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Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq

Dominik Kirst, Marc Hermes, Liron Cohen, Cezary Kaliszyk
We mechanise the undecidability of various first-order axiom systems in Coq, employing the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs.  ...  As a by-product of the undecidability of ZF formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.  ...  Undecidability and Incompleteness of First-Order Axiom Systems in Coq Proof.  ... 
doi:10.4230/lipics.itp.2021.23 fatcat:hknnxdvmlrazzbbjaqfpiitbcq

Parametric Church's Thesis: Synthetic Computability Without Choice [chapter]

Yannick Forster
2021 Lecture Notes in Computer Science  
Besides explaining the novel axioms and proofs of Rice's theorem we contribute machine-checked proofs of all results in the Coq proof assistant.  ...  In synthetic computability, pioneered by Richman, Bridges, and Bauer, one develops computability theory without an explicit model of computation.  ...  Sets in coq, coq in sets. Journal of Formalized Reasoning, 3(1):29–48, 2010. 4. A. Bauer. First steps in synthetic computability theory.  ... 
doi:10.1007/978-3-030-93100-1_6 fatcat:rfvtnrxvpra7pi6r6rzblkrjsi

Higher Structures in Homotopy Type Theory [article]

Ulrik Buchholtz
2018 arXiv   pre-print
The current repertoire of constructions, including the usual type formers and higher inductive types, suffice for many but not all of these.  ...  The intended model of the homotopy type theories used in Univalent Foundations is the infinity-category of homotopy types, also known as infinity-groupoids.  ...  Coq.  ... 
arXiv:1807.02177v1 fatcat:lgrt5uxqpvhanjj6pmw7ey7umm

Computability in constructive type theory [article]

Yannick Forster, Universität Des Saarlandes
of the Coq Library of Undecidability Proofs.  ...  We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability.  ...  Part III We introduce the notion of synthetic undecidability, which enables undecidability proofs in Coq not relying on axioms or models of computation [81, 73] .  ... 
doi:10.22028/d291-35758 fatcat:deb5muacebhsnnmk5p25mgc7ia

Polarities & Focussing: a journey from Realisability to Automated Reasoning [article]

Stéphane Graham-Lengrand
2014 arXiv   pre-print
Most of this part has been proved in the Coq proof assistant. Polarities and focussing were also introduced with applications to logic programming in mind, where computation is proof-search.  ...  The first part of this dissertation provides an introduction to these interpretations, highlighting the roles of polarities and focussing.  ...  In the third part of the thesis we explored a specific approach to theorem proving benefitting from the use of polarities and focussing.  ... 
arXiv:1412.6781v1 fatcat:sj4zl2ayu5bzfjaq76znvllsb4

An Implementation of Homotopy Type Theory in Isabelle/Pure [article]

Joshua Chen
2019 arXiv   pre-print
In this Masters thesis we present an implementation of a fragment of "book HoTT" as an object logic for the interactive proof assistant Isabelle.  ...  We also give a mathematical description of the underlying theory of the Isabelle/Pure logical framework, and discuss various issues and design decisions that arise when attempting to encode intensional  ...  Again, the order of the premises does not matter. The definitions of axiom schema and axiom for deductive systems carry over to the case of natural deduction systems.  ... 
arXiv:1911.00399v1 fatcat:vweng3xbezf6fecadvoy54l54a

A Brief Critical Introduction to the Ontological Argument and its Formalization: Anselm, Gaunilo, Descartes, Leibniz and Kant

Ricardo Sousa Silvestre
2018 IfColog journal of logics and their applications (FLAP)  
Arguments for and against the existence of God have been proposed and subjected to logical analysis in different periods of the history of philosophy.  ...  In an important sense, they all deal with the rationality of theist belief.  ...  System CO We formulate the rules and axioms of the system CO.  ... 
dblp:journals/flap/Silvestre18 fatcat:l6hbeivtpfdpfl4bog4dh7qswi

Automation of Mathematical Induction as part of the History of Logic [article]

J Strother Moore, Claus-Peter Wirth
2014 arXiv   pre-print
We review the history of the automation of mathematical induction  ...  three items, 52 and to get along with one second-order axiom for the natural numbers, or even with the first-order instances of that axiom.  ...  systems, such as Isabelle/HOL, Coq, PVS, and IsaPlanner, to name just a few.  ... 
arXiv:1309.6226v5 fatcat:i7nzngq47zfd7its4kfynlfpti

Executable Refinement Types [article]

Kenneth Knowles
2014 arXiv   pre-print
These results are useful for undecidable type systems in general.  ...  is decidable even though type checking is not, when suitably redefined to apply to undecidable type systems; (4) a novel use of existential types with dependent types to ensure that the language of logical  ...  In addition to refinement types, Sage includes dynamic types and first-class computation over types, demonstrating the breadth of interesting undecidable type systems to which our techniques apply.  ... 
arXiv:1403.3336v1 fatcat:7nbq2k2uf5a4zd3bgg3magdtvq

Algorithms in Philosophy, Informatics and Logic. A Position Manifesto 2017

Dov M. Gabbay, Jörg H. Siekmann
2017 IfColog journal of logics and their applications (FLAP)  
This mixed presentation is present everywhere: in applied logic, AI, linguistics, computer science and even philosophy.  ...  Any agent will have beliefs, belief revision, reasoning mechanisms (such as abduction) and more. Some of these features are declarative, others are algorithmic.  ...  research in Britain at a time when the promises of AI were seemingly just on the horizon.  ... 
dblp:journals/flap/GabbayS17 fatcat:wvtsqquk3vhm3ohgsxn6lzc22u

Elements of Consciousness and Cognition. Biology, Mathematic, Physics and Panpsychism: an Information Topology Perspective [article]

Pierre Baudot
2018 arXiv   pre-print
The first chapter presents the postulates and results on elementary perception at various organizational scales of the nervous system and proposes the hypothesis of an electrodynamic intrinsic nature of  ...  It provides a quantification of the structure of statistical interactions and their expression in terms of statistical physics and machine learning.  ...  Figure 1 presents the impulsional response (first order linear kernel) obtained at different organizational scales of the nervous system.  ... 
arXiv:1807.04520v2 fatcat:f2qyihx7cnadjl2ctkqn2f3j2q

An Old Discipline with a New Twist: The Course "Logic in Action"

Johan van Benthem
2017 IfColog journal of logics and their applications (FLAP)  
We are also very grateful to the Editorial Board of IfCoLog for agreeing in publishing this volume. Thanks are due to Jane Spurr, M. Antonia Huertas Sánchez, Dov Gabbay and Michael Gabbay.  ...  September 2016, Sophie Pinchinat & François Schwarzentruber Acknowledgements The author wishes to thank Matt Gray, Kahlil Hodgson, Daniel Pekevski, Nathan Ryan and Wayne Tsai for help in scripting Logic  ...  of finding new axioms to repair the incompleteness phenomenon.  ... 
dblp:journals/flap/Benthem17 fatcat:e5d54ekdhrbmjcs3ywmfzhaslm

A Survey on Network Verification and Testing with Formal Methods: Approaches and Challenges

Yahui Li, Xia Yin, Zhiliang Wang, Jiangyuan Yao, Xingang Shi, Jianping Wu, Han Zhang, Qing Wang
2018 IEEE Communications Surveys and Tutorials  
Furthermore, techniques ranging from formal modeling to verification and testing have been applied to help operators build reliable systems in electronic design automation and software.  ...  Inspired by its success, network verification has recently seen increased attention in the academic and industrial communities.  ...  Reasoning about reachability policies is undecidable because of unbounded ordered channels among middleboxes.  ... 
doi:10.1109/comst.2018.2868050 fatcat:h3op4heca5d75bpokfsbfevnwe

Formalizing Category Theory and Presheaf Models of Type Theory in Nuprl [article]

Mark Bickford
2018 arXiv   pre-print
This article is the first in a series of articles that explain the formalization of a constructive model of cubical type theory in Nuprl.  ...  So, it spells out how we make the first steps of our formalization of cubical type theory.  ...  All of this has been formalized in Coq and is well beyond the scope of this document.  ... 
arXiv:1806.06114v1 fatcat:eqie4oy5wzd2hjy2de2tzgui7i

Computational Back-And-Forth Arguments in Constructive Type Theory

Dominik Kirst, June Andronick, Leonardo de Moura
As prominent instances, we then deduce Cantor's and Myhill's isomorphism theorems on dense linear orders and one-one interreducible sets, respectively.  ...  In this proof pearl, we formalise this method abstractly in the framework of constructive type theory, emphasising the computational interpretation of the constructed isomorphisms.  ...  [5] and incompleteness [9] .  ... 
doi:10.4230/lipics.itp.2022.22 fatcat:zbx4rvtv2rfwhptn3zhcs6plsi
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