Gödel logic: from natural deduction to parallel computation

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies.

Information flow control is a key issue in the large-scale systems, it needs to consider the collaboration between multi-components, execution efficiency, system security, and other aspects, the complexity of information flow control is particularly prominent in the virtualization system. The current information flow control in virtualization system mainly use direct communication and asynchronous communication mode, but they have disadvantage such as complexity of VMM designing, increasing the risk of system security, low efficiency of data transmission. In this paper, we establish a synchronous communication mechanism based on the Collaborative-VMM, three key information flow control were designed in this VMM including I/O processing, interrupt processing, and user interface, it can eliminate the issue of processing delay between VMM and the I/O processing component. The experimental results show that the synchronous communication mechanism can simplify the design of virtualization system and does not reduce the performance of the system at the same time, it is a viable information flow control scheme.

Description logics provide powerful languages for representing and reasoning about knowledge of static application domains. The main strength of description logics is that they offer considerable expressive power going far beyond propositional logic, while reasoning is still decidable. There is a demand to bring the power and character of description logics into the description and reasoning of dynamic application domains which are characterized by actions. In this paper, based on a combination of the propositional dynamic logic PDL, a family of description logics and an action formalism constructed over description logics, we propose a family of dynamic description logics DDL(X @) for representing and reasoning about actions, where X represents well-studied description logics ranging from the to the , and X @ denotes the extension of X with the @ constructor. The representation power of DDL(X @) is reflected in four aspects. Firstly, the static knowledge of application domains is represented as RBoxes and acyclic TBoxes of the description logic X. Secondly, the states of the world and the pre-conditions of atomic actions are described by ABox assertions of the description logic X @, and the post-conditions of atomic actions are described by primitive literals of X @. Thirdly, starting with atomic actions and ABox assertions of X @, complex actions are constructed with regular program constructors of PDL, so that various control structures on actions such as the "Sequence", "Choice", "Any-Order", "Iterate", "If-Then-Else", "Repeat-While" and "Repeat-Until" can be represented. Finally, both atomic actions and complex actions are used as modal operators for the construction of formulas, so that many properties on actions can be explicitly stated by formulas. A tableau-algorithm is provided for deciding the satisfiability of DDL(X @)-formulas; based on this algorithm, reasoning tasks such as the realizability, executability and projection of actions can be effectively carried out. As a result, DDL(X @) not only offers considerable expressive power going beyond many action formalisms which are propositional, but also provides decidable reasoning services for actions described by it.

Distributed Collaborative Environments provide a common user interface that enables collaborative tasks of geographically dispersed users in a specific context. In such environments, many features can be provided, such as asynchronous and synchronous communication mechanisms, a repository for shared resources, and concurrent (synchronous) edition of content. However, it is clear that there is still a shortage concerning the security support in such environments. This becomes even more severe when the context in which these environments are inserted deals with sensitive data, such as patient data and their medical records. The present work proposes a security model for distributed collaborative environments focusing the support for medical researches.

In specifying and designing protocols, it is convenient to use an abstract synchronous communication mechanism. But in practice, asynchronous communication mechanisms cannot be avoided. This paper presents a formal approach based on CSP, and some theoretical results on the specification and design of protocols such that the correctness of the protocols is independent of the communication mechanism used in implementation.

A fully abstract model for the exchange of information in multi-agent systems

A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

14 Hybrid logics

This chapter presents a Java environment for DFCharts based design. The specification of finite state machines, synchronous dataflow graphs and communication mechanisms used in DFCharts is supported by a Java class library. Besides the DFCharts library, Ptolemy is needed for simulation of SDF graphs.

Hybrid fragments of Halpern–Shoham logic and their expressive power

The R programming language is widely used in a variety of domains. It was designed to favor an interactive style of programming with minimal syntactic and conceptual overhead. This design is well suited to data analysis, but a bad fit for tools such as compilers or program analyzers. In particular, R has no type annotations, and all operations are dynamically checked at run-time. The starting point for our work are the two questions: what expressive power is needed to accurately type R code? and which type system is the R community willing to adopt? Both questions are difficult to answer without actually experimenting with a type system. The goal of this paper is to provide data that can feed into that design process. To this end, we perform a large corpus analysis to gain insights in the degree of polymorphism exhibited by idiomatic R code and explore potential benefits that the R community could accrue from a simple type system. As a starting point, we infer type signatures for 25,215 functions from 412 packages among the most widely used open source R libraries. We then conduct an evaluation on 8,694 clients of these packages, as well as on end-user code from the Kaggle data science competition website.

The treatment of propositional logic, and of all following logical systems, is firmly based on a precise and systematic introduction of its language and syntax. Simple as the language may be, the reader will benefit from the systematic use of inductive and recursive procedures employed in this book. It would be hard and inconvenient to establish important properties of the language with a mathematical precision without recourse to the suitable induction principles. As a rule it is a rewarding experience for newcomers to see that what should be the case in formal logic, can indeed be simply proved. The semantics of propositional logic is based on the traditional truth values 0 and 1 (false and true). A rich variety of important and useful principles are established with surprising ease by inductive means. On the whole, the reader will be surprised how much milage one gets out of a well-chosen methodology. Once the notion of truth is made clear, one turns to formal proof methods. The system used here is Gentzen’s Natural Deduction. If ever the adjective “natural” was well chosen, it was here. Natural Deduction allows students to concoct their own derivations without any intricate tricks. Even the completeness theorem—true = derivable—becomes simple in the present setting. Here and in all later chapters, numerous exercises present the reader with occasions to apply the given tools.

Determinism and non-determinism in PDL