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Hierarchic Reasoning in Local Theory Extensions [chapter]

Viorica Sofronie-Stokkermans
2005 Lecture Notes in Computer Science  
We identify situations in which it is possible, for an extension T1 of a theory T0, to express the decidability and complexity of the universal theory of T1 in terms of the decidability resp. complexity  ...  We show that for special types of extensions of a base theory, which we call local, efficient hierarchic reasoning is possible.  ...  Many of the results presented here are the direct or indirect result of discussions and joint work with Harald Ganzinger during the last years.  ... 
doi:10.1007/11532231_16 fatcat:26a6233krzbillz3odqggiuw5i

On Hierarchical Reasoning in Combinations of Theories [chapter]

Carsten Ihlemann, Viorica Sofronie-Stokkermans
2010 Lecture Notes in Computer Science  
Definition 2.4.1 (Weak partial model of a theory) Let Π 0 = (Σ 0 , Pred) be a signature, Σ 1 a set of new function symbols of arity greater than 0, T 0 a Π 0 -theory, K a set of universally closed clauses  ...  Let G be a set of ground extension clauses (with additional constants), Θ a set of ground terms such that est(K, G) ⊆ Θ and A a weak partial model of T 0 , K[Θ], G with total Σ 0 -functions in which all  ...  A definitional extension is one where extension functions f only appear in the form ϕ i (x) → f (x) = t i (x) with t i being a base theory term and the ϕ i are mutually exclusive base theory clauses.  ... 
doi:10.1007/978-3-642-14203-1_4 fatcat:cqx3nkfh5fbgbbrf5uv7hc7q4q

Automated Reasoning in Some Local Extensions of Ordered Structures

Viorica Sofronie-Stokkermans, Carsten Ihlemann
2007 Proceedings - International Symposium on Multiple-Value Logic  
The map l : DM(P ∂ ) → DM(P ) ∂ , sending a set to the set of its lower bounds, is an isomorphism; its inverse is the map u . Theories and models.  ...  A Π-structure is a tuple where for every s ∈ S, M s = ∅, for all f ∈ Σ with arity a(f )=s 1 ×. . .×s n →s, f M :  ...  See www.avacs.org for more information.  ... 
doi:10.1109/ismvl.2007.10 dblp:conf/ismvl/Sofronie-StokkermansI07 fatcat:2k2s47l3i5fudniiddbfqtxrfe

Page 4699 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
By naive set theory the author means a classical first-order set the- ory with abstraction terms {x : A} for any assertion A, formalized as a Gentzen sequent calculus with the naive rules of deduction  ...  The omission of the cut rule is essential for the proof of completeness of the theory with respect to greedy models, since validity with respect to greedy models is not preserved by it.  ... 

Unifying Cubical Models of Univalent Type Theory

Evan Cavallo, Anders Mörtberg, Andrew W Swan, Michael Wagner
2020 Annual Conference for Computer Science Logic  
Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure.  ...  We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections.  ...  Licata, Andrew Pitts and Jon Sterling for helpful comments and remarks on earlier versions of this work.  ... 
doi:10.4230/lipics.csl.2020.14 dblp:conf/csl/CavalloMS20 fatcat:rgzcvxrphbf67djmhglropid4q

Constructive Set Theory from a Weak Tarski Universe [article]

Cesare Gallozzi
2014 arXiv   pre-print
Moreover, we show using the axiom of function extensionality that the type theoretic interpretation of constructive set theory generalises to homotopy type theory with a weak Tarski universe.  ...  The original part of this thesis consists in the final chapter where we introduce in the type theoretic context a definition of weak Tarski universe motivated by categorical models like the one given by  ...  construct a model of CZF with dependent choices and the regular extension axiom from homotopy type theory with a weak universe.  ... 
arXiv:1411.5591v1 fatcat:fc2yf77zkrbxlhprxpgrwhb2o4

Coherence of strict equalities in dependent type theories [article]

Rafaël Bocquet
2020 arXiv   pre-print
For a large class of type theories, we reduce the problem of the conservativity of equational extensions to more tractable acyclicity conditions.  ...  We generalize these methods to type theories without the Uniqueness of Identity Proofs principle, such as variants of Homotopy Type Theory, by introducing a notion of higher congruence over models of type  ...  thank Thorsten Altenkirch, Martin Bidlingmaier, Paolo Capriotti, Thierry Coquand, Simon Huber, Ambrus Kaposi, András Kovács, Nicolai Kraus, Chaitanya Leena Subramaniam, Christian Sattler and Bas Spitters for  ... 
arXiv:2010.14166v1 fatcat:yjqh3gy7yvcs3anj57ufk5mlfy

Page 5340 of Mathematical Reviews Vol. , Issue 92j [page]

1992 Mathematical Reviews  
The language % appropriate for a structure E is a first-order lan- guage with identity, a 2-place predicate for membership, individual constants for the terms of X and Y, respectively, and an individ-  ...  A typed language 27%. n) is also defined with variables for each of the different types, a 2-place predicate for membership (but well-formed only for terms of the appropriate types), constants for the  ... 

Simplicial sets inside cubical sets [article]

Thomas Streicher, Jonathan Weinberger
2021 arXiv   pre-print
The latter is a variant of the cubical model of type theory due to Cohen et al. for the purpose of providing a model for a variant of type theory which validates Voevodsky's Univalence Axiom and has computational  ...  Our contribution consists in constructing in 𝐜𝐒𝐞𝐭 a fibrant univalent universe for those types that are sheaves.  ...  Nevertheless, this may still model an extension of the cubical type theory of [CCHM18] providing a univalent universe for small simplicial sets the precise formulation of which we leave for future work  ... 
arXiv:1911.09594v2 fatcat:zdemuqdkdnezvi6jqo7fpg4wvu

Universal Homotopy Theories

Daniel Dugger
2001 Advances in Mathematics  
Begin with a small category C. The goal of this short note is to point out that there is such a thing as a "universal model category built from C."  ...  We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, sheaf theory, and the homotopy theory of schemes.  ...  So the homotopy theory of simplicial sets is just the universal homotopy theory on a point.  ... 
doi:10.1006/aima.2001.2014 fatcat:o4jrp7jfczfonchhmzfvyyrfoa

Large and Infinitary Quotient Inductive-Inductive Types

András Kovács, Ambrus Kaposi
2020 Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science  
We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures.  ...  We give semantics to described QIITs by modeling each signature as a finitely complete CwF (category with families) of algebras.  ...  Terms Tm Γ A : Set max(i +1, j)+1 is defined as the type of weak flCwF sections of A.  ... 
doi:10.1145/3373718.3394770 dblp:conf/lics/KovacsK20 fatcat:2zyhrfsdmjfyjkiw2mwtjxl7ea

Universal homotopy theories [article]

Daniel Dugger
2000 arXiv   pre-print
Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits.  ...  The technique of localization becomes a method for imposing 'relations' into these universal gadgets.  ...  So the homotopy theory of simplicial sets is just the universal homotopy theory on a point.  ... 
arXiv:math/0007070v1 fatcat:z6r2gjk3rjfkdfvdcj6x43d54y

Modal Dependent Type Theory and Dependent Right Adjoints [article]

Lars Birkedal, Ranald Clouston, Bassel Mannaa, Rasmus Ejlers Møgelberg, Andrew M. Pitts, Bas Spitters
2018 arXiv   pre-print
For the syntax, we introduce a dependently typed extension of Fitch-style modal lambda-calculus, show that it can be interpreted in any CwDRA, and build a term model.  ...  For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such.  ...  For the semantics, we start from Coquand's notion of a category with universes (Coquand 2012) , which covers all presheaf models of dependent type theory with universes.  ... 
arXiv:1804.05236v2 fatcat:qtf6zu5cwfd75igkaz5bw23k5m

The concept of strong and weak virtual reality

Andreas Martin Lisewski
2006 Minds and Machines  
Finally, we reformulate our characterization into a more general framework, and use Baltag's Structural Theory of Sets (STS) to show that within this general hyperset theory Sommerhoff's first and second  ...  consciousness in terms of higher cognitive functions.  ...  By a universe for ZFC set theory we mean a collection V of sets that is a model M of ZFC set theory.  ... 
doi:10.1007/s11023-006-9037-z fatcat:2oqi4zgnd5foddbgklhfz77rt4

Type theory and homotopy [article]

Steve Awodey
2010 arXiv   pre-print
type theory of Martin-L\"of into homotopy theory, resulting in new examples of higher-dimensional categories.  ...  The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive  ...  Then C is a model of Martin-Löf type theory with identity types.  ... 
arXiv:1010.1810v1 fatcat:tafuayc4pbeaxcq54lyvxrwlky
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