Local analysis for the odd order theorem pdf. SlideShare Explore Search You. Submit Search. Successfully reported this slideshow. We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime. Upcoming SlideShare. Like this document? Why not share! Local cohomology an algebraic intro Embed Size px. Start on. Show related SlideShares at end. Conflict based clause learning is known to be an important component in Modern SAT solving.

Because of the exponential blow up of the size of learnt clauses database, maintaining a relevant and polynomially bounded set of learnt clauses is crucial for the efficiency of clause learning based SAT solvers. In this paper, we first compare several criteria for selecting the most relevant learnt clauses with a simple random selection strategy. We then propose new criteria allowing us to select relevant clauses w.

Then, we use such strategies as a means to diversify the search in a portfolio based parallel solver. An experimental evaluation comparing the classical ManySAT solver with the one augmented with multiple deletion strategies, shows the interest of such approach.

### References

Constraintsatisfactionandcombinatorialoptimizationprob- lems, even when modeled with efficient metaheurisics such as local search remain computationally very intensive. Solvers stand to benefit significantly from execution on parallel systems, which are increasingly available. The architectural diversity and complexity of the latter means that these systems pose ever greater challenges in order to be effectively used, both from the point of view of the modeling effort and from that of the degree of coverage of the available computing resources.

In this article we discuss impositions and design issues for a framework to make efficient use of various parallel architectures. We propose to apply a general framework for estimating the parallel performance of Las Vegas algorithms to randomized propagation-based constraint solvers, i.

Indeed, by analyzing the runtime of the sequential process which is variable from one run to another because of the stochastic nature of the algorithm and approximating this runtime distribution with statistical methods, the runtime behavior of the parallel process can be predicted by a model based on order statistics.

We apply this approach to study the behavior of a randomized version of the Gecode solver on three classical CSPs problems, and compare the predicted performances to the results of an actual experimentation on parallel hardware up to cores. Many network design problems arising in areas as diverse as VLSI circuit design, QoS routing, traffic engineering, and computational sustainability require clients to be connected to a facility under path-length constraints and budget limits. These networks are vulnerable to a failure. Therefore, it is often important to ensure that all clients are connected to two or more facilities via edge-disjoint paths.

Traditional way of extending a sequential algorithm to run in parallel is to either perform portfolio-based search in parallel or to perform parallel neighbourhood search. We rather exploit the semantics of the constraints of the problem to perform multiple moves in parallel by ensuring that they are mutually independent. The ideas presented in this paper are general and can be adapted to any other problem. The effectiveness of our approach is demonstrated by experimenting with a set of problem instances taken from real-world passive optical network deployments in Ireland and the UK.

Results show that performing moves in parallel can significantly reduce the time required by our local-search approach. We demonstrate how to use a tiny domain-specific language for the description of APIs make it possible to perform black-box testing of APIs. Given an API, our testing framework behaves like regular client code and combines functions exported by the API to build elements of the data-types that are exported by the API.

In the submitted abstract, we use the running example of data-structures APIs, but the very same ideas apply to the smart-fuzzing of security APIs. The goal of the paper is to develop a software for automated analysis of PKCS 11 tokens with focus on the security of stored cryptographic keys.

Then the paper describes the mechanics of cryptographic key protection in Cryptoki. It analyses the individual functions relevant to key protection and defines the first set of tests. Then the paper performs an analysis of the previously published attacks against keys stored on the device and also outlines the second set of tests with focus on discovering attack vulnerabilities.

Afterwards the paper examines various ways to improve the security of Cryptoki by altering the standard. The discussion of the proposed restrictions takes into consideration their impact on practical usability of the tokens. In order to further evaluate the security of the tokens against possible attacks, the paper also utilizes model checker software. The results of the previously defined tests are used to compose a model that is sent to a model checker for further analysis. Since the developed software is written in Java, the paper also examines the Java PKCS 11 wrapper and discusses the advantages and disadvantages against using C for implementation.

The most obvious advantage is that the software can be easily deployed on other operating systems, which is particularly useful for testing tokens that only work on Windows. The disadvantage is that the used wrapper does not implement all the Cryptoki functions, but the paper shows that all these limitations either do not matter or they can be worked around.

The paper also presents the developed tool called Caetus. The tool first connects into the token, discovers the basic capabilities of the device and based on these capabilities it performs in-depth testing of Cryptoki functions. First part of the tests focuses on adherence to the PKCS 11 standard, which means the discovered issues do not necessarily mean security risks. The other set of tests attempts to discover vulnerability of the token to the previously published attacks and other trivial attacks.

Finally the tool can use the discovered capabilities in order to compose a model specification which is then sent to the model checker.

The tool is currently optimized for analysis of software tokens. The last part of the paper consists of tables with results of token testing and discussion of the results. Nonmonotonic Description Logic DL- programs couple nonmonotonic logic programs with DL-ontologies through queries in a loose way which may lead to inconsistency, i. Recently defined repair answer sets remedy this but a straightforward computation method lacks practicality. This leads to significant performance gains towards inconsistency management in practice.

The semantic web is an open and distributed environment in which it is hard to guarantee consistency of knowledge and information. Under the standard two-valued semantics everything is entailed if knowledge and information is inconsistent. The semantics of the paraconsistent logic LP offers a solution. However, if the available knowledge and information is consistent, the set of conclusions entailed under the three-valued semantics of the paraconsistent logic LP is smaller than the set of conclusions entailed under the two-valued semantics. Preferring conflict-minimal three-valued interpretations eliminates this difference.

Preferring conflict-minimal interpretations introduces non-monotonicity. To handle the non-monotonicity, this paper proposes an assumption-based argumentation system. Assumptions needed to close branches of a semantic tableaux form the arguments. Stable extensions of the set of derived arguments correspond to conflict minimal interpretations and conclusions entailed by all conflict-minimal interpretations are supported by arguments in all stable extensions. Recently several inconsistency-tolerant semantics have been introduced for querying inconsistent description logic knowledge bases.

Most of these semantics rely on the notion of a repair, defined as an inclusion-maximal subset of the facts ABox which is consistent with the ontology TBox.

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## A Machine-Checked Proof of the Odd Order Theorem | SpringerLink

In this paper, we study variants of two popular inconsistency-tolerant semantics obtained by replacing classical repairs by various types of preferred repair. Unsurprisingly, query answering is intractable in all cases, but we nonetheless identify one notion of preferred repair, based upon priority levels, whose data complexity is "only" coNP-complete. This leads us to propose an approach combining incomplete tractable methods with calls to a SAT solver.

An experimental evaluation of the approach shows good scalability on realistic cases. With the proliferation of multicore systems, the design of concurrent algorithms and concurrent data structures to support them has becomes critical. Wait-free data structures provide a very basic and natural progress guarantee, assuring that each thread always makes progress when given enough CPU cycles.

Only the weaker lock-free guarantee, which allows almost all threads to starve, was achievable in practice. In a series of recent papers, we have shown that this pessimistic belief was false and wait-freedom is actually achievable efficiently in practice. We consider the problem of provably verifying that an asynchronous message-passing system satisfies its local assertions.

We present a novel reduction scheme for asynchronous event-driven programs that finds almost-synchronous invariants invariants consisting of global states where message buffers are close to empty. The reduction finds almost-synchronous invariants and simultaneously argues that they cover all local states.

We show that asynchronous programs often have almost-synchronous invariants and that we can exploit this to build natural proofs that they are correct. We implement our reduction strategy, which is sound and complete, and show that it is more effective in proving programs correct as well as more efficient in finding bugs in several programs, compared to current search strategies which almost always diverge.

The high point of our experiments is that our technique can prove the Windows Phone USB Driver written in P correct for the receptiveness property, which was hitherto not provable using state-of-the-art model-checkers. Solvers for Satisfiability Modulo Theories SMT combine the ability of fast Boolean satisfiability solvers to find solutions for complex propositional formulas with specialized theory solvers. Theory solvers for linear real and integer arithmetic reason about systems of simultaneous inequalities.

These solvers either find a feasible solution or prove that no such solution exists. Linear programming LP solvers come from the tradition of optimization, and are designed to find feasible solutions that are optimal with respect to some optimization function. Typical LP solvers are designed to solve large systems quickly using floating point arithmetic. Because floating point arithmetic is inexact, rounding errors can lead to incorrect results, making these solvers inappropriate for direct use in theorem proving.

Previous efforts to leverage such solvers in the context of SMT have concluded that in addition to being potentially unsound, such solvers are too heavyweight to compete in the context of SMT. In this paper, we describe a technique for integrating LP solvers that dramatically improves the performance of SMT solvers without compromising correctness. This paper presents an iterative approximation refinement, called rasat loop, which solves a system of polynomial inequalities on real numbers.

The approximation scheme consists of interval arithmetic over-approximation, aiming to decide UNSAT and testing under-approximation, aiming to decide SAT. If both of them fail to decide, input intervals are refined by decomposition. We discuss three strategy design choices: dependency to set priority among atomic polynomial constraints, sensitivity to set priority among variables, and UNSAT core for reducing learned clauses and incremental UNSAT detection. Preliminary experimental observation on comparison with Z3 4.

We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified variables. Implementations of extended quantifier elimination via virtual substitution have been successfully applied to various problems in science and engineering. So far, the answers produced by these implementations included infinitesimal and infinite numbers, which are hard to interpret in practice.

We introduce here a post-processing procedure to convert, for fixed parameters, all answers into standard real numbers. The relevance of our procedure is demonstrated by applications of our implementation to various examples from the literature, where it significantly improves the quality of the results. We consider SMT-solving for linear real arithmetic. Inspired by related work for the Fourier--Motzkin method, we combine virtual substitution with learning strategies. For the first time, we present virtual substitutionincluding our learning strategiesas a formal calculus.

We prove soundness and completeness for that calculus. Some standard linear programming benchmarks computed with an experimental implementation of our calculus show that the integration of learning techniques into virtual substitution gives rise to considerable speedups. Our implementation is open-source and freely available. This talk outlines a proof-theoretic approach to developing correct and terminating monadic parsers.

Using a modified realisability interpretation, we extract provably correct and terminating programs from formal proofs. If the proof system is proven to be correct, then any extracted program is guaranteed to be so. By extracting parsers, we can ensure that they are correct, complete and terminating for any input. The work is ongoing, and is being carried out in the interactive proof system Minlog. Ontologies represented using description logics model domains of interest in terms of concepts and binary relations, and are used in a range of areas including medical science, bio-informatics, semantic web or artificial intelligence.

Often, ontologies are large and complex and cover 10,s of concepts. Uniform interpolants are restricted views of ontologies, that only use a specified set of symbols, while preserving all entailments that can be expressed using these symbols. Uniform interpolation can be used for analysing hidden relations in an ontology, removing confidential concepts from an ontology, computing logical differences between ontologies or extracting specialised ontologies for ontology reuse and has many more applications.

## Thompson transitivity theorem

We follow a resolution-based approach to make the computation of uniform interpolants of larger ontologies feasable. Uniform interpolants cannot always be represented finitely in the language of the input ontology, for which situation we offer three solutions: extending the signature with additional concepts, approximating the uniform interpolant, or using fixpoint operators.

Craig interpolation has been recently shown to be useful in a wide variety of problem domains. One use is in strategy extraction for two player games, as described in our accompanying submission. However, interpolation is not without its drawbacks. It is well-known that an interpolant may be very large and highly redundant. Subsequent use of the interpolant requires that it is transformed to CNF or DNF, which will further increase its size.

We present a new approach to handling both the size of interpolants and transformation to clausal representation. Our approach relies on the observation that in many real-world applications, interpolants are defined over a relatively small set of variables. Additionally, in most cases there likely exists a compact representation of the interpolant in CNF. For instance, in our application to games an interpolant represents a set of winning states that is likely to have a simple structure.

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We study a general framework for query rewriting in the presence of a domain independent first-order logic theory a knowledge base over a signature including database and non-database predicates, based on Craig's interpolant and Beth's definability theorem. In our framework queries are possibly open domain independent first-order formulas over the extended signature. The database predicates are meant to be closed, i. It is important to notice that all the conceptual modelling languages devised for the designing of information and database systems such as Entity-Relationship schemas, UML Class diagrams, Object-Role Modelling ORM diagrams, etc are domain independent: they can be formalised as domain independent first order theories.

Given a domain independent knowledge base and a query implicitly definable from the database signature, the framework provides precise semantic conditions to decide the existence of a domain independent first-order logically equivalent reformulation of the query called exact rewriting in terms of the database signature, and if so, it provides an effective approach to construct the reformulation based on interpolation using standard theorem proving techniques e. We are interested in domain independent reformulations of queries because their range-restricted syntax is needed to reduce the original query answering problem to a relational algebra evaluation over the original database, that is, it is effectively executable as an SQL query directly over the database.

Due to the results on the applicability of Beth's theorem and Craig's interpolant, we prove the completeness of our framework in the case of domain independent ontologies and queries expressed in any fragment of first-order logic enjoying finitely controllable determinacy, a stronger property than the finite model property of the logic. If the employed logic does not enjoy finitely controllable determinacy our approach would become sound but incomplete, but still effectively implementable using standard theorem proving techniques. Since description logics knowledge bases are not necessarily domain independent, we have syntactically characterised the domain independent fragment of several very expressive fragments of description logics by enforcing a sort of guarded negation, a very reasonable restriction from the point of view of conceptual modelling.

These fragments do also enjoy finitely controllable determinacy. We have applied this framework not only to query answering under constraints, but also to data exchange and to view update. Modern Satisfiability Modulo Theories SMT solvers are highly efficient and can generate a resolution proof in case of unsatisfiability. Some applications, such as synthesis of Boolean controllers, compute multiple coordinated interpolants from a single refutation proof. In order to do so, the proof is required to have two properties: It must be colorable and local-first.

The latter means that resolution over a literal that occurs just in one partition, has to have both premises derived from this partition. Off-the-shelf SMT solvers do not necessarily produce proofs that have these properties. In particular, proofs are usually not local-first. Hofferek et al. Our goal is to introduce a new method to directly compute a local-first, colorable resolution proof for an unsatisfiable SMT formula. This proof can then be directly used for n-interpolation.

Our approach uses a tree-based structure of SMT solvers, where every node in the tree is associated with a formula possibly empty initially , and a possibly empty set of literals. The semantics of the modular SMT problem is the conjunction of the formulas of all nodes. The set of literals associated with a node is computed recursively as follows. Every literal, which appears in more than one descendant of a parent node, is assigned to the parent node.

During solving, every node makes decisions about its associated literals only. We start at the root node and communicate the partial assignments to the child nodes until we reach a leaf. Every node has its own solver instance and tries to compute a satisfying assignment w. A blocking clause is added to the parent node if no satisfying assignment can be found.

If all child nodes find a satisfying assignment and the conjunction of them is theory-consistent, we either decide more literals, or return to the parent node if we already have a full assignment for all the literals of the current node. In case the conjunction is inconsistent within the theory, we respectively add a blocking clause for the current assignment to the children. In order to obtain this clauses while keeping the modular structure intact, we perform interpolation over the assignments of the children and use the interpolants, which are guaranteed to contain only "global" literals which can safely be added to the children without breaking the modularity , to learn a blocking clause for every child node.

The algorithm terminates if either solely full assignments are communicated to the root node and there is no literal left to decide, or when enough clauses have been learned at the root node to show inconsistency. In the latter case we are able to extract a resolution proof with the same modular structure as the original problem. We are currently working on implementing this approach in the hope that we can use it to generate colorable, local-first proofs for synthesis problems much faster than with post-processing proof transformations.

The security proofs of the modules e. This allows to compare security definitions. Connection matrices for graphs and hypergraphs are a generalization of Hankel matrices for words. They were used by L. Lovasz and A. Schrijver and their collaborators to characterize which graph parameters arise from partition functions. Lovasz also noted that they can be used to make Courcelle's Theorem, which shows that graph properties definable in Monadic Second Order Logic MSOL are in FPT on graph classes of bounded tree-width, logic-free by replacing MSOL-definability by a finiteness condition on the rank of connection matrices and allowing graph parameters with values in a field.

In this paper we extend this to graph parameters with values in a tropical semi-rings rather than a field, and graph classes of bounded clique-width. Backdoor sets for this class are studied in terms of parameterized complexity. The question whether there exist a CNF 2 -backdoor set of size k is hard for the class W[2], for both weak and strong backdoors, and in both cases it becomes fixed-parameter tractable when restricted to inputs in d-CNF for a fixed d.

As wireless networking increases, so do the risks associated with the use of such technology. We examined the role of wireless network display and choice presentation in a study involving undergraduate social science students. Another goal was to better understand the basis for decision-making. These decision justifications were associated with different network choices. The results suggest that the padlock takes different functions and meanings for the three groups which can help to better understand their security-related decision making.

We further observed significant effects for the use of colour when nudging participants towards more secure choices. We also wanted to examine the role of individual differences in relation to the choices individuals make. Perceived vulnerability and controllability of risk played a role in terms of the extent to which participants would more secure vs. This indicates that perceived risk perceptions and reasons for decisions may relate differently to the actual behavioural choices individuals make, with perceptions of risk not necessarily relating to the reasons that participants consider when making security decisions.

An enterprise's information security policy is an exceptionally important control as it provides the employees of an organisation with details of what is expected of them, and what they can expect from the organisation's security teams, as well as informing the culture within that organisation. The threat from accidental insiders is a reality across all enterprises, and can be extremely damaging to the systems, data and reputation of an organisation. Recent industry reports, and academic literature underline the fact that the risk of accidental insider compromise is potentially more pressing than that posed by a malicious insider.

In this paper we focus on the ability of enterprises' information security policies to mitigate against the accidental insider threat. Specifically we perform an analysis of real-world cases of accidental insider threat to define the key reasons, actions and impacts of these events - captured as a grounded insider threat classification scheme. This scheme is then used to perform a review of a set of organisational security policies to highlight their strengths and weaknesses when considering the prevention of incidents of accidental insider compromise.

We present a set of questions that can be used to analyse an existing security policy to help control the risk of the accidental insider threat. User constrained devices such as smart cards are commonly used in human-protocol interaction. Modelling these devices as part of human-protocol interaction is still an open problem. Examining the interaction of these devices as part of security ceremonies offers greater insight.

This paper highlights two such cases: modelling extra channels between humans and devices in the ceremony, and modelling possession when the device also acts as an agent in the ceremony. Case studies where such devices are used during authentication ceremonies are presented to demonstrate these use cases. This paper reports extensions and further analysis of a new form of singleton arc consistency, called neighbourhood SAC NSAC , where a subproblem adjacent to the variable with a reduced domain the "focal variable" is made arc consistent.

The first part of the paper presents two important extensions: 1 NSAC is generalized to k-NSAC, where k is the longest path between the focal variable and any variable in the subgraph. In this work we will only consider the just-named forms. Obviously, there is an associated dominance hierarchy with respect to level of consistency, with k-NSAC The second part presents some studies of hybrid search techniques based on NSAC and SAC, using a variety of problems.

In some cases, higher levels of consistency maintenance outperform MAC by several orders of magnitude, although with weighted degree the best tradeoff is obtained when SAC-based consistency is restricted to preprocessing. Abstract: Abduction can be defined as the process of inferring plausible explanations or hypotheses from observed facts conclusions. This form of reasoning has the potential of playing a central role in system verification, particularly for identifying bugs and providing hints to correct them.

We describe an approach to perform abductive reasoning that is based on the superposition calculus. The formulas we consider are sets of first-order clauses with equality, and the abducibles in other words the hypotheses that are allowed to be inferred are boolean combinations of equations constructed over a given finite set of ground terms.

By duality, abduction can be reduced to a consequence-finding problem. We thus show how the inference rules of the superposition calculus can be adapted to obtain a calculus that is deductive complete for ground clauses built on the considered sets of ground terms, thus guaranteeing that all abducible formulas can be generated.

This calculus enjoys the same termination properties as the superposition calculus: in particular, it is terminating on ground extensions of decidable theories of interest in software verification. The number of implicates of a given equational formula is usually huge. We describe techniques for storing sets of abduced clauses efficiently, and show how usual trie-based approaches for representing sets of propositional clauses in a compact way can be adapted and extended in order to denote equational clauses up to equivalence modulo the axioms of equality. We provide algorithms for performing redundancy pruning in an efficient way on such representations.

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We identify hints for improvements and provide lines of on-going and future research. We present an example for application of CHR to automated test data generation and model checking in verification of mission critical software for satellite control. For both data formats there exist schema languages to specify the structure of instance documents, but there is currently no opportunity to translate already existing XML Schema documents into equivalent JSON Schemas. In this paper we introduce an implementation of a language translator. Our approach is based on Prolog and CHR.

By unfolding the XML Schema document into CHR constraints, it is possible to specify the concrete translation rules in a declarative way. CHR is a declarative, concurrent and committed choice rule-based constraint programming language. We extend CHR with multiset comprehension patterns, providing the programmer with the ability to write multiset rewriting rules that can match a variable number of constraints in the store. This enables writing more readable, concise and declarative code for algorithms that coordinate large amounts of data or require aggregate operations.

We then show the soundness of this operational semantics with respect to the abstract semantics. Sometimes that allows to optimize the program or to prove its certain properties automatically. Unfolding is one of the basic operations, which is a meta-extension of one step of the abstract machine executing the program. This paper is interested in unfolding for programs based on pattern matching and manipulating the strings.

The corresponding computation model originates from Markov's normal algorithms and extends this theoretical base. Even though algorithms unfolding programs were being intensively studied for a long time in the context of variety of programming languages, as far as we know, the associative concatenation was stood at the wayside of the stream.

We define a class of term rewriting systems manipulating with strings and describe an algorithm unfolding the programs from the class. The programming language defined by this class is algorithmic complete. Given a word equation, one of the algorithms suggested in this paper results in a description of the corresponding solution set.

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The formalism of NP-nets allows for modeling multi-level multi-agent systems with dynamic structure in a natural way. In this paper we define branching processes and unfoldings for conservative NP-nets, i. We show that NP-nets unfoldings satisfy the fundamental property of unfoldings, and thus can be used for verification of conservative NP-nets in line with classical unfolding methods. A program transformation technique should terminate, return efficient output programs and be efficient itself.

For positive supercompilation ensuring termination requires memoisation of expressions, and these are subsequently used to determine when to perform generalization and folding. For a first-order language, every infinite sequence of transformation steps must include function unfolding, so it is sufficient to memoise only those expressions immediately prior to a function unfolding step. However, for a higher-order language, it is possible for an expression to have an infinite sequence of transformation steps which do not include function unfolding, so memoisation prior to a function unfolding step is not sufficient by itself to ensure termination.

But memoising additional expressions is expensive during transformation and may lead to less efficient output programs due to auxiliary functions. This additional memoisation may happen explicitly during transformation or implicitly via a pre-processing transformation as outlined in previous work by the first author. We introduce a new technique for local driving in higher-order positive supercompilation which obliviates the need for memoising other expressions than function unfolding steps, thereby improving efficiency of both the transformation and the generated programs.

The technique has proven useful on a host of examples. This talk will first give a brief high-level overview of the formal verification of the seL4 microkernel before showing some of its proof techniques in more detail. The aim will be to show examples of libraries and tactics for refinement, invariant, and security proofs for operating systems. JavaScript supports a powerful mix of object-oriented and functional programming, which provides flexibility for the programmers but also makes it difficult to reason about the behavior of the programs without actually running them.

One of the main challenging for program analysis tools is to handle the complex programming patterns that are found in widely used libraries, such as jQuery, without losing critical precision. Another challenge is the use of dynamic language features, such as 'eval'. This talk presents an overview of the challenges and the techniques used in the TAJS research project that aims to develop sound and effective program analysis techniques for JavaScript web applications. JavaScript is a popular, powerful, and highly dynamic programming language. It is the arguably the most widely used and ubiquitous programming language, has a low barrier to entry, and has vast amounts of code in the wild.

JavaScript has grown from a language used primarily to add small amounts of dynamism to web pages into one used for large-scale applications both in and out of the browser--including operating systems and compilers. As such, automated program analysis tools for the language are increasingly valuable.

Almost all of the research to date targets ECMAScript 3, a standard that was succeeded by the most recent version, 5. Much of the research targets well-behaved subsets of JavaScript, eliding the darker corners of the language the bad parts. In this work, we demonstrate how to statically analyze full, modern JavaScript, focusing on uses of the language's so-called bad parts. In particular, we highlight the analysis of scoping, strict mode, property and object descriptors, getters and setters, and eval.

Speed, precision, and soundness are the basic requirements of any static analysis. To obtain soundness, we began with LambdaS5, a small, functional language developed by Guha et al. Thompson states that every group of odd order is solvable, and the proof of this has roughly two parts. The present book provides the character-theoretic second part and completes the proof.

Thomas Peterfalvi also offers a revision of a theorem of Suzuki on split BN-pairs of rank one, a prerequisite for the classification of finite simple groups. This is a systematic mathematical study of differential and more general self-adjoint operators. The famous theorem of Feit and Thompson states that every group of odd order is solvable. This book provides the character-theoretic second part and thus completes the proof. Part I.