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A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types 语境类型的范畴论视角:从简单类型到依赖类型
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-10-20 DOI: https://dl.acm.org/doi/10.1145/3545115
Jason Z. S. Hu, Brigitte Pientka, Ulrich Schöpp

We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann’s work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad ♭ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et al.

我们描述了Cocon的简单类型变体和简化的依赖类型变体的范畴语义,这是一种上下文模态类型理论,其中框模态介于用于表示高阶抽象语法(HOAS)树的弱函数空间和描述关于它们的(递归)计算的强函数空间之间。Cocon与标准类型理论的不同之处在于,它使用一级上下文和上下文对象来描述语法树,这些语法树相对于给定的假设上下文是封闭的。遵循M. Hofmann的工作,我们使用预层模型来表征HOAS树。令人惊讶的是,这个模型已经提供了必要的结构来为con建模。特别地,我们可以用一个将预页限制在其封闭元素上的共通点来捕获上下文对象。这给出了上下文类型不变量(例如替换不变量)的简单语义特征,并将Cocon标识为预表模型的类型论语法。我们进一步将这一特征扩展到使用具有家族的类别的依赖类型,并表明我们可以在Birkedal等人提出的fitch风格依赖模态类型理论中对没有递归的Cocon片段进行建模。
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引用次数: 0
Syntactic Completeness of Proper Display Calculi 适当显示演算的句法完备性
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-10-20 DOI: https://dl.acm.org/doi/10.1145/3529255
Jinsheng Chen, Giuseppe Greco, Alessandra Palmigiano, Apostolos Tzimoulis

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analyticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e., those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (a.k.a. unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination, and the subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e., by showing that a (cut-free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far, this proof strategy for syntactic completeness has been implemented on a case-by-case base and not in general. In this article, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut-free derivation can be effectively generated that has a specific shape, referred to as pre-normal form.

最近在结构证明理论方面的一项研究旨在探索解析演算的概念(即,那些支持切消的一般和模证明策略的演算),并确定可以根据这些演算捕获的逻辑类。在这种背景下,Wansing引入了适当显示演算的概念,作为证明演算的一种可能的设计框架,其中所需的分析性以一种特别透明的方式实现。近年来,在广义Sahlqvist理论(又称统一对应)的基础上,发展了适当显示逻辑理论(即那些可以用某种适当显示演算等价表示的逻辑)。具体地说,适当显示的逻辑在句法上被表征为由解析归纳公理公理化的逻辑,这些公理可以等价地和算法地转化为解析结构规则,从而得到适当显示的演算具有一组基本性质:健全性、完备性、保守性、切消性和子公式性质。在这种情况下,证明给定的演算是完全的,而不是原始的逻辑,通常是在句法上进行的,即,通过表明在基本系统中逻辑的每个给定公理的(无切割)推导存在,该基本系统已添加了由给定公理算法生成的分析结构规则。然而,到目前为止,这种语法完整性的证明策略是在逐个案例的基础上实现的,而不是一般的。在本文中,我们通过证明任何正态(分布)格展开签名中正确显示逻辑的语法完备性来解决这一差距。具体地说,我们证明了对于每个解析归纳公理,可以有效地生成具有特定形状的无切割导数,称为前范式。
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引用次数: 0
Unifying Operational Weak Memory Verification: An Axiomatic Approach 统一操作弱记忆验证:一种公理化方法
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-10-20 DOI: https://dl.acm.org/doi/10.1145/3545117
Simon Doherty, Sadegh Dalvandi, Brijesh Dongol, Heike Wehrheim

In this article, we propose an approach to program verification using an abstract characterisation of weak memory models. Our approach is based on a hierarchical axiom scheme that captures the observational properties of a memory model. In particular, we show that it is possible to prove correctness of a program with respect to a particular axiom scheme, and we show this proof to suffice for any memory model that satisfies the axioms. Our axiom scheme is developed using a characterisation of weakest liberal preconditions for weak memory. This characterisation naturally extends to Hoare logic and Owicki-Gries reasoning by lifting weakest liberal preconditions (defined over read/write events) to the level of programs. We study three memory models (SC, TSO, and RC11-RAR) as example instantiations of the axioms, then we demonstrate the applicability of our reasoning technique on a number of litmus tests. The majority of the proofs in this article are supported by mechanisation within Isabelle/HOL.

在本文中,我们提出了一种使用弱内存模型的抽象表征来进行程序验证的方法。我们的方法是基于一个层次公理方案,它捕获了一个内存模型的观察属性。特别地,我们证明了它是可能的证明一个程序的正确性关于一个特定的公理方案,我们证明了这个证明足以满足任何内存模型的公理。我们的公理方案是利用弱记忆的最弱自由先决条件的特征来发展的。通过将最弱的自由前提条件(定义在读/写事件上)提升到程序级别,这种特征自然扩展到Hoare逻辑和Owicki-Gries推理。我们研究了三种记忆模型(SC、TSO和RC11-RAR)作为公理的实例,然后我们在许多石蕊测试中展示了我们的推理技术的适用性。本文中的大多数证明都是由Isabelle/HOL内部的机械化支持的。
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引用次数: 0
Witnesses for Answer Sets of Logic Programs 逻辑程序答案集的见证
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-10-20 DOI: 10.1145/3568955
Yisong Wang, Thomas Eiter, Yuanlin Zhang, Fangzhen Lin
In this article, we consider Answer Set Programming (ASP). It is a declarative problem solving paradigm that can be used to encode a problem as a logic program whose answer sets correspond to the solutions of the problem. It has been widely applied in various domains in AI and beyond. Given that answer sets are supposed to yield solutions to the original problem, the question of “why a set of atoms is an answer set” becomes important for both semantics understanding and program debugging. It has been well investigated for normal logic programs. However, for the class of disjunctive logic programs, which is a substantial extension of that of normal logic programs, this question has not been addressed much. In this article, we propose a notion of reduct for disjunctive logic programs and show how it can provide answers to the aforementioned question. First, we show that for each answer set, its reduct provides a resolution proof for each atom in it. We then further consider minimal sets of rules that will be sufficient to provide resolution proofs for sets of atoms. Such sets of rules will be called witnesses and are the focus of this article. We study complexity issues of computing various witnesses and provide algorithms for computing them. In particular, we show that the problem is tractable for normal and headcycle-free disjunctive logic programs, but intractable for general disjunctive logic programs. We also conducted some experiments and found that for many well-known ASP and SAT benchmarks, computing a minimal witness for an atom of an answer set is often feasible.
在本文中,我们考虑答案集编程(ASP)。它是一种说明性的问题解决范例,可用于将问题编码为逻辑程序,其答案集对应于问题的解决方案。它已广泛应用于人工智能等各个领域。假设答案集应该产生原始问题的解决方案,那么“为什么一组原子是答案集”这个问题对于语义理解和程序调试都变得很重要。对于一般的逻辑程序,它已经得到了很好的研究。然而,对于作为普通逻辑程序的实质扩展的析取逻辑程序来说,这个问题并没有得到太多的解决。在本文中,我们提出了析取逻辑程序的约简概念,并展示了它如何为上述问题提供答案。首先,我们展示了对于每个答案集,它的约简为其中的每个原子提供了分辨率证明。然后,我们进一步考虑最小规则集,这些规则集将足以为原子集提供分辨率证明。这些规则集将被称为证人,是本文的重点。我们研究了计算各种证人的复杂性问题,并提供了计算这些证人的算法。特别地,我们证明了这个问题对于普通的和无头循环的析取逻辑程序是可处理的,而对于一般的析取逻辑程序是难以处理的。我们还进行了一些实验,发现对于许多著名的ASP和SAT基准测试,计算答案集原子的最小见证值通常是可行的。
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引用次数: 2
Precise Subtyping for Asynchronous Multiparty Sessions 异步多方会话的精确子类型
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-10-20 DOI: https://dl.acm.org/doi/10.1145/3568422
Silvia Ghilezan, Jovanka Pantović, Ivan Prokić, Alceste Scalas, Nobuko Yoshida

Session subtyping is a cornerstone of refinement of communicating processes: a process implementing a session type (i.e., a communication protocol) T can be safely used whenever a process implementing one of its supertypes T′ is expected, in any context, without introducing deadlocks nor other communication errors. As a consequence, whenever TT′ holds, it is safe to replace an implementation of T′ with an implementation of the subtype T, which may allow for more optimised communication patterns. We present the first formalisation of the precise subtyping relation for asynchronous multiparty sessions. We show that our subtyping relation is sound(i.e., guarantees safe process replacement, as outlined above) and also complete: any extension of the relation is unsound. To achieve our results, we develop a novel session decomposition technique, from fullsession types (including internal/external choices) into single input/output session trees (without choices). We cover multiparty sessions with asynchronousinteraction, where messages are transmitted via FIFO queues (as in the TCP/IP protocol), and prove that our subtyping is both operationally and denotationally precise. Our session decomposition technique expresses the subtyping relation as a composition of refinement relations between single input/output trees, and providing a simple reasoning principle for asynchronous message optimisations.

会话子类型是改进通信进程的基础:在任何上下文中,只要进程期望实现会话类型(即通信协议)T之一,就可以安全地使用实现会话类型T的进程,而不会引入死锁或其他通信错误。因此,只要T≤T '成立,就可以安全地将T '的实现替换为子类型T的实现,这可能允许更优化的通信模式。我们提出了异步多方会话的精确子类型关系的第一个形式化。我们证明我们的子类型关系是健全的(即。,保证安全的过程替换,如上所述),并且是完整的:关系的任何扩展都是不健全的。为了实现我们的结果,我们开发了一种新的会话分解技术,从全会话类型(包括内部/外部选择)到单个输入/输出会话树(没有选择)。我们讨论了异步交互的多方会话,其中消息通过FIFO队列传输(如TCP/IP协议),并证明了我们的子类型在操作和表意上都是精确的。我们的会话分解技术将子类型关系表示为单个输入/输出树之间的细化关系的组合,并为异步消息优化提供了一个简单的推理原则。
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引用次数: 0
A Generalized Realizability and Intuitionistic Logic 广义可实现性与直觉逻辑
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-09-29 DOI: https://dl.acm.org/doi/10.1145/3565367
Aleksandr Yu. Konovalov

Let V be a set of number-theoretical functions. We define a notion of V-realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this paper we prove that Intuitionistic Predicate Calculus is sound with respect to the semantics of V-realizability if and only if some natural conditions for V hold.

设V是一组数论函数。我们定义了谓词公式的V可实现性的概念,用V中函数的指标来解释其蕴涵和全称量词。本文证明了直觉谓词演算对于V可实现性的语义是健全的,当且仅当V的一些自然条件成立。
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引用次数: 0
Reducible Theories and Amalgamations of Models 可还原理论与模型的合并
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-09-29 DOI: 10.1145/3565364
Bahar Aameri, M. Grüninger
Within knowledge representation in artificial intelligence, a first-order ontology is a theory in first-order logic that axiomatizes the concepts in some domain. Ontology verification is concerned with the relationship between the intended models of an ontology and the models of the axiomatization of the ontology. In particular, we want to characterize the models of an ontology up to isomorphism and determine whether or not these models are equivalent to the intended models of the ontology. Unfortunately, it can be quite difficult to characterize the models of an ontology up to isomorphism. In the first half of this article, we review the different metalogical relationships between first-order theories and identify which relationship is needed for ontology verification. In particular, we will demonstrate that the notion of logical synonymy is needed to specify a representation theorem for the class of models of one first-order ontology with respect to another. In the second half of the article, we discuss the notion of reducible theories and show we can specify representation theorems by which models are constructed by amalgamating models of the constituent ontologies.
在人工智能的知识表示中,一阶本体是一阶逻辑中的一种理论,它将某些领域中的概念公理化。本体验证涉及本体的预期模型与本体的公理化模型之间的关系。特别地,我们希望刻画本体的模型直到同构,并确定这些模型是否等价于本体的预期模型。不幸的是,很难对同构的本体模型进行刻画。在本文的前半部分,我们回顾了一阶理论之间不同的元逻辑关系,并确定了本体验证需要哪种关系。特别地,我们将证明需要逻辑同义的概念来指定一个一阶本体的模型类相对于另一个的表示定理。在文章的后半部分,我们讨论了可约理论的概念,并表明我们可以指定表示定理,通过将组成本体的模型合并来构建模型。
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引用次数: 0
Hardness Characterisations and Size-width Lower Bounds for QBF Resolution 硬度特性和QBF分辨率的尺寸宽度下限
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-09-28 DOI: 10.1145/3565286
Olaf Beyersdorff, Joshua Blinkhorn, M. Mahajan, Tomás Peitl
We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) via circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems [16], but leaving open the most important case of QBF resolution. Different from the Frege case, our characterisation uses a new version of decision lists as its circuit model, which is stronger than the CNFs the system works with. Our decision list model is well suited to compute countermodels for QBFs. Our characterisation works for both Q-Resolution and QU-Resolution. Using our characterisation, we obtain a size-width relation for QBF resolution in the spirit of the celebrated result for propositional resolution [4]. However, our result is not just a replication of the propositional relation—intriguingly ruled out for QBF in previous research [12]—but shows a different dependence between size, width, and quantifier complexity. An essential ingredient is an improved relation between the size and width of term decision lists; this may be of independent interest. We demonstrate that our new technique elegantly reproves known QBF hardness results and unifies previous lower-bound techniques in the QBF domain.
我们通过电路复杂性为量化布尔公式(QBF)的分辨率提供了证明大小的严格表征。这种表征以前是为QBF Frege系统的层次结构[16]获得的,但保留了QBF分辨率的最重要情况。与Frege案例不同,我们的描述使用了新版本的决策列表作为其电路模型,该模型比系统使用的CNF更强。我们的决策列表模型非常适合计算QBF的反模型。我们的特征描述适用于Q-Resolution和QU Resolution。使用我们的表征,我们根据命题分辨率的著名结果[4]的精神,获得了QBF分辨率的大小-宽度关系。然而,我们的结果不仅是命题关系的复制——有趣的是,在之前的研究[12]中,QBF被排除在外——而且显示了大小、宽度和量词复杂性之间的不同依赖性。一个重要因素是改进了任期决定清单的大小和宽度之间的关系;这可能具有独立的利益。我们证明,我们的新技术完美地再现了已知的QBF硬度结果,并统一了QBF领域中以前的下界技术。
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引用次数: 0
Extensible Proof Systems for Infinite-State Systems 无限状态系统的可扩展证明系统
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-07-26 DOI: 10.48550/arXiv.2207.12953
J. Keiren, R. Cleaveland
This paper revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.
本文回顾了证明无限状态标记转移系统中的状态集满足模态模微积分公式的证明系统的完备性和完备性,以便开发允许在该逻辑中无缝包含新特征的证明技术。我们的方法依赖于格理论中的新结果,这些结果给出了完全格上单调函数的最大不动点和最小不动点的构造特征。我们展示了如何使用这些结果来推理Bradfield和Stirling的完整表格法。我们还展示了我们的格理论基础的灵活性如何简化了对可选系统类的基于表的证明策略的推理。特别地,我们将模态微积分推广到时间模态,并证明了所得到的表法对于时间跃迁系统是健全完备的。
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引用次数: 1
On Proof Complexity of Resolution over Polynomial Calculus 多项式微积分解析度的证明复杂性
IF 0.5 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2022-07-22 DOI: https://dl.acm.org/doi/10.1145/3506702
Erfan Khaniki
<p>The proof system <sans-serif>Res (PC</sans-serif><sub><i>d,R</i></sub>) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree <i>d</i> multivariate polynomials over a ring <i>R</i> with Boolean variables. Proving super-polynomial lower bounds for the size of <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub>1,<i>R</i></sub>)-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub>1,𝔽</sub>) when 𝔽 is a finite field, such as 𝔽<sub>2</sub>. In this article, we investigate <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d,R</i></sub>) and tree-like <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d,R</i></sub>) and prove size-width relations for them when <i>R</i> is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows:<p><table border="0" list-type="ordered" width="95%"><tr><td valign="top"><p>(1)</p></td><td colspan="5" valign="top"><p>We prove almost quadratic lower bounds for <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d</i></sub>,𝔽)-refutations for every fixed <i>d</i>. The new lower bounds are for the following CNFs:</p><p><table border="0" list-type="ordered" width="95%"><tr><td valign="top"><p>(a)</p></td><td colspan="5" valign="top"><p>Mod <i>q</i> Tseitin formulas (<i>char</i>(𝔽)≠ <i>q</i>) and Flow formulas,</p></td></tr><tr><td valign="top"><p>(b)</p></td><td colspan="5" valign="top"><p>Random <i>k</i>-CNFs with linearly many clauses.</p></td></tr></table></p></td></tr><tr><td valign="top"><p>(2)</p></td><td colspan="5" valign="top"><p>We also prove super-polynomial (more than <i>n</i><sup><i>k</i></sup> for any fixed <i>k</i>) and also exponential (2<i><sup>nϵ</sup></i> for an ϵ > 0) lower bounds for tree-like <sans-serif>Res</sans-serif>(<sans-serif>PC</sans-serif><sub><i>d</i>,𝔽</sub>)-refutations based on how big <i>d</i> is with respect to <i>n</i> for the following CNFs:</p><p><table border="0" list-type="ordered" width="95%"><tr><td valign="top"><p>(a)</p></td><td colspan="5" valign="top"><p>Mod <i>q</i> Tseitin formulas (<i>char</i>(𝔽)≠ <i>q</i>) and Flow formulas,</p></td></tr><tr><td valign="top"><p>(b)</p></td><td colspan="5" valign="top"><p>Random <i>k</i>-CNFs of suitable densities,</p></td></tr><tr><td valign="top"><p>(c)</p></td><td colspan="5" valign="top"><p>Pigeonhole principle and Counting mod <i>q</i> principle.</p></td></tr></table></p></td></tr></table></p> The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case <i>d</i>=1. The lower bounds for the tree-like systems were known for the case <i>d</i>=1 (except for the Counting mod <i>q</i> principle, in which
证明系统Res (PCd,R)是分辨力证明系统的自然扩展,该证明系统用布尔变量环R上d次多元多项式的析取来代替文字的析取。Res(PC1,R)大小的超多项式下界的证明是命题证明复杂性中的一个重要问题。当Res(PC1,∈)是有限域时,这种下界的存在性是开放的,例如𝔽2。本文研究了R (PCd,R)和树状R (PCd,R),并证明了它们在R是有限环时的大小-宽度关系。作为一个应用,我们证明了每个有限域的新下界和一些已知的下界,如下:(1)我们证明了Res(PCd)的几乎二次下界,对于每个固定的d,我们证明了Res(PCd)-反驳的近似二次下界。新的下界适用于以下cnf:(a)Mod q tseittin公式(char(∈)≠q)和Flow公式,(b)具有线性多子句的随机k- cnf。(2)我们还证明了超多项式(对于任何固定k大于nk)和指数(对于一个λ >0)以下CNFs的树状Res(PCd,∈)-基于d相对于n的多大的反驳的下界:(a)模q tsetin公式(char(∈)≠q)和Flow公式,(b)合适密度的随机k-CNFs,(c)鸽洞原理和计数模q原理。类dag系统的下界是这些系统的第一个非平凡下界,包括d=1的情况。在d=1的情况下,树形系统的下界是已知的(计数模q原则除外,在这种情况下,d>我也很出名)。我们的下界将这些结果扩展到d>并给出d=1的新证明。
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引用次数: 0
期刊
ACM Transactions on Computational Logic
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