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Hierarchic Reasoning in Local Theory Extensions
[chapter]

2005
*
Lecture Notes in Computer Science
*

We identify situations in which it is possible,

doi:10.1007/11532231_16
fatcat:26a6233krzbillz3odqggiuw5i
*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. ...##
###
On Hierarchical Reasoning in Combinations of Theories
[chapter]

2010
*
Lecture Notes in Computer Science
*

Definition 2.4.1 (

doi:10.1007/978-3-642-14203-1_4
fatcat:cqx3nkfh5fbgbbrf5uv7hc7q4q
*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. ...##
###
Automated Reasoning in Some Local Extensions of Ordered Structures

2007
*
Proceedings - International Symposium on Multiple-Value Logic
*

The map l : DM(P ∂ ) → DM(P ) ∂ , sending

doi:10.1109/ismvl.2007.10
dblp:conf/ismvl/Sofronie-StokkermansI07
fatcat:2k2s47l3i5fudniiddbfqtxrfe
*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. ...##
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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

2020
*
Annual Conference for Computer Science Logic
*

Generalizing earlier work of Sattler

doi:10.4230/lipics.csl.2020.14
dblp:conf/csl/CavalloMS20
fatcat:rgzcvxrphbf67djmhglropid4q
*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. ...##
###
Constructive Set Theory from a Weak Tarski Universe
[article]

2014
*
arXiv
*
pre-print

Moreover, we show using the axiom of function extensionality that the type theoretic interpretation of constructive

arXiv:1411.5591v1
fatcat:fc2yf77zkrbxlhprxpgrwhb2o4
*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*. ...##
###
Coherence of strict equalities in dependent type theories
[article]

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*...

##
###
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]

2021
*
arXiv
*
pre-print

The latter is

arXiv:1911.09594v2
fatcat:zdemuqdkdnezvi6jqo7fpg4wvu
*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 ...##
###
Universal Homotopy Theories

2001
*
Advances in Mathematics
*

Begin

doi:10.1006/aima.2001.2014
fatcat:o4jrp7jfczfonchhmzfvyyrfoa
*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. ...##
###
Large and Infinitary Quotient Inductive-Inductive Types

2020
*
Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science
*

We do this by

doi:10.1145/3373718.3394770
dblp:conf/lics/KovacsK20
fatcat:2zyhrfsdmjfyjkiw2mwtjxl7ea
*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*. ...##
###
Universal homotopy theories
[article]

2000
*
arXiv
*
pre-print

Given

arXiv:math/0007070v1
fatcat:z6r2gjk3rjfkdfvdcj6x43d54y
*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. ...##
###
Modal Dependent Type Theory and Dependent Right Adjoints
[article]

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*. ...

##
###
The concept of strong and weak virtual reality

2006
*
Minds and Machines
*

Finally, we reformulate our characterization into

doi:10.1007/s11023-006-9037-z
fatcat:2oqi4zgnd5foddbgklhfz77rt4
*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*. ...##
###
Type theory and homotopy
[article]

2010
*
arXiv
*
pre-print

type

arXiv:1010.1810v1
fatcat:tafuayc4pbeaxcq54lyvxrwlky
*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. ...
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