Decidability by Resolution for Propositional Modal Logics

The paper shows that satisfiability in a range of popular propositional modal systems can be decided by ordinary resolution procedures. This follows from a general result that resolution combined with condensing, and possibly some additional form of normalization, is a decision procedure for the satisfiability problem in certain so-called path logics. Path logics arise from normal propositional modal logics by the optimized functional translation method. The decision result provides an alternative method of proving decidability for modal logics, as well as closely related systems of artificial intelligence. This alone is not interesting. A more far-reaching consequence of the result has practical value, namely, many standard first-order theorem provers that are based on resolution are suitable for facilitating modal reasoning.     

Decidability and Combination Results for Two Notions of Knowledge in Security Protocols

In formal approaches, messages sent over a network are usually modeled by terms together with an equational theory, axiomatizing the properties of the cryptographic functions (encryption, exclusive or, ...). The analysis of cryptographic protocols requires a precise understanding of the attacker knowledge. Two standard notions are usually considered: deducibility and indistinguishability. Those notions are well-studied and several decidability results already exist to deal with a variety of equational theories. Most of the existing results are dedicated to specific equational theories and only few results, especially in the case of indistinguishability, have been obtained for equational theories with associative and commutative properties (AC)(\textsf{AC}). In this paper, we show that existing decidability results can be easily combined for any disjoint equational theories: if the deducibility and indistinguishability relations are decidable for two disjoint theories, they are also decidable for their union. We also propose a general setting for solving deducibility and indistinguishability for an important class (called monoidal) of equational theories involving AC\textsf{AC} operators. As a consequence of these two results, new decidability and complexity results can be obtained for many relevant equational theories.     

Single Step Tableaux for Modal Logics

Single Step Tableaux (SST) are the basis of a calculus for modal logics that combines different features of sequent and prefixed tableaux into a simple, modular, strongly analytic, and effective calculus for a wide range of modal logics.The paper presents a number of the computational results about SST (confluence, decidability, space complexity, modularity, etc.) and compares SST with other formalisms such as translation methods, modal resolution, and Gentzen-type tableaux. For instance, it discusses the feasibility and infeasibility of deriving decision procedures for SST and translation-based methods by replacing loop checking techniques with simpler termination checks.The complexity of searching for validity and logical consequence with SST and other methods is discussed. Minimal conditions on SST search strategies are proven to yield Pspace (and NPtime for S5 and KD45) decision procedures. The paper also presents the methodology underlying the construction of the correctness and completeness proofs.     

模态逻辑推理的翻译方法

文中研究了模态逻辑推理的翻译法，即把模态逻辑公式按照一定的规则翻译成经典逻辑公式，再用传统的定理器进行推理，文中指出，该方法在理论上保持了正规命题模态逻辑的可判定性，还给出了一些试验结果，说明该方法实际可行的。     

SAT-Based Decision Procedures for Classical Modal Logics

We present a set of SAT-based decision procedures for various classical modal logics. By SAT based, we mean built on top of a SAT solver. We show how the SAT-based approach allows for a modular implementation for these logics. For some of the logics we deal with, we are not aware of any other implementation. For the others, we define a testing methodology that generalizes the 3CNF _K methodology by Giunchiglia and Sebastiani. The experimental evaluation shows that our decision procedures perform better than or as well as other state-of-the-art decision procedures.     

Computing Knowledge in Security Protocols Under Convergent Equational Theories

The analysis of security protocols requires reasoning about the knowledge an attacker acquires by eavesdropping on network traffic. In formal approaches, the messages exchanged over the network are modelled by a term algebra equipped with an equational theory axiomatising the properties of the cryptographic primitives (e.g. encryption, signature). In this context, two classical notions of knowledge, deducibility and indistinguishability, yield corresponding decision problems. We propose a procedure for both problems under arbitrary convergent equational theories. Since the underlying problems are undecidable we cannot guarantee termination. Nevertheless, our procedure terminates on a wide range of equational theories. In particular, we obtain a new decidability result for a theory we encountered when studying electronic voting protocols. We also provide a prototype implementation.     

Quantifier Elimination and Provers Integration

We exploit quantifier elimination in the global design of combined decision and semi-decision procedures for theories over non-disjoint signatures, thus providing in particular extensions of Nelson-Oppen results.     

Undecidability of Multi-modal Hybrid Logics

This paper establishes undecidability of satisfiability for multi-modal logic equipped with the hybrid binder ↓, with respect to frame classes over which the same language with only one modality is decidable. This is in contrast to the usual behaviour of many modal and hybrid logics, whose uni-modal and multi-modal versions do not differ in terms of decidability and, quite often, complexity. The results from this paper apply to a wide range of frame classes including temporally and epistemically relevant ones.     

An Optimal Decision Procedure for Right Propositional Neighborhood Logic

Propositional interval temporal logics are quite expressive temporal logics that allow one to naturally express statements that refer to time intervals. Unfortunately, most such logics turn out to be (highly) undecidable. In order to get decidability, severe syntactic or semantic restrictions have been imposed to interval-based temporal logics to reduce them to point-based ones. The problem of identifying expressive enough, yet decidable, new interval logics or fragments of existing ones that are genuinely interval-based is still largely unexplored. In this paper, we focus our attention on interval logics of temporal neighborhood. We address the decision problem for the future fragment of Neighborhood Logic (Right Propositional Neighborhood Logic, RPNL for short), and we positively solve it by showing that the satisfiability problem for RPNL over natural numbers is NEXPTIME-complete. Then, we develop a sound and complete tableau-based decision procedure, and we prove its optimality.     

Building Decision Procedures for Modal Logics from Propositional Decision Procedures: The Case Study of Modal K(m)

The goal of this paper is to propose a new technique for developing decision procedures for propositional modal logics. The basic idea is that propositional modal decision procedures should be developed on top of propositional decision procedures. As a case study, we consider satisfiability in modal K(m), that is modal K with m modalities, and develop an algorithm, called K , on top of an implementation of the Davis–Putnam–Longemann–Loveland procedure. K is thoroughly tested and compared with various procedures and in particular with the state-of-the-art tableau-based system K . The experimental results show that K outperforms K and the other systems of orders of magnitude, highlight an intrinsic weakness of tableau-based decision procedures, and provide partial evidence of a phase transition phenomenon for K(m).     

自动机,逻辑与博弈

讨论了特定的自动机、自动机的识别能力、逻辑的表达能力和博弈思想的关系。使用博弈思想可以比较容易地证明一元二阶逻辑(S1S和S2S)的可决定性。主要考虑线性时间(linear time)与分支时间(branch time)两种情况，通过这些逻辑与别的时态逻辑的表达能力的等价性可以证明其它逻辑也具有决定性，可以设计相应的自动机去解决模型检查(Model Checking)问题。     

A decision procedure for combinations of propositional temporal logic and other specialized theories

We present a decision procedure for formulae of discrete linear time propositional temporal logic whose propositional part may include assertions in a specialized theory. The combined decision procedure may be viewed as an extension of known decision procedures for quantifier-free theories to theories including temporal logic connectives. A combined decision procedure given by Pratt restricted to linear time temporal logic, runs in polynomial space relative to an oracle for the underlying theory. Our procedure differs from this one in that it can handle assertions containing arbitrary mixtures of global variables, whose values cannot change with time, and state variables, whose values can change with time. This new procedure can also handle assertions containing functions and predicates whose interpretations do not change with time. However, it requires the computation of least and greatest fixed points and has a worse asymptotic running time than that of Pratt. This new procedure has been implemented and seems efficient enough to be practical on simple formulae, although an upper bound derived for the worst case running time is triple exponential in the length of the formula. The techniques used appear to apply to logics other than temporal logic which have decision procedures based on tableaux. Most of this work was done while the author was at SRI International, Menlo Park, California, and at the University of Illinois, Urbana, Illinois, U.S.A. This work was partially supported by the National Science Foundation under grants MCS 81-09831 and MCS 83-07755.     

面向无穷数据的形式模型综述

无穷数据广泛存在于计算机程序和数据库系统中.受到形式验证与数据库两方面的应用需求的推动,面向无穷数据的形式模型已经成为理论计算机科学的一个研究热点.本论文对面向无穷数据的形式模型(逻辑与自动机)进行了相对全面详细的总结.本论文主要按照不同自动机模型对无穷数据的处理方式来组织,主要关注相关判定问题,即自动机的非空性问题、语言包含问题,以及逻辑的可满足性问题,的可判定性与复杂性.     

Model-checking games for fixpoint logics with partial order models

In this paper, we introduce model-checking games that allow local second-order power on sets of independent transitions in the underlying partial order models where the games are played. Since the interleaving semantics of such models is not considered, some problems that may arise when using interleaving representations are avoided and new decidability results for partial order models of concurrency are achieved. The games are shown to be sound and complete, and therefore determined. While in the interleaving case they coincide with the local model-checking games for the μ-calculus, in a partial order setting they verify properties of a number of fixpoint modal logics that can specify, in concurrent systems with partial order semantics, several properties not expressible with the μ-calculus. The games underpin a novel decision procedure for model-checking all temporal properties of a class of infinite and regular event structures, thus improving, in terms of temporal expressive power, previous results in the literature.     

Internal axioms for domain semirings

New axioms for domain operations on semirings and Kleene algebras are proposed. They generalise the relational notion of domain-the set of all states that are related to some other state-to a wide range of models. They are internal since the algebras of state spaces are induced by the domain axioms. They are simpler and conceptually more appealing than previous two-sorted external approaches in which the domain algebra is determined through typing. They lead to a simple and natural algebraic approach to modal logics based on equational reasoning. The axiomatisations have been developed in a new style of computer-enhanced mathematics by automated theorem proving, and the approach itself is suitable for automated systems analysis and verification. This is demonstrated by a fully automated proof of a modal correspondence result for Löb’s formula that has applications in termination analysis.     

Fibring Non-Truth-Functional Logics: Completeness Preservation

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This completeness transfer result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established.     

Structured proof procedures

In this paper, we address the problem of enriching an interactive theorem prover with complex proof procedures. We show that the approach of building complex proof procedures out of deciders for (decidable) quantifier-free theories has many advantages: (i) deciders for quantifier-free theories provide powerful, high level functionalities which greatly simplify the activity of designing and implementing complex and higher level proof procedures; (ii) this approach is of wide applicability since most of the proof procedures are composed by steps of propositional reasoning intermixed with steps carrying out higher level strategical functionalities; (iii) decidability and efficiency are retained on important (decidable) subclasses, while they are often sacrificed by uniform proof strategies for the sake of generality; and finally (iv), from a software engineering perspective, the modularity of the procedures guarantees that any modification in the implementation can be accomplished locally. As a case study, we present and discuss a proof procedure for the existential fragment of first-order logic (prenex existential formulas without function symbols) built on top of a propositional decider.