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Varieties of topological quasi-Boolean algebras in the vicinity of pre-rough algebras [28, 29] are expanded to residuated algebraic structures by introducing a new implication operation and its residual in these structures. Sequent calculi for some classes of residuated algebraic structures are established. These sequent calculi have the strong finite model property which yields the decidability of the word problem for corresponding classes of algebraic structures.

A finitely alternative normal tense logic $$T_{n,m}$$ is a normal tense logic characterized by frames in which every point has at most n future alternatives and m past alternatives. The structure of the lattice $$\Lambda (T_{1,1})$$ is described. There are $$\aleph _0$$ logics in $$\Lambda (T_{1,1})$$ without the finite model property (FMP), and only one pretabular logic in $$\Lambda (T_{1,1})$$ . There are $$2^{\aleph _0}$$ logics in $$\Lambda (T_{1,1})$$ which are not finitely axiomatizable. For $$nm\ge 2$$ , there are $$2^{\aleph _0}$$ logics in $$\Lambda (T_{n,m})$$ without the FMP, and infinitely many pretabular extensions of $$T_{n,m}$$ .

The language of Belnap–Dunn modal logic $${\mathscr {L}}_0$$ expands the language of Belnap–Dunn four-valued logic (having constant symbols for the values 0 and 1) with the modal operator $$\Box $$ . We introduce the polarity semantics for $${\mathscr {L}}_0$$ and its two expansions $${\mathscr {L}}_1$$ and $${\mathscr {L}}_2$$ with value operators. The local finitary consequence relation $$\models _4^k$$ in the language $${\mathscr {L}}_k$$ with respect to the class of all frames is axiomatized by a sequent system $$\mathsf {S}_k$$ where $$k=0, 1, 2$$ . We prove by using translations between sequents and formulas that these languages under the polarity semantics have the same expressive power on the level of frames with the language $${\mathscr {L}}_0$$ under the relational semantics for classical modal logic.

Through theoretical calculation, the stress and deformation of surrounding rock can be analyzed, which can provide guidance for support design and optimization of soft rock roadway. In this paper, theoretical solutions for both the optimal support pressure and the allowable maximum displacement of surrounding rock are derived based on the Drucker-Prager (DP) yield criteria and the steady creep criterion expressed by the third invariant of deviator stress. The DP criteria with different parameters are compared and analyzed by an engineering example. Then, based on the calculation results, the effects of long-term strength, cohesion and internal friction angle of soft rock on the maximum plastic zone radius and allowable maximum displacement of roadway are discussed. The results show that the optimal support solution of soft rock roadway based on DP criteria can not only reflect the intermediate principal stress reasonably, but also can compare and discuss the influence of different DP criteria on the calculation results. The higher the long-term strength of the roadway surrounding rock is, the smaller the optimal support force is and the larger the allowable maximum displacement is. When the calculated long-term strength of soft rock can ensure that the deformation of the roadway does not exceed the allowable maximum displacement, the roadway can maintain long-term stability without support. With the increase of the cohesion or internal friction angle of soft rock, the radius of plastic zone decreases gradually, and the allowable maximum displacement is reduced by degrees. Through grouting and other means to improve the strength of surrounding rock can effectively reduce the roadway deformation and save support costs.

The minimal weakening $${{\textsf {N}}}_0$$ of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of $${{\textsf {N}}}_0$$ with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of $${{\textsf {N}}}_0$$ . These logics have the finite model property and they are decidable.

Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted byPC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to presentPCas a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which is a residuation, inPC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection betweenPCand the alpha system.

The global consequence relation of a normal modal logic $$\Lambda $$ is formulated as a global sequent calculus which extends the local sequent theory of $$\Lambda $$ with global sequent rules. All global sequent calculi of normal modal logics admits global cut elimination. This property is utilized to show that decidability is preserved from the local to global sequent theories of any normal modal logic over $$\mathsf {K4}$$ . The preservation of Craig interpolation property from local to global sequent theories of any normal modal logic is shown by proof-theoretic method.

A deterministic weakening $$\mathsf {DW}$$ of the Belnap–Dunn four-valued logic $$\mathsf {BD}$$ is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic $$\mathsf {DW}$$ is formalized as a sequent calculus. The completeness and decidability of $$\mathsf {DW}$$ with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for $$\mathsf {DW}$$ is described, and found to be a subvariety of Berman’s variety $$\mathcal {K}_{1,2}$$ . Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that $$\mathsf {DW}$$ is the logic of a nine-element distributive lattice with a negation. Moreover, $$\mathsf {BD}$$ is embedded into $$\mathsf {DW}$$ by Glivenko’s double-negation translation.

We describe Peirce’s 1903 system of modal gamma graphs, its transformation rules of inference, and the interpretation of the broken-cut modal operator. We show that Peirce proposed the normality rule in his gamma system. We then show how various normal modal logics arise from Peirce’s assumptions concerning the broken-cut notation. By developing an algebraic semantics we establish the completeness of fifteen modal logics of gamma graphs. We show that, besides logical necessity and possibility, Peirce proposed an epistemic interpretation of the broken-cut modality, and that he was led to analyze constructions of knowledge in the style of epistemic logic.

Charles Peirce's alpha system $$\mathfrak {S}_\alpha $$S? is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of $$\mathfrak {S}_\alpha $$S? is given in terms of two embedding theorems: the system $$\mathfrak {S}_\alpha $$S? and Brunnler's deep inference system for classical propositional logic can be embedded into each other; and the system $$\mathfrak {S}_\alpha $$S? and Gentzen sequent calculus $$\mathbf {G3c}^*$$G3c? can be embedded into each other.

Normal update logic is the temporalization of normal conditional logic. Sequent calculi for the least normal update logic \(\mathbf {UCK}\) by Andreas Herzig (1998) and some of its extensions are developed. The subformula property of these sequent calculi is shown by Takano’s semantic method. Consequently we prove the finite model property and decidability of these sequent calculi.

In order to research the effects of headway changes with memory and backward looking effect on traffic flow, in the light of the full velocity difference model (FVDM), a novel car-following model that considers backward looking effect and headway changes with memory is proposed. The use of linear stability theory is to obtain the stability criterion that can judge traffic flow stability. In addition, on the basis of nonlinear theory, the time-dependent Ginzburg–Lan (TDGL) equation and the modified Korteweg–de Vries (mKdV) equation near the critical stability point are inferred. The kink–antikink solutions of the above two equations can be used to describe the traffic jam. In the end, numerical simulation is carried out, and the results further demonstrate that the novel model can stabilize traffic flow and eliminate traffic jam better.

Two conjectures on the covering-based rough set semantics for modal logics in 35 are answered. The C 2 and C 5 semantics give rise to the same modal system S4. There are Galois connections between C 2 and C 5 which lead to the covering-based semantics for the temporal logic system S4t. The P 1 and C 4 semantics give rise to the same modal system KTB.

The minimal non-normal modal logic C2 is temporalized, and the minimal non-normal temporal logic C2t is obtained. The Hilbert-style axiomatic system HC2t, which is sound and complete, is established for C2t. The logic C2t is more general than the minimal temporal logic Kt. The proof theory of C2t is approached by introducing the labelled sequent calculus GC2t. The calculus GC2t is sound and complete, and involves cut elimination. The decidability of GC2t is obtained, and the proof search can be performed in GC2t.

A sequent calculus \(\mathbf {wG5}\) is introduced for the variety of partition topological quasi-Boolean algebras. The sequent calculus \(\mathbf {wG5}\) has the cut elimination property, i.e., every sequent derivable in \(\mathbf {wG5}\) has a cut-free derivation. Furthermore, a sequent calculus \(\mathbf {wG4}_t\) is introduced for the variety of topological quasi-Boolean algebras with tense operators, and it is a conservative extension of a sequent calculus \(\mathbf {wG4}\) for the variety of topological quasi-Boolean algebras.

Some classes of topological quasi-Boolean algebras, including algebraic structures related with rough sets, are enriched with residuated and adjoint pairs. The strong finite model property for these classes of algebraic structures is established. The decidability of equational theories of these classes of algebras is derived from the finite model property.

Peirce introduced the Alpha part of the logic of Existential Graphs (egs) as a diagrammatic syntax and graphical system corresponding to classical propositional logic. The logic of quasi-Boolean algebras (De Morgan algebras) is a weakening of classical propositional logic. We develop a graphical system of weak Alpha graphs for quasi-Boolean algebras, and show its soundness and completeness with respect to this algebra. Weak logical graphs arise with only minor modifications to the transformation rules of the original theory of egs. Implications of these modifications to the meaning of the sheet of assertion are then also examined.

A contraction-free and cut-free labelled sequent calculus \(\mathsf {GInqL}\) for inquisitive logic is established. Labels are defined by a set-theoretic syntax. The completeness of \(\mathsf {GInqL}\) is shown by the equivalence between the Hilbert-style axiomatic system and sequent system.

A contraction-free and cut-free sequent calculus $$\mathsf {G3SDM}$$ for semi-De Morgan algebras, and a structural-rule-free and single-succedent sequent calculus $$\mathsf {G3DM}$$ for De Morgan algebras are developed. The cut rule is admissible in both sequent calculi. Both calculi enjoy the decidability and Craig interpolation. The sequent calculi are applied to prove some embedding theorems: $$\mathsf {G3DM}$$ is embedded into $$\mathsf {G3SDM}$$ via Godel–Gentzen translation. $$\mathsf {G3DM}$$ is embedded into a sequent calculus for classical propositional logic. $$\mathsf {G3SDM}$$ is embedded into the sequent calculus $$\mathsf {G3ip}$$ for intuitionistic propositional logic.

Residuated boolean algebras are introduced by Jonsson and Tsinakis [10] as generalizations of relation algebras. Jispen [12] proved that the equational theory of residuated boolean algebras with unit, and that of many relative classes of algebras are decidable. Buszkowski [2] showed the finite embeddability property for residuated boolean algebras, which yields the decidability of the universal theory of residuated boolean algebras. In this paper, we study the complexity of the equational theory of residuated boolean algebras. The main result is that the equational theory of residuated boolean algebras is PSPACE-complete.

Traffic congestion in freeway network is a prevalent transportation problem and leads to a strong degradation of freeway network facilities. This paper proposes a new coordinated optimization approach for freeway network, aiming to relieve the traffic congestion in freeway network equipped with the coordinated traffic controllers. The challenge is how to solve the problem between the traffic control goals of each path and those of the network. In this paper, the bi-level programming theory is used to address this problem. More exactly, in the path level, the optimal control model is designed to relieve congestion in each path of the test network. Furthermore, in the network level, the optimal model considers the traffic equilibrium and cost optimization. The efficiency of the proposed approach is tested on a well-known traffic network calibrated by the field test data of freeway network. The results show that the proposed approach has positive effects in balancing traffic load and improving the service level of freeway networks.

International audience; This paper presents a tableau calculus for two semantic interpretations of public announcements over monotone neighbourhood models: the intersection and the subset semantics, developed by Ma and Sano. We show that, without employing reduction axioms, both calculi are sound and complete with respect to their corresponding semantic interpretations and, moreover, we establish that the satisfiability problem of this public announcement extensions is NP-complete in both cases. The tableau calculi has been implemented in Lotrecscheme.

Visser's basic propositional logic $$\mathbf {BPL}$$ is the subintuitionistic logic determined by the class of all transitive Kripke frames, and his formal provability logic $$\mathbf {FPL}$$, an extension of $$\mathbf {BPL}$$, is determined by the class of all irreflexive and transitive finite Kripke frames. While Visser showed that $$\mathbf {FPL}$$ is embeddable into the modal logic $$\mathbf {GL}$$, we first show that $$\mathbf {BPL}$$ is embeddable into the modal logic $$\mathbf {wK4}$$, which is determined by the class of all weakly transitive Kripke frames, and we also show that $$\mathbf {BPL}$$ is characterized by the same frame class. Second, we introduce the proper successor semantics under which we prove that $$\mathbf {BPL}$$ is characterized by the class of weakly transitive frames, transitive frames, pre-ordered frames, and partially ordered frames. Third, we introduce topological semantics by interpreting implication in terms of the co-derived set operator and prove that $$\mathbf {BPL}$$ is characterized by the class of all topological spaces, $$T_0$$-spaces and $$T_d$$-spaces. Finally, we establish the topological completeness of $$\mathbf {FPL}$$ with respect to the class of scattered spaces.

Background and objectiveGenetic factors including TNF-β have been considered as important components in the aetiology of migraine. Many studies have investigated the association between TNF-β 252A>G polymorphism and migraine risk, with debatable results. This study was designed to examine whether the TNF-β 252A>G polymorphism confers genetic susceptibility to migraine in diverse populations.MethodStudies eligible for this meta-analysis were searched in the PubMed, Embase, and CNKI by using the keywords "tumor necrosis factor", "TNF", "252A>G", "rs909253", "polymorphism", "polymorphisms", "variant", "SNP", combined with "migraine" or "migraine with aura (MA)" or "migraine without aura (MO)". Pooled ORs and 95% CI were appropriately calculated using the fixed-effect model.ResultsWe finally included a total of seven studies, providing 5 557 migraineurs and 20 543 unrelated healthy controls. Meta-analysis results showed no statistical evidence of a significant association between TNF-β 252A>G polymorphism and overall migraine risk. Stratified analyses by type of migraine and gender revealed similar results. Interestingly, an OR with 95% CI representing an increased migraine risk was indicated in Asians under the recessive model (GG vs. AG + AA: OR, 1.38; 95%CI, 1.04-1.84; P for heterogeneity, 0.665).ConclusionsOur findings appear to support the hypothesis that genetic variability of 252A>G polymorphism in TNF region may modulate risk of migraine in the population of Asian ancestry.

This paper studies two ways of updating neighborhood models: update by taking the intersection of the neighborhoods with the announced proposition, and update by selecting all the neighborhoods that can entail the announced proposition. For each of these two ways, we establish reduction axioms and some basic model-theoretic results on public announcement logic and dynamic epistemic logic of product update. We also study various notions of group knowledge such as common and distributed knowledge over neighborhood models.

The unified correspondence theory for distributive lattice expansion logics (DLE-logics) is specialized to strict implication logics. As a consequence of a general semantic consevativity result, a wide range of strict implication logics can be conservatively extended to Lambek Calculi over the bounded distributive full non-associative Lambek calculus (BDFNL). Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based algorithm $\msf{ALBA}$. Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed., Comment: This is a Pre-publication version of a submission to the Journal of Logic and Computation

We explore logics of belief over weighted structures under the supposition that everything believed by an agent has a weight in the range of agent's belief. We first define static graded belief logics which are complete with respect to the class of all weighted frames. Furthermore, we discuss their public announcement and dynamic epistemic extensions. We may also define notions of plausible belief by comparing weights of formulas at the current state in a weighted model. This approach is not a new one but we provide new logics and their dynamic extensions which can capture some intuitive notions of belief and their dynamics.

We study some mathematical aspects of public announcement logic (PAL) and its several variants. By a model-theoretic approach, we explore van Benthem's result that uses recursion axioms to characterize the submodel operation, and show some model-theoretic results on the respecting phenomena. The second approach to understand public announcements is algebraic. Based on a joint work with A. Palmigiano and M. Sadrzadeh, we treat public announcements as devices for getting a new quotient algebra updated by an element in the original one. Then we show the algebraic soundness and completeness result for PAL and generalize this approach to PAL extension of epistemic intuitionistic modal logic. Finally, we give some observations on the PAL extensions of first-order logic as well as epistemic predicate modal logic.

Following the approach given by Nina Gierasimczuk [2009] to bridge dynamic epistemic logic and learning theory, the logic of finite identification in the limit in the process of learning has been explored in this paper. The main results are two complete axiomatic systems. One is defined in terms of public announcements, and the other in terms of update by event models.