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Local Search For Satisfiability Modulo Integer Arithmetic Theories 可满足模整数算法理论的局部搜索
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-07-25 DOI: https://dl.acm.org/doi/10.1145/3597495
Shaowei Cai, Bohan Li, Xindi Zhang

Satisfiability Modulo Theories (SMT) refers to the problem of deciding the satisfiability of a formula with respect to certain background first-order theories. In this article, we focus on Satisfiablity Modulo Integer Arithmetic, which is referred to as SMT(IA), including both linear and non-linear integer arithmetic theories. Dominant approaches to SMT rely on calling a CDCL-based SAT solver, either in a lazy or eager flavour. Local search, a competitive approach to solving combinatorial problems including SAT, however, has not been well studied for SMT. We develop the first local-search algorithm for SMT(IA) by directly operating on variables, breaking through the traditional framework. We propose a local-search framework by considering the distinctions between Boolean and integer variables. Moreover, we design a novel operator and scoring functions tailored for integer arithmetic, as well as a two-level operation selection heuristic. Putting these together, we develop a local search SMT(IA) solver called LocalSMT. Experiments are carried out to evaluate LocalSMT on benchmark sets from SMT-LIB. The results show that LocalSMT is competitive and complementary with state-of-the-art SMT solvers, and performs particularly well on those formulae with only integer variables. A simple sequential portfolio with Z3 improves the state-of-the-art on satisfiable benchmark sets from SMT-LIB.

可满足模理论(SMT)是指在一定背景一阶理论下确定公式可满足性的问题。在本文中,我们将重点讨论可满足模整数算法,即SMT(IA),包括线性和非线性整数算法理论。SMT的主要方法依赖于调用基于cdcl的SAT求解器,要么是懒惰的,要么是急切的。局部搜索是解决包括SAT在内的组合问题的一种竞争性方法,但在SMT中尚未得到很好的研究。突破传统框架,直接对变量进行操作,开发了SMT(IA)的首个局部搜索算法。通过考虑布尔变量和整数变量之间的区别,我们提出了一个局部搜索框架。此外,我们还设计了一种适合整数运算的算子和评分函数,以及一种两级操作选择启发式算法。将这些组合在一起,我们开发了一个名为LocalSMT的本地搜索SMT(IA)求解器。在SMT-LIB的基准集上进行了LocalSMT的评估实验。结果表明,LocalSMT与最先进的SMT求解器具有竞争性和互补性,并且在只有整数变量的公式上表现得特别好。使用Z3的简单顺序组合提高了SMT-LIB中可满足基准集的性能。
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引用次数: 0
Inputs, Outputs, and Composition in the Logic of Information Flows 信息流逻辑中的输入、输出和构成
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-07-15 DOI: https://dl.acm.org/doi/10.1145/3604553
Heba Aamer, Bart Bogaerts, Dimitri Surinx, Eugenia Ternovska, Jan Van den Bussche

The logic of information flows (LIF) is a general framework in which tasks of a procedural nature can be modeled in a declarative, logic-based fashion. The first contribution of this paper is to propose semantic and syntactic definitions of inputs and outputs of LIF expressions. We study how the two relate and show that our syntactic definition is optimal in a sense that is made precise. The second contribution is a systematic study of the expressive power of sequential composition in LIF. Our results on composition tie in the results on inputs and outputs, and relate LIF to first-order logic (FO) and bounded-variable LIF to bounded-variable FO.

This paper is the extended version of a paper presented at KR 2020 [2].

信息流逻辑(LIF)是一个通用框架,在该框架中,可以以声明性的、基于逻辑的方式对具有过程性质的任务进行建模。本文的第一个贡献是提出了LIF表达式输入和输出的语义和语法定义。我们研究了这两者之间的关系,并表明我们的句法定义在某种意义上是最优的,它是精确的。第二个贡献是对LIF中顺序构图表达能力的系统研究。我们关于组合的结果与输入和输出的结果相联系,并将LIF与一阶逻辑(FO)和有界变量LIF与有界变量FO联系起来。本文是在KR 2020[2]上发表的一篇论文的扩展版本。
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引用次数: 0
Interpolation Results for Arrays with Length and MaxDiff 具有Length和MaxDiff的数组的插值结果
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-06-09 DOI: https://dl.acm.org/doi/10.1145/3587161
Silvio Ghilardi, Alessandro Gianola, Deepak Kapur, Chiara Naso

In this article, we enrich McCarthy’s theory of extensional arrays with a length and a maxdiff operation. As is well-known, some diff operation (i.e., some kind of difference function showing where two unequal arrays differ) is needed to keep interpolants quantifier free in array theories. Our maxdiff operation returns the max index where two arrays differ; thus, it has a univocally determined semantics.

The length function is a natural complement of such a maxdiff operation and is needed to handle real arrays. Obtaining interpolation results for such a rich theory is a surprisingly hard task. We get such results via a thorough semantic analysis of the models of the theory and of their amalgamation and strong amalgamation properties. The results are modular with respect to the index theory; we show how to convert them into concrete interpolation algorithms via a hierarchical approach realizing a polynomial reduction to interpolation in linear arithmetics endowed with free function symbols.

在本文中,我们用length和maxdiff操作丰富了McCarthy的扩展数组理论。众所周知,在数组理论中,需要一些差分操作(即某种差分函数显示两个不相等数组的不同之处)来保持插值量词的自由。maxdiff操作返回两个数组不同处的最大索引;因此,它具有惟一确定的语义。length函数是maxdiff操作的自然补充,需要处理实际数组。为这样一个丰富的理论获得插值结果是一项非常困难的任务。我们通过对理论模型及其合并和强合并特性进行深入的语义分析,得出了上述结论。所得结果相对于指标理论是模性的;我们展示了如何通过分层方法将它们转换为具体的插值算法,实现了赋予自由函数符号的线性算法中的多项式约简插值。
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引用次数: 0
A Decidable Fragment of First Order Modal Logic: Two Variable Term Modal Logic 一阶模态逻辑的可判定片段:两变量项模态逻辑
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-06-09 DOI: https://dl.acm.org/doi/10.1145/3593584
Anantha Padmanabha, R. Ramanujam

First order modal logic (𝖥𝖮𝖬𝖫) is built by extending First Order Logic (𝖥𝖮) with modal operators. A typical formula is of the form (forall x exists y Box P(x,y)). Not only is 𝖥𝖮𝖬𝖫 undecidable, even simple fragments like that of restriction to unary predicate symbols, guarded fragment and two variable fragment, which are all decidable for 𝖥𝖮 become undecidable for 𝖥𝖮𝖬𝖫. In this paper we study Term Modal logic (𝖳𝖬𝖫) which allows modal operators to be indexed by terms. A typical formula is of the form (forall x exists y~Box _x P(x,y)). There is a close correspondence between 𝖳𝖬𝖫 and 𝖥𝖮𝖬𝖫 and we explore this relationship in detail in the paper.

In contrast to 𝖥𝖮𝖬𝖫, we show that the two variable fragment (without constants, equality) of 𝖳𝖬𝖫 is decidable. Further, we prove that adding a single constant makes the two variable fragment of 𝖳𝖬𝖫 undecidable. On the other hand, when equality is added to the logic, it loses the finite model property.

一阶模态逻辑(𝖥𝖮𝖬𝖫)是通过使用模态运算符扩展一阶逻辑(𝖥𝖮)来构建的。一个典型的公式是(forall x exists y Box P(x,y))。不仅𝖥𝖮𝖬𝖫是不可判定的,就连对于𝖥𝖮来说都是可判定的限制一元谓词符号、保护片段和双变量片段这样的简单片段,对于𝖥𝖮𝖬𝖫来说都是不可判定的。在本文中,我们研究了项模态逻辑(𝖳𝖬𝖫),它允许模态运算符按项索引。一个典型的公式是(forall x exists y~Box _x P(x,y))。𝖳𝖬𝖫和𝖥𝖮𝖬𝖫之间有密切的对应关系,我们在本文中详细探讨了这种关系。与𝖥𝖮𝖬𝖫相反,我们证明了𝖳𝖬𝖫的两个变量片段(没有常数,相等)是可决定的。进一步,我们证明了添加单个常数使𝖳𝖬𝖫的两个变量片段不可确定。另一方面,当在逻辑中加入等式时,它就失去了有限模型的性质。
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引用次数: 0
Living Without Beth and Craig: Definitions and Interpolants in Description and Modal Logics with Nominals and Role Inclusions 没有贝丝和克雷格的生活:名词和角色包含的描述和模态逻辑中的定义和插入
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-06-03 DOI: https://dl.acm.org/doi/10.1145/3597301
Alessandro Artale, Jean Christoph Jung, Andrea Mazzullo, Ana Ozaki, Frank Wolter

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as (mathcal {ALCO} ), (mathcal {ALCH} ) and (mathcal {ALCHOI} ) and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime-complete in the presence of an ontology or the universal modality, and coNExpTime-complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions.

克雷格插值性质(CIP)表明,一个隐含的插值存在,只要它是有效的。投影贝丝可定义性(PBDP)表明,如果一个表述隐式可定义性的公式有效,则存在显式定义。因此,CIP和PBDP将潜在的硬存在问题减少到底层逻辑的蕴涵。带有标称和/或角色包含的描述(和模态)逻辑不享受CIP和PBDP,但是插值和显式定义有许多应用,特别是在概念学习、本体工程和基于本体的数据管理中。在这篇文章中,我们证明,即使没有Beth和Craig,在含有标称和/或角色包含的描述逻辑(如(mathcal {ALCO} ), (mathcal {ALCH} )和(mathcal {ALCHOI} ))和相应的混合模态逻辑中,插值和显式定义的存在是可决定的。然而,没有Beth和Craig的生活使这些问题比蕴涵更难:存在问题在本体或普遍模态的存在下变成2ExpTime-complete,否则变成coNExpTime-complete。如果所有的符号(除了被定义的符号)在定义中被承认,我们也分析了显式定义的存在性。在这种情况下,复杂性取决于是否考虑单个名称或概念名称。最后,我们考虑计算插值和显式定义的问题,如果它们存在,并将复杂度上界证明转化为计算它们的算法,至少对于具有角色包含的描述逻辑。
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引用次数: 0
Faster Property Testers in a Variation of the Bounded Degree Model 一种有界度模型变化中的快速性能测试仪
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-05-10 DOI: https://dl.acm.org/doi/10.1145/3584948
Isolde Adler, Polly Fahey

Property testing algorithms are highly efficient algorithms that come with probabilistic accuracy guarantees. For a property P, the goal is to distinguish inputs that have P from those that are far from having P with high probability correctly, by querying only a small number of local parts of the input. In property testing on graphs, the distance is measured by the number of edge modifications (additions or deletions) that are necessary to transform a graph into one with property P. Much research has focused on the query complexity of such algorithms, i. e., the number of queries the algorithm makes to the input, but in view of applications, the running time of the algorithm is equally relevant.

In (Adler, Harwath, STACS 2018), a natural extension of the bounded degree graph model of property testing to relational databases of bounded degree was introduced, and it was shown that on databases of bounded degree and bounded tree-width, every property that is expressible in monadic second-order logic with counting (CMSO) is testable with constant query complexity and sublinear running time. It remains open whether this can be improved to constant running time.

In this article we introduce a new model, which is based on the bounded degree model, but the distance measure allows both edge (tuple) modifications and vertex (element) modifications. We show that every property that is testable in the classical model is testable in our model with the same query complexity and running time, but the converse is not true. Our main theorem shows that on databases of bounded degree and bounded tree-width, every property that is expressible in CMSO is testable with constant query complexity and constant running time in the new model. Our proof methods include the semilinearity of the neighborhood histograms of databases having the property and a result by Alon (Proposition 19.10 in Lovász, Large networks and graph limits, 2012) that states that for every bounded degree graph (mathcal {G}) there exists a constant size graph (mathcal {H}) that has a similar neighborhood distribution to (mathcal {G}).

It can be derived from a result in (Benjamini et al., Advances in Mathematics 2010) that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the classical model (and hence in the new model). Using our methods, we give an alternative proof that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the new model.

We argue that our model is natural and our meta-theorem showing constant-time CMSO testability supports this.

属性测试算法是一种具有概率准确性保证的高效算法。对于属性P,目标是通过仅查询输入的一小部分局部部分来区分具有P的输入和具有高概率的输入。在图的属性测试中,距离是通过将图转换为具有属性p的图所必需的边修改(添加或删除)的数量来测量的。许多研究都集中在这类算法的查询复杂性上,即算法对输入的查询次数,但从应用角度来看,算法的运行时间同样相关。在(Adler, Harwath, STACS 2018)中,将属性测试的有界度图模型自然扩展到有界度关系数据库,并证明了在有界度和有界树宽的数据库上,每个可用一元二阶计数逻辑(CMSO)表示的属性都是可测试的,查询复杂度不变,运行时间次线性。是否可以将其改进为恒定的运行时间仍然是开放的。在本文中,我们引入了一个新的模型,该模型基于有界度模型,但距离度量允许边(元组)修改和顶点(元素)修改。我们证明了在经典模型中可测试的每一个属性在我们的模型中都是可测试的,并且具有相同的查询复杂度和运行时间,但反之并非如此。我们的主要定理表明,在有界度和有界树宽的数据库上,在新模型中,在查询复杂度和运行时间不变的情况下,CMSO中可表示的所有属性都是可测试的。我们的证明方法包括数据库的邻域直方图的半线性,具有以下性质和Alon的结果(Lovász, Large networks and graph limits, 2012中的命题19.10),该结果表明,对于每个有界度图(mathcal {G}),存在一个恒定大小的图(mathcal {H}),其邻域分布与(mathcal {G})相似。在经典模型中(因此在新模型中),超有限遗传特性在恒定的查询复杂度和恒定的运行时间下是可测试的。利用我们的方法,我们给出了一种替代证明,证明在新模型中,超有限遗传性质在查询复杂度和运行时间不变的情况下是可测试的。我们认为我们的模型是自然的,我们的元定理显示恒定时间CMSO可测试性支持这一点。
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引用次数: 0
Parameterized Complexity of Logic-based Argumentation in Schaefer’s Framework Schaefer框架中基于逻辑论证的参数化复杂性
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-05-10 DOI: https://dl.acm.org/doi/10.1145/3582499
Yasir Mahmood, Arne Meier, Johannes Schmidt

Argumentation is a well-established formalism dealing with conflicting information by generating and comparing arguments. It has been playing a major role in AI for decades. In logic-based argumentation, we explore the internal structure of an argument. Informally, a set of formulas is the support for a given claim if it is consistent, subset-minimal, and implies the claim. In such a case, the pair of the support and the claim together is called an argument. In this article, we study the propositional variants of the following three computational tasks studied in argumentation: ARG (exists a support for a given claim with respect to a given set of formulas), ARG-Check (is a given set a support for a given claim), and ARG-Rel (similarly as ARG plus requiring an additionally given formula to be contained in the support). ARG-Check is complete for the complexity class DP, and the other two problems are known to be complete for the second level of the polynomial hierarchy (Creignou et al. 2014 and Parson et al., 2003) and, accordingly, are highly intractable. Analyzing the reason for this intractability, we perform a two-dimensional classification: First, we consider all possible propositional fragments of the problem within Schaefer’s framework (STOC 1978) and then study different parameterizations for each of the fragments. We identify a list of reasonable structural parameters (size of the claim, support, knowledge base) that are connected to the aforementioned decision problems. Eventually, we thoroughly draw a fine border of parameterized intractability for each of the problems showing where the problems are fixed-parameter tractable and when this exactly stops. Surprisingly, several cases are of very high intractability (para-NP and beyond).

论证是一种完善的形式主义,通过产生和比较论证来处理相互矛盾的信息。几十年来,它一直在人工智能领域发挥着重要作用。在基于逻辑的论证中,我们探索论证的内部结构。非正式地说,如果一组公式是一致的、子集最小的,并且暗示了该断言,那么它就是对给定断言的支持。在这种情况下,支持和主张一起被称为论点。在本文中,我们研究了在论证中研究的以下三个计算任务的命题变体:ARG(相对于给定的一组公式存在对给定断言的支持),ARG- check(给定的一组是对给定断言的支持)和ARG- rel(类似于ARG +需要在支持中包含额外的给定公式)。ARG-Check对于复杂度类DP是完全的,而已知其他两个问题对于多项式层次的第二级是完全的(Creignou et al. 2014 and Parson et al., 2003),因此是高度棘手的。为了分析这种难解性的原因,我们进行了二维分类:首先,我们在Schaefer的框架(STOC 1978)中考虑问题的所有可能的命题片段,然后研究每个片段的不同参数化。我们确定了与上述决策问题相关的合理结构参数(索赔的大小、支持、知识库)的列表。最后,我们彻底地为每个问题绘制了参数化难处理的精细边界,显示了问题在哪里是固定参数可处理的,以及这种情况何时停止。令人惊讶的是,有几个病例的难治性非常高(para-NP及以上)。
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引用次数: 0
Mixed Iterated Revisions: Rationale, Algorithms, and Complexity 混合迭代修订:基本原理、算法和复杂性
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-05-10 DOI: https://dl.acm.org/doi/10.1145/3583071
Paolo Liberatore

Several forms of iterable belief change exist, differing in the kind of change and its strength: some operators introduce formulae, others remove them; some add formulae unconditionally, others only as additions to the previous beliefs; some only relative to the current situation, others in all possible cases. A sequence of changes may involve several of them: for example, the first step is a revision, the second a contraction and the third a refinement of the previous beliefs. The ten operators considered in this article are shown to be all reducible to three: lexicographic revision, refinement, and severe withdrawal. In turn, these three can be expressed in terms of lexicographic revision at the cost of restructuring the sequence. This restructuring needs not to be done explicitly: an algorithm that works on the original sequence is shown. The complexity of mixed sequences of belief change operators is also analyzed. Most of them require only a polynomial number of calls to a satisfiability checker, some are even easier.

存在几种可迭代信念变化的形式,它们在变化的种类和强度上有所不同:一些算子引入公式,另一些算子去掉公式;有些人无条件地添加公式,有些人只是对先前的信念进行补充;有些只与目前的情况有关,有些则适用于所有可能的情况。一系列变化可能包括其中的几个:例如,第一步是修正,第二步是收缩,第三步是对先前信念的改进。本文中考虑的十个操作符都可简化为三个:词典修订、细化和严重退出。反过来,这三个可以用字典修订来表示,代价是重组序列。这种重组不需要显式地完成:这里展示了一种对原始序列起作用的算法。分析了信念变换算子混合序列的复杂性。它们中的大多数只需要对可满足性检查器进行多项式次数的调用,有些甚至更容易。
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引用次数: 0
A Decidable Fragment of First Order Modal Logic: Two Variable Term Modal Logic 一阶模态逻辑的可判定片段:两变量项模态逻辑
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-04-19 DOI: 10.1145/3593584
A. Padmanabha, R. Ramanujam
First order modal logic (𝖥𝖮𝖬𝖫) is built by extending First Order Logic (𝖥𝖮) with modal operators. A typical formula is of the form (forall x exists y Box P(x,y)) . Not only is 𝖥𝖮𝖬𝖫 undecidable, even simple fragments like that of restriction to unary predicate symbols, guarded fragment and two variable fragment, which are all decidable for 𝖥𝖮 become undecidable for 𝖥𝖮𝖬𝖫. In this paper we study Term Modal logic (𝖳𝖬𝖫) which allows modal operators to be indexed by terms. A typical formula is of the form (forall x exists y~Box _x P(x,y)) . There is a close correspondence between 𝖳𝖬𝖫 and 𝖥𝖮𝖬𝖫 and we explore this relationship in detail in the paper. In contrast to 𝖥𝖮𝖬𝖫, we show that the two variable fragment (without constants, equality) of 𝖳𝖬𝖫 is decidable. Further, we prove that adding a single constant makes the two variable fragment of 𝖳𝖬𝖫 undecidable. On the other hand, when equality is added to the logic, it loses the finite model property.
一阶模态逻辑(𝖥𝖮𝖬𝖫)是通过使用模态运算符扩展一阶逻辑(𝖥𝖮)来构建的。一个典型的公式是(forall x exists y Box P(x,y))。不仅𝖥𝖮𝖬𝖫是不可判定的,就连对于𝖥𝖮来说都是可判定的限制一元谓词符号、保护片段和双变量片段这样的简单片段,对于𝖥𝖮𝖬𝖫来说都是不可判定的。在本文中,我们研究了项模态逻辑(𝖳𝖬𝖫),它允许模态运算符按项索引。一个典型的公式是(forall x exists y~Box _x P(x,y))。𝖳𝖬𝖫和𝖥𝖮𝖬𝖫之间有密切的对应关系,我们在本文中详细探讨了这种关系。与𝖥𝖮𝖬𝖫相反,我们证明了𝖳𝖬𝖫的两个变量片段(没有常数,相等)是可决定的。进一步,我们证明了添加单个常数使𝖳𝖬𝖫的两个变量片段不可确定。另一方面,当在逻辑中加入等式时,它就失去了有限模型的性质。
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引用次数: 1
Interval Temporal Logic for Visibly Pushdown Systems 可见下推系统的间隔时间逻辑
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2023-04-17 DOI: https://dl.acm.org/doi/10.1145/3583756
Laura Bozzelli, Angelo Montanari, Adriano Peron

In this article, we introduce and investigate an extension of Halpern and Shoham’s interval temporal logic HS for the specification and verification of branching-time context-free requirements of pushdown systems under a state-based semantics over Kripke structures enforcing visibility of the pushdown operations. The proposed logic, called nested BHS, supports branching-time both in the past and in the future and is able to express non-regular properties of linear and branching behaviours of procedural contexts in a natural way. It strictly subsumes well-known linear time context-free extensions of LTL such as CaRet [4] and NWTL [2]. The main result is the decidability of the visibly pushdown model-checking problem against nested BHS. The proof exploits a non-trivial automata-theoretic construction.

在本文中,我们引入并研究了Halpern和Shoham区间时间逻辑HS的扩展,用于规范和验证下推系统在Kripke结构上基于状态语义的分支时间上下文无关需求,从而增强下推操作的可见性。所提出的逻辑称为嵌套BHS,支持过去和未来的分支时间,能够以自然的方式表达过程上下文的线性和分支行为的非规则属性。它严格地包含了众所周知的线性时间上下文无关的LTL扩展,如CaRet[4]和NWTL[2]。主要结果是针对嵌套BHS的可见下推模型检查问题的可判定性。这个证明利用了一个非平凡的自动机理论构造。
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引用次数: 0
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ACM Transactions on Computational Logic
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