IA Scholar Query: Easy Proofs of Löwenheim-Skolem Theorems by Means of Evaluation Games.
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Internet Archive Scholar query results feedeninfo@archive.orgMon, 17 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Forcing the Π^1_n-Uniformization Property
https://scholar.archive.org/work/zxsqjmso6zc4ddn74jhxfgcbra
We generically construct a model in which the Π^1_3-uniformization property is true, thus lowering the best known consistency strength from the existence of M_1^# to just 𝖹𝖥𝖢. The forcing construction can be adapted to work over canonical inner models with Woodin cardinals, which yields, for the first time, universes where the Π^1_2n-uniformization property holds for n >1, thus producing models which contradict the natural 𝖯𝖣-induced pattern. It can also be used to obtain models for the Π^1_1-uniformization property in the generalized Baire space.Stefan Hoffelnerwork_zxsqjmso6zc4ddn74jhxfgcbraMon, 17 Oct 2022 00:00:00 GMTOn Synthesizing Computable Skolem Functions for First Order Logic
https://scholar.archive.org/work/6b7ngxb5efhzpcfqcbmn7aoxdy
Skolem functions play a central role in the study of first order logic, both from theoretical and practical perspectives. While every Skolemized formula in first-order logic makes use of Skolem constants and/or functions, not all such Skolem constants and/or functions admit effectively computable interpretations. Indeed, the question of whether there exists an effectively computable interpretation of a Skolem function, and if so, how to automatically synthesize it, is fundamental to their use in several applications, such as planning, strategy synthesis, program synthesis etc. In this paper, we investigate the computability of Skolem functions and their automated synthesis in the full generality of first order logic. We first show a strong negative result, that even under mild assumptions on the vocabulary, it is impossible to obtain computable interpretations of Skolem functions. We then show a positive result, providing a precise characterization of first-order theories that admit effective interpretations of Skolem functions, and also present algorithms to automatically synthesize such interpretations. We discuss applications of our characterization as well as complexity bounds for Skolem functions (interpreted as Turing machines).Supratik Chakraborty, S. Akshay, Stefan Szeider, Robert Ganian, Alexandra Silvawork_6b7ngxb5efhzpcfqcbmn7aoxdyMon, 22 Aug 2022 00:00:00 GMTOn synthesizing Skolem functions for first order logic formulae
https://scholar.archive.org/work/qmqrsxhjy5e23itimqto4mj7lm
Skolem functions play a central role in the study of first order logic, both from theoretical and practical perspectives. While every Skolemized formula in first-order logic makes use of Skolem constants and/or functions, not all such Skolem constants and/or functions admit effectively computable interpretations. Indeed, the question of whether there exists an effectively computable interpretation of a Skolem function, and if so, how to automatically synthesize it, is fundamental to their use in several applications, such as planning, strategy synthesis, program synthesis etc. In this paper, we investigate the computability of Skolem functions and their automated synthesis in the full generality of first order logic. We first show a strong negative result, that even under mild assumptions on the vocabulary, it is impossible to obtain computable interpretations of Skolem functions. We then show a positive result, providing a precise characterization of first-order theories that admit effective interpretations of Skolem functions, and also present algorithms to automatically synthesize such interpretations. We discuss applications of our characterization as well as complexity bounds for Skolem functions (interpreted as Turing machines).S. Akshay, Supratik Chakrabortywork_qmqrsxhjy5e23itimqto4mj7lmThu, 04 Aug 2022 00:00:00 GMTThe Chromatic Nullstellensatz
https://scholar.archive.org/work/2t4ua45dl5hkrm7zpurjqwxmoa
We show that Lubin–Tate theories attached to algebraically closed fields are characterized among T(n)-local 𝔼_∞-rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every T(n)-local 𝔼_∞-ring R, the collection of 𝔼_∞-ring maps from R to such Lubin-Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero T(n)-local 𝔼_∞-ring R admits an 𝔼_∞-ring map to such a Lubin-Tate theory. As consequences, we construct 𝔼_∞ complex orientations of algebraically closed Lubin-Tate theories, compute the strict Picard spectra of such Lubin-Tate theories, and prove redshift for the algebraic K-theory of arbitrary 𝔼_∞-rings.Robert Burklund, Tomer M. Schlank, Allen Yuanwork_2t4ua45dl5hkrm7zpurjqwxmoaWed, 20 Jul 2022 00:00:00 GMTFirst-order logic with self-reference
https://scholar.archive.org/work/xp4blmlkf5dtpjwbb5gji3x4tu
We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two closely related but different logics. We provide a natural deduction system that is complete for validities for both of these logics, and we also investigate a range of related basic decision problems. For example, the validity problems of the two-variable fragments of the logics are shown coNexpTime-complete, which is in stark contrast with the high undecidability of two-variable logic extended with least fixed points. We also argue for the naturalness and benefits of the investigated approach to recursion and self-reference by, for example, relating the new logics to Lindstrom's Second Theorem.Reijo Jaakkola, Antti Kuusistowork_xp4blmlkf5dtpjwbb5gji3x4tuFri, 15 Jul 2022 00:00:00 GMTStrongly First Order, Domain Independent Dependencies: the Union-Closed Case
https://scholar.archive.org/work/o7povnhctndapkj5ilukw73ft4
Team Semantics generalizes Tarski's Semantics by defining satisfaction with respect to sets of assignments rather than with respect to single assignments. Because of this, it is possible to use Team Semantics to extend First Order Logic via new kinds of connectives or atoms - most importantly, via dependency atoms that express dependencies between different assignments. Some of these extensions are more expressive than First Order Logic proper, while others are reducible to it. In this work, I provide necessary and sufficient conditions for a dependency atom that is domain independent (in the sense that its truth or falsity in a relation does not depend on the existence in the model of elements that do not occur in the relation) and union closed (in the sense that whenever it is satisfied by all members of a family of relations it is also satisfied by their union) to be strongly first order, in the sense that the logic obtained by adding them to First Order Logic is no more expressive than First Order Logic itself.Pietro Gallianiwork_o7povnhctndapkj5ilukw73ft4Thu, 30 Jun 2022 00:00:00 GMTProceedings of the SNS Logic Colloquium March 1990
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Republication of the proceedings of the Informal Logic Colloquium held in March 1990 at the Seminar für natürlichsprachliche Systeme (SNS) of the University of Tübingen.Peter Schroeder-Heister, Universitaet Tuebingenwork_myebfuw7yfaw7a6vspe24lhx4uMon, 27 Jun 2022 00:00:00 GMTRegister Automata with Extrema Constraints, and an Application to Two-Variable Logic
https://scholar.archive.org/work/iikdqag3ijafjl4h3nqlmgstse
We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register values in subtrees. We show that the emptiness problem for these automata is decidable. As an application, we prove decidability of the countable satisfiability problem for two-variable logic in the presence of a tree order, a linear order, and arbitrary atoms that are MSO definable from the tree order. As a consequence, the satisfiability problem for two-variable logic with arbitrary predicates, two of them interpreted by linear orders, is decidable.Szymon Toruńczyk, Thomas Zeumework_iikdqag3ijafjl4h3nqlmgstseMon, 21 Mar 2022 00:00:00 GMTZero-One Laws and Almost Sure Valuations of First-Order Logic in Semiring Semantics
https://scholar.archive.org/work/v3ezhrklejh5vliz4fdwsmrnsa
Semiring semantics evaluates logical statements by values in some commutative semiring K. Random semiring interpretations, induced by a probability distribution on K, generalise random structures, and we investigate here the question of how classical results on first-order logic on random structures, most importantly the 0-1 laws of Glebskii et al. and Fagin, generalise to semiring semantics. For positive semirings, the classical 0-1 law implies that every first-order sentence is, asymptotically, either almost surely evaluated to 0 by random semiring interpretations, or almost surely takes only values different from 0. However, by means of a more sophisticated analysis, based on appropriate extension properties and on algebraic representations of first-order formulae, we can prove much stronger results. For many semirings K, the first-order sentences can be partitioned into classes F(j) for all semiring values j in K, such that every sentence in F(j) evaluates almost surely to j under random semiring interpretations. Further, for finite or infinite lattice semirings, this partition actually collapses to just three classes F(0), F(1), and F(e), of sentences that, respectively, almost surely evaluate to 0, 1, and to the smallest non-zero value e. The problem of computing the almost sure valuation of a first-order sentence on finite lattice semirings is PSPACE-complete. An important semiring where the analysis is somewhat different is the semiring of natural numbers. Here, both addition and multiplication are increasing with respect to the natural semiring order and the classes F(j), for natural numbers j, no longer cover all FO-sentences, but have to be extended by the class of sentences that almost surely evaluate to unboundedly large values.Erich Grädel, Hayyan Helal, Matthias Naaf, Richard Wilkework_v3ezhrklejh5vliz4fdwsmrnsaMon, 07 Mar 2022 00:00:00 GMTThis Week's Finds in Mathematical Physics (1-50)
https://scholar.archive.org/work/7dqichdfjraltdavtghiqngis4
These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of n-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).John C. Baezwork_7dqichdfjraltdavtghiqngis4Mon, 28 Feb 2022 00:00:00 GMTSome Fundamental Theorems in Mathematics
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An expository hitchhikers guide to some theorems in mathematics.Oliver Knillwork_6lqit72adje3zlo54s5zpgviemFri, 04 Feb 2022 00:00:00 GMTModel Theoretic Characterizations of Large Cardinals Revisited
https://scholar.archive.org/work/tzt4kuj47jhetk7j32z6vi2kh4
In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.Will Boney, Stamatis Dimopoulos, Victoria Gitman, Menachem Magidorwork_tzt4kuj47jhetk7j32z6vi2kh4Tue, 01 Feb 2022 00:00:00 GMTZero-One Laws and Almost Sure Valuations of First-Order Logic in Semiring Semantics
https://scholar.archive.org/work/74fbfoxjqvblveskup5vptj3m4
Semiring semantics evaluates logical statements by values in some commutative semiring (K, +, •, 0, 1). Random semiring interpretations, induced by a probability distribution on K, generalise random structures, and we investigate here the question of how classical results on first-order logic on random structures, most importantly the 0-1 laws of Glebskii et al. and Fagin, generalise to semiring semantics. For positive semirings, the classical 0-1 law implies that every first-order sentence is, asymptotically, either almost surely evaluated to 0 by random semiring interpretations, or almost surely takes only values different from 0. However, by means of a more sophisticated analysis, based on appropriate extension properties and on algebraic representations of first-order formulae, we can prove much stronger results. For many semirings K the first-order sentences in FO(τ ) can be partitioned into classes (Φj)j∈K such that for each j ∈ K, every sentence in Φj evaluates almost surely to j under random semiring interpretations. Further, for finite or infinite lattice semirings, this partition actually collapses to just three classes Φ0, Φ1, and Φε, of sentences that, respectively, almost surely evaluate to 0, 1, and to the smallest value ε ̸ = 0. For all other values j ∈ K we have that Φj = ∅. The problem of computing the almost sure valuation of a first-order sentence on finite lattice semirings is Pspace-complete. An important semiring where the analysis is somewhat different is the natural semiring (N, +, •, 0, 1). Here, both addition and multiplication are increasing with respect to the natural semiring order and the classes (Φj) j∈N no longer cover all FO(τ )-sentences, but have to be extended by Φ∞, the class of sentences that almost surely evaluate to unboundedly large values.Erich Grädel, Hayyan Helal, Matthias Ferdinand Naaf, Richard Wilkework_74fbfoxjqvblveskup5vptj3m4