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We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper- and lower-bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: we prove (1) that the reachability problem for COCA with global upper- and lower-bound tests is in NC 2 ; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is NP-complete for COCA with parametric counter updates and bound tests.

Similar to the construction of ordinal sums of overlap functions, Baczynski et al. introduced two new kinds of ordinal sums of fuzzy implications without additional restrictions on summands in 2017. In this paper, based on the ordinal sum of fuzzy implications with the first method, we discuss its basic properties, such as iterative Boolean law, right ordering property and strong boundary condition. Meanwhile, we give characterizations of such ordinal sum of fuzzy implications that is a QL-implication constructed from tuples ( O , G , N ⊤ ) or a D-implication derived from grouping function G, and show some conclusions about its relations with ( G , N ) - and R O -implications. Moreover, we study the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. More specifically, we give necessary and sufficient conditions under which this ordinal sum of fuzzy implications is distributive over overlap and grouping functions satisfying some conditions.

Recently, Baczynski et al. have introduced two new methods of constructing ordinal sums of fuzzy implications. These methods are based on a construction of ordinal sum of overlap functions and extend some type of algebraic structures. In this paper, we explore the family of ordinal sum implications constructed by one of the method presented in (M. Baczynski, P. Drygaś, A. Krol, R. Mesiar (2017), [7] ) and examine intersections of this new family of fuzzy implications with (S, N)-, (T, N)-, QL-, D-, R-, Yager's f- and g-implications, respectively. Concretely, this new family of fuzzy implications is shown to be different from the above mentioned implications. Moreover, the necessary and sufficient conditions under which these new ordinal sum implications are distributive over t-norms and t-conorms are given.

The characterization of ( S , N ) -implications when N is a non-continuous negation has remained one of the most significant open problems in fuzzy logic for the last decades. This paper constitutes the first progress in this topic. Namely, a general characterization of this family of fuzzy implication functions is presented, in which the central property is the existence of a completion of a binary function defined on a certain subregion of [ 0 , 1 ] 2 to a t-conorm. In this paper, the dual problem of finding a completion of a binary function defined on a subregion of [ 0 , 1 ] 2 to a continuous t-norm is studied and solved for the minimum and a cancellative function. These results are the basis for the novel axiomatic characterizations of ( S , N ) -implications in the case when N has one point of discontinuity and S is equal to the maximum t-conorm in a certain subregion of [ 0 , 1 ] 2 or a strict t-conorm.

Our aim is to obtain a suitable characterization of certain topological properties of a random closed set through its capacity functional. The main technique mixes two different fields: on the one hand, the abstract simplicial complex associated to a covering, whose topology coincides with the topology of the original set thanks to the nerve theorem. On the other hand, the celebrated Choquet-Kendall-Matheron theorem, which states that a random closed set is characterized by its capacity functional.

This study aims to develop the notion of general fuzzy automata (GFA) to a new one which is known as “ L B -valued general fuzzy automata”. Instead of the term L B -valued general fuzzy automata, for simplicity, L B -valued GFA is used where B is regarded as a set of propositions about the GFA, in which its underlying structure has been a complete infinitely distributive lattice. Further, L B -valued GFA is scrutinized via different operators and also the interrelationship among these operators is examined. Specifically, it is shown that L B -valued successor, L B -valued predecessor and L B -valued residuated operators play an important role in the algebraic study of L B -valued general fuzzy automaton and that under certain conditions these operators are interrelated. In addition, the concept of a homomorphism between L B -valued general fuzzy automata is introduced and studied. Finally, the concepts of equivalence and congruence are defined, the quotient L B -valued GFA with respect to congruence is formulated and the equivalence between L B -valued GFA and its quotient automaton is proved. To clarify the notions and the results obtained in this study, some examples are submitted as well.

A fuzzy interval is a fuzzy set in R such that its α-level sets are bounded closed intervals in R . The purpose of this article is to generate fuzzy intervals using gradual intervals and gradual numbers by following the mathematical expression in decomposition theorem. On the other hand, the concepts of elements and subsets of fuzzy intervals are proposed on the basis of the concept of gradual numbers. Each gradual number can induce a corresponding fuzzy set in R . We also compare the fuzziness between the generated fuzzy interval and the fuzzy set generated by the corresponding gradual number.

Several operators have emerged in the framework of Choquet integral with the purpose of simultaneously generalizing weighted means and ordered weighted averaging (OWA) operators. However, on many occasions, not enough attention has been paid to whether the constructed operators behaved similarly to the weighted means and OWA operators that have been generalized. In this sense, it seems necessary that these new operators preserve the weights assigned to the information sources (which are established through the weighting vector associated with the weighted mean) and that they are able to rule out extreme values (which is an important characteristic of OWA operators). In this paper we analyze a family of operators recently introduced in the literature through the Crescent Method. First, we introduce a broad class of weighting vectors that allow us to guarantee that the games generated with the Crescent Method are capacities. Next we analyze the conjunctive/disjunctive character of the Choquet integrals associated with these capacities and we also give closed-form expressions of some tolerance and importance indices such as k-conjunctiveness/disjunctiveness indices, the veto and favor indices, and the Shapley values. Finally, we give two examples to show the usefulness of the results obtained.

The problem of determining whether a term is an instance of another in the simply typed lambda -calculus, i.e. of solving the equation a=b where a and b are simply typed lambda -terms and b is ground, is addressed. An algorithm that decides whether a matching problem in which all the variables are at most third order has a solution is given. The main idea is that if the problem a=b has a solution, then it also has a solution whose depth is bounded by some integer s depending only on the problem a=b, so a simple enumeration of the substitutions whose depth is bounded by s gives a decision algorithm. This result can also be used to bound the depth of the search tree in Huet's semi-decision algorithm and thus to turn it into an always-terminating algorithm. The problems that occur in trying to generalize the proof given to higher-order matching are discussed. >

In this paper we give a new method of constructing partial order on the set of formal balls of a KM-fuzzy metric space, which overcomes a key defect that the partial order on the set of formal balls depends on the minimum t-norm ∧ in the case of [19] , [21] . Further, based on a continuous t-norm, we will establish a connection between KM-fuzzy metric space and domain theory, which is that a KM-fuzzy metric space is T-complete if and only if its poset of formal balls is a domain.

Gated sets in metric spaces play an important role in the study of convexity. In this paper, we investigate the notion of gated sets in the context of betweenness of fuzzy metric spaces. For this purpose, we analyze the validity of postulates presented by Huntington and Kline to the betweenness in GV fuzzy metric spaces. The postulates are different from the usual properties of metric betweenness, amounting to thirteen. Interestingly, the validity of most of these postulates to the betweenness need to be considered in fuzzy normed spaces. In particular, two of the postulates for this relation are valid when the space is 1-dimensional. Five of the postulates for this relation are centrally studied in connection with strict convexity. As a consequence, several characterizations on strict convexity in a GV fuzzy normed space under the minimum t-norm are established.

We establish some new Opial type inequalities for interval-valued functions, and obtain some generalizations and refinements of interval Opial type inequalities derived by Costa et al. (2019) [12] . Also, some examples are presented to illustrate our results.

In this paper, we introduce crisp utility functions with range in the class of fuzzy numbers. These utility functions can be employed to prove a generalization of the classical result of Cantor using partially ordered sets as domain of the utility functions. In particular, for every partially ordered countable set, we prove that it can be represented by an order preserving function (utility).

This paper is a sequel to \cite{Tz10}, where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and $\Pi_1^1$-indescribable models, were considered. By analogy we refer to such models as "large models", and the properties in question as "large model properties". Continuing here in the same spirit we consider further large model properties, that resemble stronger large cardinals, namely, "elementarily embeddable", "extendible" and "strongly extendible", "critical" and "strongly critical", "self-critical'' and "strongly self-critical", the definitions of which involve elementary embeddings. Each large model property $\phi$ gives rise to a localization axiom $Loc^{\phi}({\rm ZFC})$ saying that every set belongs to a transitive model of ZFC satisfying $\phi$. The theories ${\rm LZFC}^\phi={\rm LZFC}$+$Loc^{\phi}({\rm ZFC})$ are local analogues of the theories ZFC+"there is a proper class of large cardinals $\psi$", where $\psi$ is a large cardinal property. If $sext(x)$ is the property of strong extendibility, it is shown that ${\rm LZFC}^{sext}$ proves Powerset and $\Sigma_1$-Collection. In order to refute $V=L$ over LZFC, we combine the existence of strongly critical models with an axiom of different flavor, the Tall Model Axiom ($TMA$). $V=L$ can also be refuted by $TMA$ plus the axiom $GC$ saying that "there is a greatest cardinal", although it is not known if $TMA+GC$ is consistent over LZFC. Finally Vop\v{e}nka's Principle ($VP$) and its impact on LZFC are examined. It is shown that ${\rm LZFC}^{sext}+VP$ proves Powerset and Replacement, i.e., ZFC is fully recovered. The same is true for some weaker variants of ${\rm LZFC}^{sext}$. Moreover the theories LZFC$^{sext}$+$VP$ and ZFC+$VP$ are shown to be identical., Comment: 32 pages

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by $Loc({\rm ZFC})$, says that every set belongs to a transitive model of ZFC. LZFC consists of $Loc({\rm ZFC})$ plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All $\Pi_2$ consequences of ZFC are provable in ${\rm LZFC}$. LZFC strongly extends Kripke-Platek (KP) set theory minus $\Delta_0$-Collection and minus $\in$-induction scheme. ZFC+``there is an inaccessible cardinal'' proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define $\alpha$-Mahlo models and $\Pi_1^1$-indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form $Loc({\rm ZFC}+\phi)$ are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved., Comment: 38 pages

In this note, we point out that Theorem 3 (a version of Egoroff's theorem for monotone set-valued measures) shown in the paper “Lusin's theorem for monotone set-valued measures on topological spaces” (Fuzzy Sets and Systems 364 (2019) 111-123) is not valid, and we present two revised versions. We also point out that the proofs of Lemma 2 and Theorem 1 are defective, thus the truth of their conclusions cannot be affirmed. Thereby, an open problem for the characteristics of monotone measures is raised.

The thin set theorem for n-tuples and k colors ( $\operatorname {\mathrm {\sf {TS}}}^n_k$ ) states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$ , nor the free set theorem ( $\operatorname {\mathrm {\sf {FS}}}^n$ ) imply the Erdős–Moser theorem ( $\operatorname {\mathrm {\sf {EM}}}$ ) whenever k is sufficiently large (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). Given a problem $\mathsf {P}$ , a computable instance of $\mathsf {P}$ is universal iff its solution computes a solution of any other computable $\mathsf {P}$ -instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance (answering a question of Patey).

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

We present the concepts of the limit superior and the limit inferior of a double sequence of fuzzy numbers and obtain several properties of these concepts. By using these concepts we also characterize the core of a double sequence of fuzzy numbers as a closed interval. Furthermore, we define Γ-convergence and the endograph metric convergence for a double sequence of fuzzy numbers, which are new types of convergence and investigate the relations between these types of convergence and the core of the sequence.

Destruction of the roads and disruption in transportation networks are the aftermath of natural disasters, particularly if they are of great magnitude. As a version of the network capacity reliability problem, this work researches a post-disaster transportation network, where the reliability and operational capacity of links are uncertain. Uncertainty theory is utilized to develop a model of and solve the uncertain maximum capacity path (UMCP) problem to ensure that the maximum amount of relief materials and rescue vehicles arrive at areas impacted by the disaster. We originally present two new problems of $$\alpha$$ -maximum capacity path ( $$\alpha$$ -MCP), which aims to determine paths of highest capacity under a given confidence level $$ \alpha$$ , and most maximum capacity path (MMCP), where the objective is to maximize the confidence level under a given threshold of capacity value. We utilize these auxiliary programming models to explicate the method to, in an uncertain network, achieve the uncertainty distribution of the MCP value. A novel approach is additionally suggested to confront, in the framework of uncertainty programming, the stability analysis problem. We explicitly enunciate the method of computing the links’ tolerances in $${\mathcal{O}}\left( m \right)$$ time or $${\mathcal{O}}\left( {\left| {P^{*} } \right|m} \right)$$ time (where $$m$$ indicates the number of links in the network and $$\left| {{\text{P}}^{*} } \right|$$ the number of links on the given MCP $${\text{P}}^{*}$$ ). After all, the practical performance of the method and optimization model is illustrated by adopting two network samples from a real case study to show how our approach works in realistic contexts.

We advocate the use of de Bruijn's universal abstraction $\lambda^\infty$ for the quantification of schematic variables in the predicative setting and we present a typed $\lambda$-calculus featuring the quantifier $\lambda^\infty$ accompanied by other practically useful constructions like explicit substitutions and expected type annotations. The calculus stands just on two notions, i.e., bound rt-reduction and parametric validity, and has the expressive power of $\lambda\rightarrow$. Thus, while not aiming at being a logical framework by itself, it does enjoy many desired invariants of logical frameworks including confluence of reduction, strong normalization, preservation of type by reduction, decidability, correctness of types and uniqueness of types up to conversion. This calculus belongs to the $\lambda\delta$ family of formal systems, which borrow some features from the pure type systems and some from the languages of the Automath tradition, but stand outside both families. In particular, the calculus includes and evolves two earlier systems of this family. Moreover, a machine-checked specification of its theory is available., Comment: 37 pages, minor revision

This paper shows that the eight propositions and the four corollaries about four types of fuzzy covering-based rough set models in Yang and Hu (2017) [1] are incorrect through counter-examples, and the eight propositions are corrected.

In this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$ , $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated with this ordering, the Levin–V’yugin degrees (or $\mathrm {LV}$ -degrees), can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin–V’yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V’yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V’yugin’s about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree.

Several results about the game of cops and robbers on infinite graphs are analyzed from the perspective of computability theory. Computable robber-win graphs are constructed with the property that no computable robber strategy is a winning strategy, and such that for an arbitrary computable ordinal $$\alpha $$ , any winning strategy has complexity at least $$0^{(\alpha )}$$ . Symmetrically, computable cop-win graphs are constructed with the property that no computable cop strategy is a winning strategy. Locally finite infinite trees and graphs are explored. The Turing computability of a binary relation used to classify cop-win graphs is studied, and the computational difficulty of determining the winner for locally finite computable graphs is discussed.

In recent years, some theorists have argued that the clogics are not only defined by their inferences, but also by their metainferences. In this sense, logics that coincide in their inferences, but not in their metainferences were considered to be different. In this vein, some metainferential logics have been developed, as logics with metainferences of any level, built as hierarchies over known logics, such as $$\mathbf {ST}, \mathbf {LP}, \mathbf {K_3}$$ , and $$\mathbf {TS}$$ . What is distinctive of these metainferential logics is that they are mixed, i.e. the standard for the premises and the conclusion is not necessarily the same. However, so far, all of these systems have been presented following a semantical standpoint, in terms of valuations based on the Strong Kleene truth-tables. In this article, we provide sound and complete sequent-calculi for the valid inferences and the invalid inferences of the logics $$\mathbf {ST}, \mathbf {LP}, \mathbf {K_3}$$ and $$\mathbf {TS}$$ , and introduce an algorithm that allows obtaining sound and complete sequent-calculi for the global validities and the global invalidities of any metainferential logic of any level.

Let $\mathcal {I}$ be an ideal on $\omega $ . For $f,\,g\in \omega ^{\omega }$ we write $f \leq _{\mathcal {I}} g$ if $f(n) \leq g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal {I}$ . Moreover, we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ (in particular, $\mathcal {D}_{\mathrm {Fin}}$ denotes the family of all finite-to-one functions).We examine cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ and $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq _{\mathcal {I}}$ sets in $\mathcal {D}_{\mathrm {Fin}}$ and $\mathcal {D}_{\mathcal {I}}$ , respectively. For a maximal ideal $\mathcal {I}$ , these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.We show that $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$ for all ideals $\mathcal {I}$ with the Baire property and that $\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$ for all coanalytic weak P-ideals (this class contains all $\bf {\Pi ^0_4}$ ideals). What is more, we give examples of Borel (even $\bf {\Sigma ^0_2}$ ) ideals $\mathcal {I}$ with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$ as well as with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$ .We also study cardinals $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$ describing the smallest sizes of sets in $\mathcal {D}_{\mathcal {K}}$ not bounded from below with respect to the preorder $\leq _{\mathcal {I}}$ by any member of $\mathcal {D}_{\mathcal {J}}\!$ . Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN.

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

It is known that the satisfiability problems of the product logics K4 × S5 and S4 × S5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. Furthermore, it is known that the satisfiability problems of these logics are in N2EXPTIME. We improve the lower and the upper bounds for the complexity of these problems by showing that all three problems are in ESPACE and are EXPSPACE-complete under logspace reduction.

In this paper, for a real number ϵ ∈ [ 0 , 1 ] , we put forward the notion of fuzzy ϵ-approximate regular languages and study their properties, especially the Pumping lemma in the context of this kind of languages. By comparing sets of fuzzy ϵ-approximate regular languages under the order of set inclusion, we get an (infinite) hierarchy of fuzzy languages, the smallest is the set of fuzzy regular languages and the biggest is the set of fuzzy languages, and the sets of ϵ-approximate regular languages are different for different ϵ ∈ ( 0 , 1 2 ) . We also investigate whether operations closed in the set of fuzzy regular languages are still closed in the set of fuzzy ϵ-approximate regular languages. Furthermore, for a fuzzy ϵ-approximate regular language f, ϵ-approximate equivalence relations for f are characterized in order to construct deterministic fuzzy automata ϵ-accepting f. If a fuzzy ϵ-approximate regular language f is also a fuzzy regular language and accepted by a given accessible deterministic fuzzy automaton A, then we give a polynomial-time algorithm to construct at least one minimal deterministic fuzzy automaton ϵ-accepting f by means of A. Finally, we point out that the number of states of minimal deterministic fuzzy automata ϵ-accepting a fuzzy regular language is smaller than or equal to that of minimal deterministic fuzzy automata accepting this language.

Fix \(2 < n < \omega\). Let \(L_n\) denote first order logic restricted to the first n variables. Using the machinery of algebraic logic, positive and negative results on omitting types are obtained for \(L_n\) and for infinitary variants and extensions of \(L_{\omega, \omega}\).

In this paper, we shall give another proof of the faithfulness of Blass translation (for short, B-translation) of the propositional fragment $$\mathbf{L}_1$$ of Leśniewski’s ontology in the modal logic $$\mathbf{K}$$ by means of Hintikka formula. And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a result of observing the proofs we shall give general theorems on the faithfulness of B-translation with respect to normal modal logics complete to certain sets of well-known accessibility relations with a restriction that transitivity and symmetry are not set at the same time. As an application of the theorems, for example, B-translation is faithful for the provability logic $$\mathbf{PrL}$$ (= $$\mathbf{GL}$$ ), that is, $$\mathbf{K}$$ $$+$$ $$\Box (\Box \phi \supset \phi ) \supset \Box \phi $$ . The faithfulness also holds for normal modal logics, e.g., $$\mathbf{KD}$$ , $$\mathbf{K4}$$ , $$\mathbf{KD4}$$ , $$\mathbf{KB}$$ . We shall conclude this paper with the section of some open problems and conjectures.

This paper investigates a first-order extension of GL called \(\textup{ML}^3\). We outline briefly the history that led to \(\textup{ML}^3\), its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \(\textup{ML}^3\) is arithmetically complete. As expanded below, \(\textup{ML}^3\) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―"\(\Box\)" simulating the the informal classical "\(\vdash\)"―is also arithmetically complete in the Solovay sense.

This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned.

The general theory developed by Ben Yaacov for metric structures provides Fraïssé limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.

We present a generalization of the algebra-valued models of $$\mathrm {ZF}$$ where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get intermediate values, in a set of designated values. Under this generalization there are many algebras which are neither Boolean, nor Heyting, but that still validate $$\mathrm {ZF}$$ .

This paper answers several open questions around structures with o-minimal open core. We construct an expansion of an o-minimal structure $\mathcal{R}$ by a unary predicate such that its open core is a proper o-minimal expansion of $\mathcal{R}$. We give an example of a structure that has an o-minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o-minimal open core and definable Skolem functions, but is not o-minimal.

The t-norm fuzzy sets is a class of fuzzy sets equipped with a pair of algebraic operation, a notion of T-convexity and a metric, where all these notions are based on a strict triangular norm T. In this paper we introduce in this class a notion of T-convex hull of a fuzzy set. We prove theorem that binds the T-convex hull of an upper semicontinuous fuzzy set with the convex hull of a (crisp) set. We further show two applications of this result. First, we prove that the operation of forming T-convex hull behaves well with respect to algebraic operations. Second, we show an analogue of Shapley-Folkman theorem. Shapley-Folkman theorem is a well-known result that provides an upper bound on the distance between Minkowski sum of sets and the convex hull of this sum. In this paper we show that the distance between the sum of upper semicontinuous fuzzy subsets of a finite dimensional Euclidean space and the T-convex hull of this sum has an upper bound. As a consequence of this result, we present an iterative procedure for forming T-convex hull of an upper semicontinuous fuzzy set. As an application of the results regarding t-norm fuzzy sets we show an example from a game theoretic setting, where t-norm fuzzy sets were used to handle uncertainty in a repeated two player game.

In the first part of this paper we consider the α-cut of a sum of two fuzzy sets. We show that the α-cut of such a sum can be written as one closed interval determined by the minimum and maximum of some special functions, over a set uniquely determined by the considered t-norm. In the second part of the paper we show that not only α-cut of a sum of two fuzzy intervals has this property, but also α-cut of a continuous function of two variables, strictly increasing in each argument as well. Two examples of the presented theory are considered.

We will give a two-sort system which axiomatizes winning strategies for the combinatorial game Node Kayles. It is shown that our system captures alternating polynomial time reasonings in the sense that the provably total functions of the theory corresponds to those computable in APTIME. We will also show that our system is equivalently axiomatized by Sprague–Grundy theorem which states that any Node Kayles position is provably equivalent to some NIM heap.

We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system.

We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by Gödel (1931, 1934) and Kleene (1936, Math. Ann., 112, 727–742). We show that this class of functions can also be characterized by MS-machines, which are Turing machine-like devices. The proof of the characterization gives a normal form theorem in the style of Kleene (1936, Math. Ann., 112, 727–742). Furthermore, this characterization is a natural combination of two most influential theories of computation over real numbers, namely the type-two theory of effectivity (see, e.g. Weihrauch (2000, Springer)) and the Blum–Shub–Smale (1989, Bull. Amer. Math. Soc. (N.S.), 21, 1–46) model of computation. Under this notion of computability, the recursive (or computable) subsets of real numbers are exactly effective $\varDelta ^0_2$ sets.

We prove neighbourhood canonicity and strong completeness for the logics $\mathbf {EK}$ and $\mathbf {ECK}$ , obtained by adding axiom (K), resp. adding both (K) and (C), to the minimal modal logic $\textbf {E}$ . In contrast to an earlier proof in [10], ours is constructive. More precisely, we construct minimal characteristic models for both logics and do not rely on compactness of first order logic. The proof involves a specific circumscription technique and quite some set-theoretic maneuvers to establish that the models satisfy the appropriate frame conditions. After giving both proofs, we briefly spell out how they generalize to four stronger logics and to the extensions of the resulting six logics with a global modality.

Based on the random vector ( X + Z , Y + Z ) we study the perturbation C X + Z , Y + Z of the copula C X , Y of the random vector ( X , Y ) when the random noise Z is independent of both X and Y. We also propose a bivariate copula-based modification of the Irwin-Hall distribution and use it to extend the results of Mesiar et al. (2019) and present several examples for illustration.