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Introduction to Mathematical Logic, Edition 2021

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NEW EDITION 2021: more and better motivations, chapter about tableaux method added, improved treatment of resolution method. [[[[[]]]]]Textbook for students in mathematical logic. Part 1. CONTENTS. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem. [[[[[[]]]]]] For Part 2 see my book at https://www.researchgate.net/publication/306112247_What_is_Mathematics_Godel's_Theorem_and_Around
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Fault Scenario Identification (FSI) is a challenging task that aims to automatically identify the fault types in communication networks from massive alarms to guarantee effective fault recoveries. Existing methods are developed based on rules, which are not accurate enough due to the mismatching issue. In this paper, we propose an effective method named Knowledge-Enhanced Graph Neural Network (KE-GNN), the main idea of which is to integrate the advantages of both the rules and GNN. This work is the first work that employs GNN and rules to tackle the FSI task. Specifically, we encode knowledge using propositional logic and map them into a knowledge space. Then, we elaborately design a teacher-student scheme to minimize the distance between the knowledge embedding and the prediction of GNN, integrating knowledge and enhancing the GNN. To validate the performance of the proposed method, we collected and labeled three real-world 5 G fault scenario datasets. Extensive evaluation conducted on these datasets indicates that our method achieves the best performance compared with other representative methods, improving the accuracy by up to 8.10%. Furthermore, the proposed method achieves the best performance against a small dataset setting and can be effectively applied to a new carrier site with a different topology structure.
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this report. These are: 1. The one literal rule also known as the unit rule
52 2.1. Proving Formulas Containing Implication only
  • ................................................................................. . Propositional Logic
Propositional Logic......................................................................................52 2.1. Proving Formulas Containing Implication only...................................52 2.2. Proving Formulas Containing Conjunction..........................................53 2.3. Proving Formulas Containing Disjunction...........................................56 2.4. Formulas Containing Negation -Minimal Logic.................................58 2.5. Formulas Containing Negation -Constructive Logic..........................63 2.6. Formulas Containing Negation -Classical Logic................................65 2.7. Constructive Embedding: Glivenko's Theorem....................................69 2.8. Axiom Independence. Using Computers in Mathematical Proofs........71
Russian translation available: "Kib. sbornik (novaya seriya)
  • J A Robinson
J. A. Robinson. A machine-oriented logic based on the resolution principle, "Jour. Assoc. Comput. Mach.", vol.12, N1, January 1965, pp.23-41 (available online, Russian translation available: "Kib. sbornik (novaya seriya)", 7, 1970, pp.194-218)
Computational Logic: Memories of the Past and Challenges for the Future. Computational Logic -CL
  • J A Robinson
J. A. Robinson. Computational Logic: Memories of the Past and Challenges for the Future. Computational Logic -CL 2000, First International Conference, London, UK, 24-28 July, 2000, Proceedings, Springer, Lecture Notes in Computer Science, 2000, Vol. 1861, pp. 1-24 (online copy).
├ C', then, c) If [L 1 -L 11
  • Now
Now, if [L 1 -L 8, f→B, MP]:├ C', then, c) If [L 1 -L 11, MP]:├ C, then [L 1 -L 9, MP]: B 1,..., B n ├ C, where hypotheses are instances of the axioms L 10 and L 11. Return to case (b). Q.E.D.
Herbrand completed his Ph.D. thesis in 1929. In the same 1929 Gödel completed his doctoral dissertation about completeness (see Section 4.3). In fact, Herbrand's method allows proving of Gödel's Completeness Theorem, but he (Herbrand) "did not notice it
Unlike the proof presented below, the original proof of Herbrand's Theorem does not depend on Gödel's Completeness Theorem (or Model Existence Theorem). Herbrand completed his Ph.D. thesis in 1929. In the same 1929 Gödel completed his doctoral dissertation about completeness (see Section 4.3). In fact, Herbrand's method allows proving of Gödel's Completeness Theorem, but he (Herbrand) "did not notice it". Why? See Samuel R. Buss. On Herbrand's Theorem. "Lecture Notes in Computer Science", Vol. 960, 1995, Springer-Verlag, pp.195-209 (available online).