Grigori Mints, Thomas Studer
We establish syntactic cut-elimination for the one-variable fragment of the modal mu-calculus. Our method is based on a recent cut-elimination technique by Mints that makes use of Buchholz' Omega-rule.
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On the cut-elimination of the modal $μ$-calculus: Linear Logic to the rescue
本文借助线性逻辑及其指数模态构造线性化系统 \muLLmodinf{}，并通过线性翻译证明非良基模态 $μ$-演算的语形切割消除。
Variable Elimination as Rewriting in a Linear Lambda Calculus
本文在理想化概率函数式语言（线性简单类型λ-演算）中，将经典精确推理算法变量消元（VE）形式化为项重写过程，通过交换和嵌套let-in表达式将全局定义转为局部定义，并关键利用线性类型，以理解VE作为程序上的算法。
Cut-Elimination for the Bimodal Logic GR
本文提出GR双模逻辑的超序列演算，证明了切消定理。
We give a linear nested sequent calculus for the basic normal tense logic Kt. We show that the calculus enables backwards proof-search, counter-model construction and syntactic cut-elimination. Linear nested sequents thus provide the minimal amount of nesting necessary to provide an adequate proof-theory for modal logics containing converse. As a bonus, this yields a cut-free calculus for symmetric modal logic KB.
We show how Leibnitz.s indiscernibility principle and Gentzen's original work lead to extensions of the sequent calculus to first order logic with equality and investigate the cut elimination property. Furthermore we discuss and improve the nonlengthening property of Lifshitz and Orevkov.
A Simple Proof That Super-Consistency Implies Cut Elimination
本文提出超一致性在模演绎中蕴含消切割的简化直接证明，并与涉及 V-复形的高阶逻辑定理比较。
Urban and Bierman introduced a calculus of proof terms for the sequent calculus LK with a strongly normalizing reduction relation. We extend this calculus to simply-typed higher-order logic with inferences for induction and equality, albeit without strong normalization. We implement thiscalculus in GAPT, our library for proof transformations. Evaluating the normalization on both artificial and real-world benchmarks, we show that this algorithm is typically several orders of magnitude faster than the existing Gentzen-like cut-reduction, and an order of magnitude faster than any other cut-elimin…
We study cut elimination for a multifocused variant of full linear logic in the sequent calculus. The multifocused normal form of proofs yields problems that do not appear in a standard focused system, related to the constraints in grouping rule instances in focusing phases. We show that cut elimination can be performed in a sensible way even though the proof requires some specific lemmas to deal with multifocusing phases, and discuss the difficulties arising with cut elimination when considering normal forms of proofs in linear logic.
We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditio…
Confluence as a cut elimination property
本文比较重写系统的合流性与证明论的切消概念，证明项重写系统的合流性质等价于其关联逻辑系统的切消性质，但该等价性不适用于直接重写原子命题的系统。
Cut-restriction: from cuts to analytic cuts
针对多数非经典逻辑相继式演算不满足切消定理的问题，提出切限制方法，将任意切割限制为解析切割，适用于满足易检条件的演算，统一涵盖现有结果并建立新的解析子公式性质。
We give a new proof of the decidability of reachability in alternating pushdown systems, showing that it is a simple consequence of a cut-elimination theorem for some natural-deduction style inference systems. Then, we show how this result can be used to extend an alternating pushdown system into a complete system where for every configuration $A$, either $A$ or $\neg A$ is provable.
Axioms vs. rewrite rules: from completeness to cut elimination
本文研究自动定理证明中证明搜索与重写结合时的完备性问题，指出其关键在于为演绎与计算结合的扩展证明概念建立切割消去，从而揭示自动定理证明与证明论之间的新联系。
A new calculus for intuitionistic Strong Löb logic: strong termination and cut-elimination, formalised
本文为直觉主义强Löb逻辑提出新相继式演算，实现强终止的向后搜索与句法消割，并在Coq中形式化验证。
Cut-elimination is the bedrock of proof theory. It is the algorithm that eliminates cuts from a sequent calculus proof that leads to cut-free calculi and applications. Cut-elimination applies to many logics irrespective of their semantics. Such is its influence that whenever cut-elimination is not provable in a sequent calculus the invariable response has been a move to a richer proof system to regain it. In this paper we investigate a radically different approach to the latter: adapting age-old cut-elimination to restrict the shape of the cut-formulas when elimination is not possible. We tack…
Cut elimination for the alternation-free modal mu-calculus
本文为模态μ-演算的无交替片段提出一种基于循环证明系统的语法切消去过程，利用多切和良拟序理论直接将带切证明转换为无切证明并保证终止。
IMELL Cut Elimination with Linear Overhead
本研究针对直觉主义乘法指数线性逻辑的切消问题，为指数替换演算引入抽象机，完善了已有结果，证明该机器实现好策略且能在线性开销内得到无切证明。
Syntactic Cut-Elimination for Provability Logic GL via Nested Sequents
本文针对可证性逻辑GL切割消去终止性证明困难的问题，提出基于嵌套矢列式的语法化切割消去方法，并借鉴Kushida（2020）的思想，仅用直接归纳避免额外度量和复杂重写，从而消除Poggiolesi（2009）证明中的歧义。
We associate to every proof structure in multiplicative linear logic an ideal which represents the logical content of the proof as polynomial equations. We show how cut-elimination in multiplicative proof nets corresponds to instances of the Buchberger algorithm for computing Gröbner bases in elimination theory.
Infinitary Cut-Elimination for Non-Wellfounded Parsimonious Linear Logic
本文研究节俭线性逻辑的非良基证明系统，提出基于有限逼近的无穷消切，证明其在进展准则下产生良定义证明并保持正则性，且给出关系模型指称语义。
A uniform cut-elimination theorem for linear logics with fixed points and super exponentials
针对轻量线性逻辑变体切消定理证明冗余的问题，结合归约翻译法与充分条件判定法，为带不动点的参数化线性逻辑系统提出统一切消定理，覆盖多类指数联结词系统。
Making progress: Reducibility Candidates and Cut Elimination in the Ill-founded Realm
本文提出两种基于可约候选者技术的非良基μMALL切消论证，通过定义性质保证进步性，利用拓扑内部闭集概念完成第二个论证。
Phase Semantic Cut-elimination for Intuitionistic Linear Logic with Least and Greatest Fixed Points
研究具有最小和最大不动点的直觉主义命题乘加线性逻辑（$\mu$IMALL）的消去切割定理，通过定义相位语义，证明其可靠性与无切割完全性，改进前人论证方法来达成该定理的证明。
Undecidability of the Emptiness Problem of Deterministic Propositional While Programs with Graph Loop: Hypothesis Elimination Using Loops
本文证明带图循环的确定性命题while程序的空性问题不可判定且为 $\mathrmΠ^{0}_{1}$-完全，通过引入基于循环的假设消除方法从周期多米诺问题补问题进行归约，并进一步得出正关系演算等式理论的不可判定性及存在关系演算空性问题的 PSPACE-完全性。
We consider an extension of bi-intuitionistic logic with the traditional modalities from tense logic Kt. Proof theoretically, this extension is obtained simply by extending an existing sequent calculus for bi-intuitionistic logic with typical inference rules for the modalities used in display logics. As it turns out, the resulting calculus, LBiKt, seems to be more basic than most intuitionistic tense or modal logics considered in the literature, in particular, those studied by Ewald and Simpson, as it does not assume any a priori relationship between the diamond and the box modal operators. We…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles are based on a proof theoretic (rather than set-theoretic) notion of definition. Definitions are akin to logic programs, where the left and right rules for defined atoms allow one to view theories as "closed" or defining fixed points. The use of definitions and free equality makes it possible to reason intentionally about syntax. We add in a consistent way …
On the Cut Elimination of Weak Intuitionistic Tense Logic
本文提出一种通过分裂收缩规则和切割规则来证明弱直觉时态逻辑切割消去的新方法，并展望其未来理论扩展与应用。
An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets
本文研究MLL证明网并行消去的组合性质，通过路径约束保证无限加权和归约的有限性，并证明该约束适用于MELL泰勒展开中的微分网，为线性逻辑定量语义提供充分条件。
The failure of cut-elimination in cyclic proof for first-order logic with inductive definitions
本文证明带归纳定义的一阶逻辑循环证明系统中切割消除不成立，通过构造一个必须依赖切割规则才能证明的相继式实例，证实了该系统无法消除切割的猜想。
Cut elimination for Cyclic Proofs: A Case Study in Temporal Logic
本文为扩展了“最终”时序算子的模态逻辑的循环矢列演算提供切消过程，将归约切消方法适配至循环演算，能直接将循环证明转化为无切循环证明而无需正则化机制。
This paper presents a cut-elimination proof for the logic $LG^ω$, which is an extension of a proof system for encoding generic judgments, the logic $\FOLDNb$ of Miller and Tiu, with an induction principle. The logic $LG^ω$, just as $\FOLDNb$, features extensions of first-order intuitionistic logic with fixed points and a ``generic quantifier'', $\nabla$, which is used to reason about the dynamics of bindings in object systems encoded in the logic. A previous attempt to extend $\FOLDNb$ with an induction principle has been unsuccessful in modeling some behaviours of bindings in inductive specif…
We present a cut elimination argument that witnesses the conservativity of the compositional axioms for truth (without the extended induction axiom) over any theory interpreting a weak subsystem of arithmetic. In doing so we also fix a critical error in Halbach's original presentation. Our methods show that the admission of these axioms determines a hyper-exponential reduction in the size of derivations of truth-free statements.
Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ``cut-free'' sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic…
Compression for Coinductive Rewriting and the Cut-Elimination of Non-Wellfounded Proofs
本文提出了一种通用方法，用于处理混合归纳与共归纳的语法对象，应用于无穷重构及非非良好基础证明的切消。
We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a semantic version of Friedman's A-translation, showing that it preserves the structure of pseudo-Heyting algebra, our semantic framework. Then we relate the interpretation of a theory in the A-translated a…
Exponentials as Substitutions and the Cost of Cut Elimination in Linear Logic
本文提出指数替换演算（ESC），基于显式替换思想为IMELL切割消除提供新表示，设计了具有子项性质的切割消除策略，首次为无约束指数的切割消除提供多项式代价模型，并证明了无类型合流与有类型强规范化。
Cut elimination for Zermelo set theory
本文通过将策梅洛集合论重构为指向图理论并将公理转化为重写规则，在演绎模中实现了其直觉主义片段的正规化，并证明了所得理论是原集合论的保守扩张。
Proof schemata are a variant of LK-proofs able to simulate various induction schemes in first-order logic by adding so called proof links to the standard first-order LK-calculus. Proof links allow proofs to reference proofs thus giving proof schemata a recursive structure. Unfortunately, applying reductive cut- elimination is non-trivial in the presence of proof links. Borrowing the concept of lazy instantiation from functional programming, we evaluate proof links locally allowing reductive cut-elimination to proceed past them. Though, this method cannot be used to obtain cut-free proof schema…
A study of cut-elimination for a non-labelled cyclic proof system for propositional dynamic logics
本文研究了命题动态逻辑中非标记循环证明系统的消去问题，提出了序数演算和循环证明系统，并证明了其正确性和完备性，同时展示了消去失败及限制后的消去性质。
A Semantic Proof of Generalised Cut Elimination for Deep Inference
本文提出一种语义方法，简洁证明深度推理系统MAV的广义消割性质，并在Agda中机械化证明，同时给出可靠性与完备性证明。
This paper explores the connection between two central results in the proof theory of classical logic: Gentzen's cut-elimination for the sequent calculus and Herbrands "fundamental theorem". Starting from Miller's expansion-tree-proofs, a highly structured way presentation of Herbrand's theorem, we define a calculus of weakening-free proof nets for (prenex) first-order classical logic, and give a weakly-normalizing cut-elimination procedure. It is not possible to formulate the usual counterexamples to confluence of cut-elimination in this calculus, but it is nonetheless nonconfluent, lending c…
It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the alge- braic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $μ$-calculus based on intuitionistic logic is trivial, every $μ$-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm…
In this paper we present a constructive proof of cut elimination for a system of full second order logic with the structural rules absorbed and using sets instead of sequences. The standard problem of the cutrank growth is avoided by using a new parameter for the induction, the cutweight. This technique can also be applied to first order logic.
A study for recovering the cut-elimination property in cyclic proof systems by restricting the arity of inductive predicates
本文研究循环证明系统中消切性质失效的原因，证明即使将语言限制为一元归纳谓词和一元函数，消切性质在简单循环证明系统中仍不成立，改进了先前需三元谓词的反例。
Inducing syntactic cut-elimination for indexed nested sequents
本文通过建立标记相继式与索引嵌套相继式的对应关系，为索引嵌套相继式演算引入句法切割消去证明，并首次为中间逻辑构造了无切割的索引嵌套相继式演算。
This paper employs the linear nested sequent framework to design a new cut-free calculus LNIF for intuitionistic fuzzy logic--the first-order Gödel logic characterized by linear relational frames with constant domains. Linear nested sequents--which are nested sequents restricted to linear structures--prove to be a well-suited proof-theoretic formalism for intuitionistic fuzzy logic. We show that the calculus LNIF possesses highly desirable proof-theoretic properties such as invertibility of all rules, admissibility of structural rules, and syntactic cut-elimination.
We describe an algorithmic method of proof compression based on the introduction of Pi_2-cuts into a cut-free LK-proof. The current approach is based on an inversion of Gentzen s cut-elimination method and extends former methods for introducing Pi_1-cuts. The Herbrand instances of a cut-free proof pi of a sequent S are described by a grammar G which encodes substitutions defined in the elimination of quantified cuts. We present an algorithm which, given a grammar G, constructs a Pi_2-cut formula A and a proof phi of S with one cut on A. It is shown that, by this algorithm, we can achieve an ex…
It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $μ$-calculus based on intuitionistic logic is trivial, every $μ$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to …
Partial Redundancy in Saturation
本文提出基于部分子句和冗余公式的新冗余概念以改进饱和证明搜索，设计了反驳完备的叠加演算PaRC并在Vampire中实现，成功解决了24个此前未被任何证明器解决的TPTP问题。
Cut elimination for propositional cyclic proof systems with fixed-point operators
本文研究带不动点算子或归纳定义的命题循环证明系统，证明其与无穷式系统可证明性等价，并因此具有无切割完备性。
The logic of bunched implications (BI) is a substructural logic that forms the backbone of separation logic, the much studied logic for reasoning about heap-manipulating programs. Although the proof theory and metatheory of BI are mathematically involved, the formalization of important metatheoretical results is still incipient. In this paper we present a self-contained formalized, in the Coq proof assistant, proof of a central metatheoretical property of BI: cut elimination for its sequent calculus. The presented proof is *semantic*, in the sense that is obtained by interpreting sequents in a…
In the modal mu-calculus, a formula is well-formed if each recursive variable occurs underneath an even number of negations. By means of De Morgan's laws, it is easy to transform any well-formed formula into an equivalent formula without negations -- its negation normal form. Moreover, if the formula is of size n, its negation normal form of is of the same size O(n). The full modal mu-calculus and the negation normal form fragment are thus equally expressive and concise. In this paper we extend this result to the higher-order modal fixed point logic (HFL), an extension of the modal mu-calculus…
Public announcement logic(PAL) is an extension of epistemic logic (EL) with some reduction axioms. In this paper, we propose a cut-free labelled sequent calculus for PAL, which is an extension of that for EL with sequent rules adapted from the reduction axioms. This calculus admits cut and allows terminating proof search.
As an attempt to uncover the topological nature of composition of strategies in game semantics, we present a ``topological'' game for Multiplicative Additive Linear Logic without propositional variables, including cut moves. We recast the notion of (winning) strategy and the question of cut elimination in this context, and prove a cut elimination theorem. Finally, we prove soundness and completeness. The topology plays a crucial role, in particular through the fact that strategies form a sheaf.
A MALL Geometry of Interaction Based on Indexed Linear Logic
本文为乘法-加法线性逻辑构建基于索引线性逻辑的交互几何，扩展了Haghverdi-Scott范畴框架，利用索引处理加法消割的叠加、擦除和收缩，并证明执行公式沿消割不变，收敛到无割证明的指称。
This paper presents general syntactic conditions ensuring the strong normalization and the logical consistency of the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. On the one hand, the Calculus of Constructions is a powerful type system in which one can formalize the propositions and natural deduction proofs of higher-order logic. On the other hand, rewriting is a simple and powerful computation paradigm. The combination of both allows, among other things, to develop formal proofs with a r…
Buchholz' Omega-rule is a way to give a syntactic, possibly ordinal-free proof of cut elimination for various subsystems of second order arithmetic. Our goal is to understand it from an algebraic point of view. Among many proofs of cut elimination for higher order logics, Maehara and Okada's algebraic proofs are of particular interest, since the essence of their arguments can be algebraically described as the (Dedekind-)MacNeille completion together with Girard's reducibility candidates. Interestingly, it turns out that the $Ω$-rule, formulated as a rule of logical inference, finds its algebra…
The cut-elimination method CERES (for first- and higher-order classical logic) is based on the notion of a characteristic clause set, which is extracted from an LK-proof and is always unsatisfiable. A resolution refutation of this clause set can be used as a skeleton for a proof with atomic cuts only (atomic cut normal form). This is achieved by replacing clauses from the resolution refutation by the corresponding projections of the original proof. We present a generalization of CERES (called CERESs) to first-order proof schemata and define a schematic version of the sequent calculus called LK…
This paper is a structured introduction to Light Affine Logic, and to its intuitionistic fragment. Light Affine Logic has a polynomially costing cut elimination (P-Time correctness), and encodes all P-Time Turing machines (P-Time completeness). P-Time correctness is proved by introducing the Proof nets for Intuitionistic Light Affine Logic. P-Time completeness is demonstrated in full details thanks to a very compact program notation. On one side, the proof of P-Time correctness describes how the complexity of cut elimination is controlled, thanks to a suitable cut elimination strategy that exp…
In previous work, an attempt was made to apply the schematic CERES method [8] to a formal proof with an arbitrary number of Π 2 cuts (a recursive proof encapsulating the infinitary pigeonhole principle) [5]. However the derived schematic refutation for the characteristic clause set of the proof could not be expressed in the formal language provided in [8]. Without this formalization a Herbrand system cannot be algorithmically extracted. In this work, we provide a restriction of the proof found in [5], the ECA-schema (Eventually Constant Assertion), or ordered infinitary pigeonhole principle, w…
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