Matroids, Cyclic Flats, and Polyhedra

<p>Matroids have a wide variety of distinct, cryptomorphic axiom systems that are capable of defining them. A common feature of these is that they are able to be efficiently tested, certifying whether a given input complies with such an axiom system in polynomial time. Joseph Bonin and Anna de Mier, rediscovering a theorem first proved by Julie Sims, developed an axiom system for matroids in terms of their cyclic flats and the ranks of those cyclic flats. As with other matroid axiom systems, this is able to be tested in polynomial time. Distinct, non-isomorphic matroids may each have the same lattice of cyclic flats, and so matroids cannot be defined solely in terms of their cyclic flats. We do not have a clean characterisation of families of sets that are cyclic flats of matroids. However, it may be possible to tell in polynomial time whether there is any matroid that has a given lattice of subsets as its cyclic flats. We use Bonin and de Mier’s cyclic flat axioms to reduce the problem to a linear program, and show that determining whether a given lattice is the lattice of cyclic flats of any matroid corresponds to finding integral points in the solution space of this program, these points representing the possible ranks that may be assigned to the cyclic flats. We distinguish several classes of lattice for which solutions may be efficiently found, based upon the nature of the matrix of coefficients of the linear program, and of the polyhedron it defines, and then identify families of lattice that belong to those classes. We define operations and transformations on lattices of sets by examining matroid operations, and examine how these operations affect membership in the aforementioned classes. We conjecture that it is always possible to determine, in polynomial time, whether a given collection of subsets makes up the lattice of cyclic flats of any matroid.</p>

Matroids, Cyclic Flats, and Polyhedra

<p>Matroids have a wide variety of distinct, cryptomorphic axiom systems that are capable of defining them. A common feature of these is that they are able to be efficiently tested, certifying whether a given input complies with such an axiom system in polynomial time. Joseph Bonin and Anna de Mier, rediscovering a theorem first proved by Julie Sims, developed an axiom system for matroids in terms of their cyclic flats and the ranks of those cyclic flats. As with other matroid axiom systems, this is able to be tested in polynomial time. Distinct, non-isomorphic matroids may each have the same lattice of cyclic flats, and so matroids cannot be defined solely in terms of their cyclic flats. We do not have a clean characterisation of families of sets that are cyclic flats of matroids. However, it may be possible to tell in polynomial time whether there is any matroid that has a given lattice of subsets as its cyclic flats. We use Bonin and de Mier’s cyclic flat axioms to reduce the problem to a linear program, and show that determining whether a given lattice is the lattice of cyclic flats of any matroid corresponds to finding integral points in the solution space of this program, these points representing the possible ranks that may be assigned to the cyclic flats. We distinguish several classes of lattice for which solutions may be efficiently found, based upon the nature of the matrix of coefficients of the linear program, and of the polyhedron it defines, and then identify families of lattice that belong to those classes. We define operations and transformations on lattices of sets by examining matroid operations, and examine how these operations affect membership in the aforementioned classes. We conjecture that it is always possible to determine, in polynomial time, whether a given collection of subsets makes up the lattice of cyclic flats of any matroid.</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>

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>

Performability of Actions

Action theory may be regarded as a theoretical foundation of AI, because it provides in a logically coherent way the principles of performing actions by agents. But, more importantly, action theory offers a formal ontology mainly based on set-theoretic constructs. This ontology isolates various types of actions as structured entities: atomic, sequential, compound, ordered, situational actions etc., and it is a solid and non-removable foundation of any rational activity. The paper is mainly concerned with a bunch of issues centered around the notion of performability of actions. It seems that the problem of performability of actions, though of basic importance for purely practical applications, has not been investigated in the literature in a systematic way thus far. This work, being a companion to the book as reported (Czelakowski in Freedom and enforcement in action. Elements of formal action theory, Springer 2015), elaborates the theory of performability of actions based on relational models and formal constructs borrowed from formal lingusistics. The discussion of performability of actions is encapsulated in the form of a strict logical system "Equation missing". This system is semantically defined in terms of its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since due to the nature of compound actions the system "Equation missing" is not finitary, other semantic variants of "Equation missing" are defined. The focus in on the system "Equation missing" of performability of finite compound actions. An adequate axiom system for "Equation missing" is defined. The strong completeness theorem is the central result. The role of the canonical model in the proof of the completeness theorem is highlighted. The relationship between performability of actions and dynamic logic is also discussed.

An Axiom System for Feedback Centralities

In recent years, the axiomatic approach to centrality measures has attracted attention in the literature. However, most papers propose a collection of axioms dedicated to one or two considered centrality measures. In result, it is hard to capture the differences and similarities between various measures. In this paper, we propose an axiom system for four classic feedback centralities: Eigenvector centrality, Katz centrality, Katz prestige and PageRank. We prove that each of these four centrality measures can be uniquely characterized with a subset of our axioms. Our system is the first one in the literature that considers all four feedback centralities.

A Formal Verification Method of Compilation Based on C Safety Subset

With the rapid increase in the number of wireless terminals and the openness of wireless networks, the security of wireless communication is facing serious challenges. The safety and security of computer communication have always been a research hotspot, especially the wireless communication that still has a more complex architecture which leads to more safety problems in the communication system development. In recent years, more and more wireless communication systems are applied in the safety-critical field which tends to need high safety guarantees. A compiler is an important tool for system development, and its safety and reliability have an important impact on the development of safety-critical software. As the strictest method, formal verification methods have been widely paid attention to in compiler verification, but the current formal verification methods have some problems, such as high proof complexity, weak verification ability, and low algorithm efficiency. In this paper, a compiler formal verification method based on safety C subsets is proposed. By abstracting the concept of C grammar units from safety C subsets, the formal verification of the compiler is transformed into the verification of limited C grammar units. In this paper, an axiom system of first-order logic and special axioms are introduced. On this axiom system, the semantic consistency verification of C grammar unit and target code pattern is completed by means of theorem proving, and the formal verification of the compiler is completed.

Classical Mereology

The aim of this chapter is to provide a comprehensive introduction to classical mereology. It examines this theory by providing a clear and perspicuousaxiom system that isolates several important elements of any mereological theory. The chapter examines, algebraic, and set-theoretic models of classical mereology, sketching proofs of their equivalence. The new axiom system facilitates algebraic comparisons, showing that models of these axioms are complete Boolean algebras without a bottom element. Then set-theoretic models are presented, and are shown to satisfy the axioms. The chapter explains the important relationship between models and powersets, and the role of Stone’s Representation Theorem in this connection. Finally, a number of significant rival axiom systems using different mereological primitives are introduced.

A Note on Non-deterministic Turing Machine and the "P vs. NP ( P versus NP problem )"

摘要：一个非确定型图灵机NDTM的计算过程，可以相当于其对应的确定型图灵机DTM的幂集。如果接受ZF公理系统的幂集公理，“P对NP”问题最可能的答案是：对于确定型图灵机，P≠NP。可以从另外3个角度对它进行一定的解释。ABSTRACT: The calculation process of a non-deterministic Turing machine (NDTM) can be equipotent to the power set of its corresponding deterministic Turing machine (DTM). If accepting the “Axiom of power set” of the ZF axiom system ( Zermelo–Fraenkel set theory ), the most likely answer to the "P vs. NP" ( P versus NP problem ) is: For a deterministic Turing machine, P≠NP ( P is not equal to NP ). This answer can be explained from three other perspectives.