Temporal type theory: a topos-theoretic approach to systems and behavior
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Birkhäuser
[2019]
|
| Series: | Progress in computer science and applied logic
volume 29 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | viii, 235 Seiten Illustrationen |
| ISBN: | 9783030007034 |
| ISSN: | 2297-0584 |
Staff View
MARC
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| 650 | 4 | |a Systems Theory, Control | |
| 650 | 4 | |a Mathematical Logic and Formal Languages | |
| 650 | 4 | |a Aerospace Technology and Astronautics | |
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Record in the Search Index
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| adam_text | Contents 1 Introduction......................................................... 1.1 1.2 1.3 1.4 1.5 1.6 Overview........................................................................ Behavior Types as Sheaves .................................................................. Temporal Type Theory........................................................................... Related Work.................................... Notation, Conventions, and Background.............................................. What to Expect from the Book............................................................. 2 The Interval Domain......................................................................................... 2.1 Review of Posites and (0,1)-Sheaves.................................................. 2.2 Domains and Posites............................................................................. 2.3 The Interval Domain and Its Associated Topos................................... 2.4 IK and the Upper Half-Plane................................................................ 2.5 Grothendieck Posites............................................................................. 3 Translation Invariance...................................................................................... 3.1 Construction of the Translation-Invariant Topos S............................ 3.2 IR/ as a Continuous Category............................................................ 3.3 The Subobject Classifier........................................................................ 3.4 The Behavior Type
Time...................................................................... 4 Logical Preliminaries......................................................................................... 4.1 4.2 4.3 5 Informal Introduction to Type Theory.................................................. Modalities................................................................................................ Dedekind j-Numeric Types................................................................... 1 1 4 8 9 11 12 17 17 19 22 26 32 39 39 42 43 43 47 47 61 66 87 Constant Types....................................................................................... 88 Introducing Time................................................................................... 92 Important Modalities in Temporal Type Theory................................. 98 Remaining Axiomatics........................................................................... 107 Axiomatics............................................................................................................ 5.1 5.2 5.3 5.4 vii
Contents viii 6 Categorical Semantics....................................................................... Constant Objects and Decidable Predicates..................................... Semantics of Dedekind Numeric Objects and Time....................... Semantics of the Modalities 1, @, and π .................................... Proof that Each Axiom Is Sound ..................................................... 115 115 118 121 126 128 Local Numeric Types andDerivatives........................................................ 7.1 Relationships Between Various Dedekind ƒ-Numeric Types.......... 7.2 Semantics of Numeric Types in Various Modalities........................ 7.3 Derivatives of Interval-Valued Functions......................................... 133 134 143 147 Applications................................................................................................... 157 157 163 165 166 170 175 Semantics and Soundness............................................................................ 6.1 6.2 6.3 6.4 6.5 7 8 8.1 8.2 8.3 8.4 8.5 8.6 Hybrid Sheaves................................................................................ Delays............................................................................................... Ordinary Differential Equations, Relations, and Inclusions............ Systems, Components, and Behavior Contracts .............................. Case Study: The National Airspace System..................................... Relation to Other Temporal Logics.................................................. Appendix APredomains and
Approximable Mappings................................ 179 A. 1 Predomains and Their Associated Domains.................................... 179 A.2 Approximable Mappings.................................................................. 188 A.3 Predomains in Subtoposes............................................................... 196 Appendix В IR/[ as a Continuous Category............................................... 207 B.l Review of Continuous Categories................................................... 207 B.2 B.3 B.4 The (Connected, Discrete Bifibration) Factorization System......... 209 Proof that 1 R/ Is Continuous....................................................... 215 Two Constructions of the Topos S ................................................. 217 Glossary of Symbols............................................................................................ 221 Bibliography.......................................................................................................... 225 Index........................................................................................................................ 229
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| adam_txt |
Contents 1 Introduction. 1.1 1.2 1.3 1.4 1.5 1.6 Overview. Behavior Types as Sheaves . Temporal Type Theory. Related Work. Notation, Conventions, and Background. What to Expect from the Book. 2 The Interval Domain. 2.1 Review of Posites and (0,1)-Sheaves. 2.2 Domains and Posites. 2.3 The Interval Domain and Its Associated Topos. 2.4 IK and the Upper Half-Plane. 2.5 Grothendieck Posites. 3 Translation Invariance. 3.1 Construction of the Translation-Invariant Topos S. 3.2 IR/ as a Continuous Category. 3.3 The Subobject Classifier. 3.4 The Behavior Type
Time. 4 Logical Preliminaries. 4.1 4.2 4.3 5 Informal Introduction to Type Theory. Modalities. Dedekind j-Numeric Types. 1 1 4 8 9 11 12 17 17 19 22 26 32 39 39 42 43 43 47 47 61 66 87 Constant Types. 88 Introducing Time. 92 Important Modalities in Temporal Type Theory. 98 Remaining Axiomatics. 107 Axiomatics. 5.1 5.2 5.3 5.4 vii
Contents viii 6 Categorical Semantics. Constant Objects and Decidable Predicates. Semantics of Dedekind Numeric Objects and Time. Semantics of the Modalities 1, @, and π . Proof that Each Axiom Is Sound . 115 115 118 121 126 128 Local Numeric Types andDerivatives. 7.1 Relationships Between Various Dedekind ƒ-Numeric Types. 7.2 Semantics of Numeric Types in Various Modalities. 7.3 Derivatives of Interval-Valued Functions. 133 134 143 147 Applications. 157 157 163 165 166 170 175 Semantics and Soundness. 6.1 6.2 6.3 6.4 6.5 7 8 8.1 8.2 8.3 8.4 8.5 8.6 Hybrid Sheaves. Delays. Ordinary Differential Equations, Relations, and Inclusions. Systems, Components, and Behavior Contracts . Case Study: The National Airspace System. Relation to Other Temporal Logics. Appendix APredomains and
Approximable Mappings. 179 A. 1 Predomains and Their Associated Domains. 179 A.2 Approximable Mappings. 188 A.3 Predomains in Subtoposes. 196 Appendix В IR/[ as a Continuous Category. 207 B.l Review of Continuous Categories. 207 B.2 B.3 B.4 The (Connected, Discrete Bifibration) Factorization System. 209 Proof that 1 R/ Is Continuous. 215 Two Constructions of the Topos S . 217 Glossary of Symbols. 221 Bibliography. 225 Index. 229 |
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| author | Schultz, Patrick Spivak, David I. 1978- |
| author_GND | (DE-588)1183050135 (DE-588)1063382106 |
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| dewey-full | 512.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.6 |
| dewey-search | 512.6 |
| dewey-sort | 3512.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| illustrated | Illustrated |
| index_date | 2024-07-03T16:37:53Z |
| indexdate | 2024-07-10T09:04:00Z |
| institution | BVB |
| isbn | 9783030007034 |
| issn | 2297-0584 |
| language | English |
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| physical | viii, 235 Seiten Illustrationen |
| publishDate | 2019 |
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| publisher | Birkhäuser |
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| series | Progress in computer science and applied logic |
| series2 | Progress in computer science and applied logic |
| spelling | Schultz, Patrick Verfasser (DE-588)1183050135 aut Temporal type theory a topos-theoretic approach to systems and behavior Patrick Schultz, David I. Spivak Cham Birkhäuser [2019] © 2019 viii, 235 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Progress in computer science and applied logic volume 29 2297-0584 Category Theory, Homological Algebra Mathematical Logic and Foundations Systems Theory, Control Mathematical Logic and Formal Languages Aerospace Technology and Astronautics Algebra Logic, Symbolic and mathematical Systems theory Computer science Astronautics Spivak, David I. 1978- Verfasser (DE-588)1063382106 aut Erscheint auch als Online-Ausgabe 978-3-030-00704-1 Progress in computer science and applied logic volume 29 (DE-604)BV042031016 29 Digitalisierung BSB München - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032554709&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schultz, Patrick Spivak, David I. 1978- Temporal type theory a topos-theoretic approach to systems and behavior Progress in computer science and applied logic Category Theory, Homological Algebra Mathematical Logic and Foundations Systems Theory, Control Mathematical Logic and Formal Languages Aerospace Technology and Astronautics Algebra Logic, Symbolic and mathematical Systems theory Computer science Astronautics |
| title | Temporal type theory a topos-theoretic approach to systems and behavior |
| title_auth | Temporal type theory a topos-theoretic approach to systems and behavior |
| title_exact_search | Temporal type theory a topos-theoretic approach to systems and behavior |
| title_exact_search_txtP | Temporal type theory a topos-theoretic approach to systems and behavior |
| title_full | Temporal type theory a topos-theoretic approach to systems and behavior Patrick Schultz, David I. Spivak |
| title_fullStr | Temporal type theory a topos-theoretic approach to systems and behavior Patrick Schultz, David I. Spivak |
| title_full_unstemmed | Temporal type theory a topos-theoretic approach to systems and behavior Patrick Schultz, David I. Spivak |
| title_short | Temporal type theory |
| title_sort | temporal type theory a topos theoretic approach to systems and behavior |
| title_sub | a topos-theoretic approach to systems and behavior |
| topic | Category Theory, Homological Algebra Mathematical Logic and Foundations Systems Theory, Control Mathematical Logic and Formal Languages Aerospace Technology and Astronautics Algebra Logic, Symbolic and mathematical Systems theory Computer science Astronautics |
| topic_facet | Category Theory, Homological Algebra Mathematical Logic and Foundations Systems Theory, Control Mathematical Logic and Formal Languages Aerospace Technology and Astronautics Algebra Logic, Symbolic and mathematical Systems theory Computer science Astronautics |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032554709&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV042031016 |
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