Logical Calculus Research Articles

Quantitative Weakest Hyper Pre: Unifying Correctness and Incorrectness Hyperproperties via Predicate Transformers

We present a novel weakest pre calculus for reasoning about quantitative hyperproperties over nondeterministic and probabilistic programs. Whereas existing calculi allow reasoning about the expected value that a quantity assumes after program termination from a single initial state, we do so for initial sets of states or initial probability distributions. We thus (i) obtain a weakest pre calculus for hyper Hoare logic and (ii) enable reasoning about so-called hyperquantities which include expected values but also quantities (e.g. variance) out of scope of previous work. As a byproduct, we obtain a novel strongest post for weighted programs that extends both existing strongest and strongest liberal post calculi. Our framework reveals novel dualities between forward and backward transformers, correctness and incorrectness, as well as nontermination and unreachability.

Tessellations and Sweeping Nets: Advancing the Calculus of Geometric Logic

This paper studies the advancement of geometric logic through tessellations and sweeping nets, addressing the challenge of arranging reflecting points for efficient ray tracing under limited time constraints. We introduce the concept of a sweeping subnet alongside a causal barrier to encapsulate the geometrical limitations posed by time, thereby delineating the boundary of influence for light propagation within a defined space. This work delves into the underpinnings of tessellation dynamics, revealing how the spatial arrangement and temporal evolution of tessellated patterns can be navigated and optimized through a novel algorithmic framework. Through a combination of theoretical exploration and practical implementation, including Python code for simulation and visualization, we provide a platform for approximating optimal tessellations that adapt to the constraints dictated by the causal barrier. This paper formalizes quasi-quanta notation in such a way that it may be implemented.

Prawitz's completeness conjecture: A reassessment

AbstractIn 1973, Dag Prawitz conjectured that the calculus of intuitionistic logic is complete with respect to his notion of validity of arguments. On the background of the recent disproof of this conjecture by Piecha, de Campos Sanz and Schroeder‐Heister, we discuss possible strategies of saving Prawitz's intentions. We argue that Prawitz's original semantics, which is based on the principal frame of all atomic systems, should be replaced with a general semantics, which also takes into account restricted frames of atomic systems. We discard the option of not considering extensions of atomic systems, but acknowledge the need to incorporate definitional atomic bases in the semantic framework. It turns out that ideas and results by Westerståhl on the Carnap categoricity of intuitionistic logic can be applied to Prawitz semantics. This implies that Prawitz semantics has a status of its own as a genuine, though incomplete, semantics of intuitionstic logic. An interesting side result is the fact that every formula satisfiable in general semantics is satisfiable in an axioms‐only frame (a frame whose atomic systems do not contain proper rules). We draw a parallel between this seemingly paradoxical result and Skolem's paradox in first‐order model theory.

On Combining Intuitionistic and S4 Modal Logic

We address the problem of combining intuitionistic and S4 modal logic in a non-collapsing way inspired by the recent works in combining intuitionistic and classical logic. The combined language includes the shared constructors of both logics namely conjunction, disjunction and falsum as well as the intuitionistic implication, the classical implication and the necessity modality. We present a Gentzen calculus for the combined logic defined over a Gentzen calculus for the host S4 modal logic. The semantics is provided by Kripke structures. The calculus is proved to be sound and complete with respect to this semantics. We also show that the combined logic is a conservative extension of each component. Finally we establish that the Gentzen calculus for the combined logic enjoys cut elimination.

Fuzzy-fractional modeling and simulation of electric circuits using extended He-Laplace-Carson algorithm

In literature, most of the electric circuits are found to be at integer order, which are unable to capture the involved uncertainties and memory effects. To understand these effects, the current study is focused on modeling and analysis of fuzzy-fractional L˜C˜ circuit, R˜C˜ circuit, R˜L˜ circuit, and R˜L˜C˜ circuit. Triangular fuzzy numbers (TFNs) are utilized to introduce uncertainties in circuits. The memory effect is considered through fractional derivative in Caputo sense. A hybrid homotopy perturbation method (HPM) is established for solution purpose, in which standard HPM is combined with Laplace-Carson transformation in fuzzy-Caputo sense. This leads to an excellent and well-organized approach to calculate numerical solutions in the aspect of fuzzy logic and fractional calculus. The results obtained through proposed approach are validated by comparing them with already existing results in literature. Residual errors are also calculated across the domain of r at fractional order to determine the convergence and stability of electric circuit models. To study the dynamics of inductor, resistor, and capacitor on current along with fractional and fuzzy parameters involved in the electric circuits, contour and three-dimensional plots are constructed. Several plots specifying the fuzzy membership function of models across various parameter values are also demonstrated. The results achieved through this study indicate that He-Laplace-Carson method can assist engineers and researchers when dealing with electric circuits characterized by fuzzy and fractional dynamics.

Completeness of tableau calculi for two-dimensional hybrid logics

Completeness of tableau calculi for two-dimensional hybrid logics

Attack principles in sequent-based argumentation theory

Abstract Attack principles have been introduced in semi-abstract argumentation frameworks and, in the present work, we interpret them in sequent-based argumentation frameworks. Thus, we investigate the role of minimality and consistency of the support set of an argument. Through the notion of preservation of strength, we introduce a formal criterion to sort out the attack principles; isolate the more “acceptable” ones, i.e. those easier to justify; and recover a new argumentative semantics for the non-classical logic that arises from dropping the rules $(\neg , r)$, $(\land , r)$ and $(\supset , l)$ from Gentzen’s classical sequent calculus for classical logic $\textsf{LK}$.

Meaning and identity of proofs in a bilateralist setting: A two-sorted typed Lambda-calculus for proofs and refutations

Abstract In this paper, I will develop a $\lambda $-term calculus, $\lambda ^{2Int}$, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry–Howard correspondence, which has been well-established between the simply typed $\lambda $-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing’s bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed $\lambda $-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results, I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.

Intensions and extensions of granules: A two-component treatment

The concept of a granule (of knowledge) originated from Zadeh, where granules appeared as references to words (phrases) of a natural (or an artificial) language. According to Zadeh's program, “a granule is a collection of objects drawn together by similarity or functionality and considered therefore as a whole”. Pawlak's original theory of rough sets and its different generalizations have a common property: all systems rely on a given background knowledge represented by the system of base sets. Since the members of a base set have to be treated similarly, base sets can be considered as granules. The background knowledge has a conceptual structure, and it contains information that does not appear on the level of base granules, so such information cannot be taken into consideration in approximations. A new problem arises: is there any possibility of constructing a system modeling the background knowledge better? A two-component treatment can be a solution to this problem. After giving the formal language of granules involving the tools for approximations, a logical calculus containing approximation operators is introduced. Then, a two-component semantics (treating intensions and extensions of granule expressions) is defined. The authors show the connection between the logical calculus and the two-component semantics.

A TERMINATING INTUITIONISTIC CALCULUS

Abstract A terminating sequent calculus for intuitionistic propositional logic is obtained by modifying the R $\supset $ rule of the labelled sequent calculus $\mathbf {G3I}$ . This is done by adding a variant of the principle of a fortiori in the left-hand side of the premiss of the rule. In the resulting calculus, called ${\mathbf {G3I}}_{\mathbf {t}}$ , derivability of any given sequent is directly decidable by root-first proof search, without any extra device such as loop-checking. In the negative case, the failed proof search gives a finite countermodel to the sequent on a reflexive, transitive, and Noetherian Kripke frame. As a byproduct, a direct proof of faithfulness of the embedding of intuitionistic logic into Grzegorcyk logic is obtained.

On some classes of formulas in S5 which are pre-complete relative to existential expressibility

Existential expressibility for all k-valued functions was proposed by A. V. Kuznetsov and later was investigated in more details by S. S. Marchenkov. In the present paper, we consider existential expressibility in the case of formulas defined by a logical calculus and find out some conditions for a system of formulas to be closed relative to existential expressibility. As a consequence, it has been established some pre-complete as to existential expressibility classes of formulas in some finite extensions of the paraconsistent modal logic S5.

Defining Logical Systems via Algebraic Constraints on Proofs

Abstract We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof system for a target logic by enriching a proof system for another, typically simpler, logic with an algebra of constraints that act as correctness conditions on the latter to capture the former; e.g. one may use Boolean algebra to give constraints in a sequent calculus for classical propositional logic to produce a sequent calculus for intuitionistic propositional logic. The idea behind such forms of decomposition is to obtain a tool for uniform and modular treatment of proof theory and to provide a bridge between semantics logics and their proof theory. The paper discusses the theoretical background of the project and provides several illustrations of its work in the field of intuitionistic and modal logics: including, a uniform treatment of modular and cut-free proof systems for a large class of propositional logics; a general criterion for a novel approach to soundness and completeness of a logic with respect to a model-theoretic semantics; and a case study deriving a model-theoretic semantics from a proof-theoretic specification of a logic.

Dynamic Separation Logic

This paper introduces a dynamic logic extension of separation logic. The assertion language of separation logic is extended with modalities for the five types of the basic instructions of separation logic: simple assignment, look-up, mutation, allocation, and de-allocation. The main novelty of the resulting dynamic logic is that it allows to combine different approaches to resolving these modalities. One such approach is based on the standard weakest precondition calculus of separation logic. The other approach introduced in this paper provides a novel alternative formalization in the proposed dynamic logic extension of separation logic. The soundness and completeness of this axiomatization has been formalized in the Coq theorem prover.

Importance analysis of a system based on survival signature by structural importance measures

The analysis of the influence of system components on system failure is a typical problem in reliability engineering which is known as importance analysis of a system. There are different methods for the calculation of the importance measures of the system components. In this paper, a new method for the calculation of structural importance measures that are one of the types of importance measures is presented. The novelty of the proposed method is the use of the survival signature instead of the structure function for the computation of the importance measures. The proposed method is based on the tool of logical differential calculus, in particular, direct partial logical derivatives. The advantages and practical applicability of the proposed method were confirmed in the analysis of the reliability characteristics of the fuel system of the propulsion complex of a passenger aircraft.

Sustainable development, eco-tourism carrying capacity and fuzzy algorithm-a study on Kanas in Belt and Road

In this paper, the method of fuzzy pattern recognition is adopted in more precisely evaluating the actual state of eco-tourism development regarding a given tourist destination in comparison with the three standard patterns (saturated/optimal/deficient) of development degree. The research process is as follows: Firstly, the indictors of carrying capacity of a tourist destination and the corresponding measuring factors are established; secondly, an assessment group is recruited to work out the most constraining factors among the measuring factors; thirdly, by means of field survey, numerical values of the actual state are acquired; fourthly, there comes out the membership vectors and the membership matrix of the standard patterns corresponding to the vectors of three standard patterns, threshold and the actual state; Finally, it could be identified which standard pattern that the actual state is closest to via the lattice degrees of proximity. An exemplary case study on Kanas National Nature Reserves is attached to the logic calculus. This paper is contributed to dynamically monitor the threshold of tourism carrying capacity and precisely identify which carrying capacity (spatial resource/ecological environment/economic resources/people’s psychology/socio-culture) with potential risks.

Sequent-type rejection systems for finite-valued non-deterministic logics

A rejection system, also referred to as a complementary calculus, is a proof system axiomatising the invalid formulas of a logic, in contrast to traditional calculi which axiomatise the valid ones. Rejection systems therefore introduce a purely syntactic way of determining non-validity without having to consider countermodels, which can be useful in procedures for automated deduction and proof search. Rejection calculi have first been formally introduced by Łukasiewicz in the context of Aristotelian syllogistic and subsequently rejection systems for many well-known logics have been proposed. In this paper, we deal with rejection systems for so-called non-deterministic finite-valued logics, a special case of non-deterministic many-valued logics which were introduced by Avron and Lev as a generalisation of traditional many-valued logics. More specifically, we introduce a systematic method for constructing sequent-style rejection systems for any given non-deterministic finite-valued logic. Furthermore, as special instances of our method, we provide concrete calculi for specific paraconsistent logics which can be characterised in terms of non-deterministic two- and three-valued semantics, respectively.

Logic of knowledge with infinitely many agents

Cut-free sequent calculus for logic of knowledge with infinitely many agents, based on multimodul S5n.

Linear logic in a refutational setting

Abstract Sequent-style refutation calculi with non-invertible rules are challenging to design because multiple proof-search strategies need to be simultaneously verified. In this paper, we present a refutation calculus for the multiplicative–additive fragment of linear logic ($\textsf{MALL}$) whose binary rule for the multiplicative conjunction $(\otimes )$ and the unary rule for the additive disjunction $(\oplus )$ fail invertibility. Specifically, we design a cut-free hypersequent calculus $\textsf{HMALL}$, which is equivalent to $\textsf{MALL}$, and obtained by transforming the usual tree-like shape of derivations into a parallel and linear structure. Next, we develop a refutation calculus $\overline{\textsf{HMALL}}$ based on the calculus $\textsf{HMALL}$. As far as we know, this is also the first refutation calculus for a substructural logic. Finally, we offer a fractional semantics for $\textsf{MALL}$—whereby its formulas are interpreted by a rational number in the closed interval [0, 1] —thus extending to the substructural landscape the project of fractional semantics already pursued for classical and modal logics.

Specification and verification of concurrent systems by causality and realizability

A logical theory for interface specification and verification of distributed, concurrent, interactive, real-time systems is worked out based on a semantic foundation including operational and denotational semantics. It supports a calculus for the specification and verification of concurrent interactive systems by interface assertions. Systems are composed acting concurrently and interacting via streams exchanged over their channels forming feedback loops. A denotational semantics is defined handling feedback communication by recursion and fixpoints based on strong causality and realizability instead of monotonicity. The resulting verification calculus for the specification logic is proved to be sound and relatively complete with respect to an operational semantics in terms of generalized Moore machines. Actually, two models of concurrent systems are defined, a more abstract one with communication and interaction modeled by untimed streams and a more concrete one working with timed streams. The untimed model is an abstraction of the timed model. The timed model allows expressing the laws of causality and realizability. Moreover, the timed model can be used to specify real-time properties.

Harmony and Normalisation in Bilateral Logic

In a recent paper del Valle-Inclan and Schlöder argue that bilateral calculi call for their own notion of proof-theoretic harmony, distinct from the usual (or ‘unilateral’) ones. They then put forward a specifically bilateral criterion of harmony, and present a harmonious bilateral calculus for classical logic. In this paper, I show how del Valle-Inclan and Schlöder’s criterion of harmony suggests a notion of normal form for bilateral systems, and prove normalisation for two (harmonious) bilateral calculi for classical logic, HB1 and HB2. The resulting normal derivations have the usual desirable features, like the separation and subformula properties. HB1-normal form turns out to be strictly stronger that the notion of normal form proposed by Nils Kürbis, and HB2-normal form is neither stronger nor weaker than a similar proposal by Marcello D’Agostino, Dov Gabbay, and Sanjay Modgyl.