A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

<p>This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom.</p>

A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

<p>This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom.</p>

UHPLC-ESI-QTOF-MS Analysis on Canthium Coromandelicum Stem

A rapid ultra high-performance liquid chromatography coupled with crossover triple quadrupole time of flight mass spectrometry (UHPLC- ESI -QTOF-MS/MS) method has been developed for the identification of debasement products. According to the distinctive fragmentation patterns, the presence of 79 compounds with retention time between 1.05 to 26.81 minutes was found. In Canthium coromandelicum stem, 22 amino acids, 10 fatty acids, 6 alkaloids, 6 steroids, 2 flavonoids, 2 terpenoids, 2 phenolic, 4 lipids, 3 anthraquinone glycosides, sugars, vitamins were distinguished. These outcomes demonstrated that the contemporary technique has been utilized for quality control of Canthium coromandelicum, exceptionally for recognizable proof, verification and portrayal in medication arrangements.

Complexity Assessments for Decidable Fragments of Set Theory. I: A Taxonomy for the Boolean Case*

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪ ∩ \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates ‘• = Ø’ and ‘Disj(•, •)’, meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘• ≠ Ø and ‘¬Disj(•, •)’. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time.

Graphical Methods in Device-Independent Quantum Cryptography

We introduce a framework for graphical security proofs in device-independent quantum cryptography using the methods of categorical quantum mechanics. We are optimistic that this approach will make some of the highly complex proofs in quantum cryptography more accessible, facilitate the discovery of new proofs, and enable automated proof verification. As an example of our framework, we reprove a previous result from device-independent quantum cryptography: any linear randomness expansion protocol can be converted into an unbounded randomness expansion protocol. We give a graphical proof of this result, and implement part of it in the Globular proof assistant.

STRATEGI PEMECAHAN MASALAH DALAM MATEMATIKA

The main purpose of studying mathematics is to find ways to solve problems or mathematics problems. What is meant by problems or mathematics problems is a thing that final result, or how to solve it is not known. In solving mathematical problems there are several strategies that can be used that are: direct proof, indirect proof, verification with contradiction, proof with examples of denying, reverse working strategy, pattern discovery, and the use of bird house principles.

Implementasi Zero Knowledge Proof Menggunakan Protokol Feige Fiat Shamir Untuk Verifikasi Tiket Rahasia

Cryptography is known for it’s ability to protect confidential information, but it can also be used for other purposes. One of them is for identity verification or authentication. One of the biggest disadvantages of traditional authencation method is at the end of the session, the verifier knows about secrets which is supposed to be known only by prover. In this paper, we implemented a ZeroKnowledge Proof-based secret ticket verification system using Feige Fiat Shamir protocol. The goal of this system is to help prover identified themselves to the verifier, but also prevent the verifier to understand anything about the prover’s secret information. The system is also able to prevent ticket duplication or double-use of tickets by using an interactive proof verification method. By combining it with cryptography, not only we can achieve completeness and soundness property of Zero-Knowledge Proof, but we can also achieve information security property. Index Terms - Feige Flat Shamir, Verification, Zero Knowledge Proof.