proofs are machinechecked in Coq but should transport to other foundationsthey do not rely on impredicativity, on choice principles, or on large eliminations

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens

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    26 pages, extended version of the IJCAR 2020 paper. arXiv admin note: substantial text overlap with arXiv:2004.07390International audienceWe study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. To showcase an application of Trakthenbrot's theorem, we continue our reduction chain with a many-one reduction from FSAT to separation logic. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs

    Scalable Community Detection

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    Sets, Logic, Computation: An Open Introduction to Metalogic

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    An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic

    Trakhtenbrot's Theorem in Coq, A Constructive Approach to Finite Model Theory

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    We study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs

    Trakhtenbrot’s Theorem in Coq: A Constructive Approach to Finite Model Theory

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    International audienceWe study finite first-order satisfiability (FSAT) in the constructive setting of dependent type theory. Employing synthetic accounts of enumerability and decidability, we give a full classification of FSAT depending on the first-order signature of non-logical symbols. On the one hand, our development focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as the signature contains an at least binary relation symbol. Our proof proceeds by a many-one reduction chain starting from the Post correspondence problem. On the other hand, we establish the decidability of FSAT for monadic first-order logic, i.e. where the signature only contains at most unary function and relation symbols, as well as the enumerability of FSAT for arbitrary enumerable signatures. All our results are mechanised in the framework of a growing Coq library of synthetic undecidability proofs
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