1,825 research outputs found

    Components of arithmetic theory acceptance

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    This paper ties together three threads of discussion about the following question: in accepting a system of axioms S, what else are we thereby warranted in accepting, on the basis of accepting S? First, certain foundational positions in the philosophy of mathematics are said to be epistemically stable, in that there exists a coherent rationale for accepting a corresponding system of axioms of arithmetic, which does not entail or otherwise rationally oblige the foundationalist to accept statements beyond the logical consequences of those axioms. Second, epistemic stability is said to be incompatible with the implicit commitment thesis, according to which accepting a system of axioms implicitly commits the foundationalist to accept additional statements not immediately available in that theory. Third, epistemic stability stands in tension with the idea that in accepting a system of axioms S, one thereby also accepts soundness principles for S. We offer a framework for analysis of sets of implicit commitment which reconciles epistemic stability with the latter two notions, and argue that all three ideas are in fact compatible

    Complete and easy type Inference for first-class polymorphism

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    The Hindley-Milner (HM) typing discipline is remarkable in that it allows statically typing programs without requiring the programmer to annotate programs with types themselves. This is due to the HM system offering complete type inference, meaning that if a program is well typed, the inference algorithm is able to determine all the necessary typing information. Let bindings implicitly perform generalisation, allowing a let-bound variable to receive the most general possible type, which in turn may be instantiated appropriately at each of the variable’s use sites. As a result, the HM type system has since become the foundation for type inference in programming languages such as Haskell as well as the ML family of languages and has been extended in a multitude of ways. The original HM system only supports prenex polymorphism, where type variables are universally quantified only at the outermost level. This precludes many useful programs, such as passing a data structure to a function in the form of a fold function, which would need to be polymorphic in the type of the accumulator. However, this would require a nested quantifier in the type of the overall function. As a result, one direction of extending the HM system is to add support for first-class polymorphism, allowing arbitrarily nested quantifiers and instantiating type variables with polymorphic types. In such systems, restrictions are necessary to retain decidability of type inference. This work presents FreezeML, a novel approach for integrating first-class polymorphism into the HM system, focused on simplicity. It eschews sophisticated yet hard to grasp heuristics in the type systems or extending the language of types, while still requiring only modest amounts of annotations. In particular, FreezeML leverages the mechanisms for generalisation and instantiation that are already at the heart of ML. Generalisation and instantiation are performed by let bindings and variables, respectively, but extended to types beyond prenex polymorphism. The defining feature of FreezeML is the ability to freeze variables, which prevents the usual instantiation of their types, allowing them instead to keep their original, fully polymorphic types. We demonstrate that FreezeML is as expressive as System F by providing a translation from the latter to the former; the reverse direction is also shown. Further, we prove that FreezeML is indeed a conservative extension of ML: When considering only ML programs, FreezeML accepts exactly the same programs as ML itself. # We show that type inference for FreezeML can easily be integrated into HM-like type systems by presenting a sound and complete inference algorithm for FreezeML that extends Algorithm W, the original inference algorithm for the HM system. Since the inception of Algorithm W in the 1970s, type inference for the HM system and its descendants has been modernised by approaches that involve constraint solving, which proved to be more modular and extensible. In such systems, a term is translated to a logical constraint, whose solutions correspond to the types of the original term. A solver for such constraints may then be defined independently. To this end, we demonstrate such a constraint-based inference approach for FreezeML. We also discuss the effects of integrating the value restriction into FreezeML and provide detailed comparisons with other approaches towards first-class polymorphism in ML alongside a collection of examples found in the literature

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    The umbilical cord of finite model theory

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    Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but didn't turn out that way. There is one application domain -- relational database management -- that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.Comment: To be published in the Logic in Computer Science column of the February 2023 issue of the Bulletin of the European Association for Theoretical Computer Scienc

    Generic multiplicative endomorphism of a field

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    We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative map, which we call ACFH. Among others, we prove that this theory is NSOP1_1 and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.Comment: 34 page

    Investigations into Proof Structures

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    We introduce and elaborate a novel formalism for the manipulation and analysis of proofs as objects in a global manner. In this first approach the formalism is restricted to first-order problems characterized by condensed detachment. It is applied in an exemplary manner to a coherent and comprehensive formal reconstruction and analysis of historical proofs of a widely-studied problem due to {\L}ukasiewicz. The underlying approach opens the door towards new systematic ways of generating lemmas in the course of proof search to the effects of reducing the search effort and finding shorter proofs. Among the numerous reported experiments along this line, a proof of {\L}ukasiewicz's problem was automatically discovered that is much shorter than any proof found before by man or machine.Comment: This article is a continuation of arXiv:2104.1364

    Shelah's Main Gap and the generalized Borel-reducibility

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    We answer one of the main questions in generalized descriptive set theory, Friedman-Hyttinen-Kulikov conjecture on the Borel-reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel-reducibility notions of complexity. For any κ\kappa satisfying κ=λ+=2λ\kappa=\lambda^+=2^\lambda and 2c≤λ=λω12^{\mathfrak{c}}\leq\lambda=\lambda^{\omega_1}, we show that if TT is a classifiable theory and T′T' not, then the isomorphism of models of T′T' is strictly above the isomorphism of models of TT with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, TT, the isomorphism of models of TT is either Δ11\Delta^1_1 or analytically-complete

    Positivity Problems for Reversible Linear Recurrence Sequences

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    Σ1\Sigma_1 gaps as derived models and correctness of mice

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    Assume ZF + AD + V=L(R). Let [α,β][\alpha,\beta] be a Σ1\Sigma_1 gap with Jα(R)J_\alpha(R) admissible. We analyze Jβ(R)J_\beta(R) as a natural form of ``derived model'' of a premouse PP, where PP is found in a generic extension of VV. In particular, we will have P(R)∩Jβ(R)=P(R)∩D\mathcal{P}(R)\cap J_\beta(R)=\mathcal{P}(R)\cap D, and if Jβ(R)⊨J_\beta(R)\models``Θ\Theta exists'', then Jβ(R)J_\beta(R) and DD in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of (L(R))M(L(R))^M, for ω\omega-small mice MM. We also establish some preliminary work toward this conjecture in the present paper.Comment: 128 page

    On Internal Merge

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