Intermediate Logics Research Articles

Vertically stacked, low-voltage organic ternary logic circuits including nonvolatile floating-gate memory transistors

Multi-valued logic (MVL) circuits based on heterojunction transistor (HTR) have emerged as an effective strategy for high-density information processing without increasing the circuit complexity. Herein, an organic ternary logic inverter (T-inverter) is demonstrated, where a nonvolatile floating-gate flash memory is employed to control the channel conductance systematically, thus realizing the stabilized T-inverter operation. The 3-dimensional (3D) T-inverter is fabricated in a vertically stacked form based on all-dry processes, which enables the high-density integration with high device uniformity. In the flash memory, ultrathin polymer dielectrics are utilized to reduce the programming/erasing voltage as well as operating voltage. With the optimum programming state, the 3D T-inverter fulfills all the important requirements such as full-swing operation, optimum intermediate logic value (~VDD/2), high DC gain exceeding 20 V/V as well as low-voltage operation (< 5 V). The organic flash memory exhibits long retention characteristics (current change less than 10% after 104 s), leading to the long-term stability of the 3D T-inverter. We believe the 3D T-inverter employing flash memory developed in this study can provide a useful insight to achieve high-performance MVL circuits.

Paraconsistent da Costa Weakening of Intuitionistic Negation: What does it mean?

In this paper we consider the systems of weakening of intuitionistic negation logic mZ, introduced in [1], [2], which are developed in the spirit of da Costa's approach. We take a particular attention on the philosophical considerations of the paraconsistent mZ logic w.r.t. the constructive semantics of the intuitionistic logic, and we show that mZ is a subintuitionistic logic. Hence, we present the relationship between intuitionistic and paraconsistent subintuitionistic negation used in mZ. Then we present a significant number of examples for this subintuitionistic and paraconsistent mZ logics: Logic Programming with Fiting's fixpoint semantics for paraconsistent weakening of 3-valued Kleene's and 4-valued Belnap's logics. Moreover, we provide a canonical construction of infinitary-valued mZ logics and, in particular, the paraconsistent weakening of standard Zadeh's fuzzy logic and of the Godel-Dummet t-norm intermediate logics.

EPSILON THEOREMS IN INTERMEDIATE LOGICS

AbstractAny intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s $\varepsilon $ -calculus. The first and second $\varepsilon $ -theorems for classical logic establish conservativity of the $\varepsilon $ -calculus over its classical base logic. It is well known that the second $\varepsilon $ -theorem fails for the intuitionistic $\varepsilon $ -calculus, as prenexation is impossible. The paper investigates the effect of adding critical $\varepsilon $ - and $\tau $ -formulas and using the translation of quantifiers into $\varepsilon $ - and $\tau $ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ${\varepsilon \tau }$ -calculi. The “extended” first $\varepsilon $ -theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second $\varepsilon $ -theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first $\varepsilon $ -theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic.

Finitary unification in locally tabular modal logics characterized

We provide necessary and sufficient conditions for finitary unification in locally tabular modal logics, solely in terms of Kripke models. We apply the conditions to establish the unification types of logics determined by simple finite frames. In particular, we show that unification is finitary (or unitary) in the logic determined by the fork (frame F4, see Fig. 6), the rhombus (frame F5), in GL.3m, GrzBd2, S4Bd2 and other logics; whereas it is nullary in the logic of F6, and of the pentagon FN5. In Appendix analogous results are given for superintuitionistic logics.

AN ALGEBRAIC APPROACH TO INQUISITIVE AND -LOGICS

AbstractThis article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1

Systematic Control of Negative Transconductance in Organic Heterojunction Transistor for High-Performance, Low-Power Flexible Ternary Logic Circuits.

Organic multi-valued logic (MVL) circuits can substantially improve the data processing efficiency in highly advanced wearable electronics. Organic ternary logic circuits can be implemented by utilizing the negative transconductance (NTC) of heterojunction transistors (H-TRs). To achieve high-performance organic ternary logic circuits, the range of NTC in H-TRs must be optimized in advance to ensure the well-defined intermediate logic state in ternary logic inverters (T-inverters). Herein, a simple and efficient strategy, which enables the systematic control of the range and position of NTC in H-TRs is presented. Each thickness of p-/n-type semiconductor in H-TRs is adjusted to control the channel conductivity. Furthermore, asymmetric source/drain (S/D) electrode structure is newly developed for H-TRs, which can adjust the amount of hole and electron injection, independently. Based on the semiconductor thickness variation and asymmetric S/D electrodes, the T-inverter exhibits full-swing operation with three distinguishable logic states, resulting in unprecedentedly high static noise margin (≈48% of the ideal value). Moreover, a flexible T-inverter with an ultrathin polymer dielectric is demonstrated, whose operating voltage is less than 8V. The proposed strategy is fully compatible with the conventional integrated circuit design, which is highly desirable for broad applicability and scalability for various types of T-inverter production.

On Implicational Intermediate Logics Axiomatizable by Formulas Minimal in Classical Logic: A Counter-Example to the Komori–Kashima Problem

On Implicational Intermediate Logics Axiomatizable by Formulas Minimal in Classical Logic: A Counter-Example to the Komori–Kashima Problem

One-variable fragments of intermediate logics over linear frames

A correspondence is established between one-variable fragments of (first-order) intermediate logics defined over a fixed countable linear frame and Gödel modal logics defined over many-valued equivalence relations with values in a closed subset of the real unit interval. It is also shown that each of these logics can be interpreted in the one-variable fragment of the corresponding constant domain intermediate logic, which is equivalent to a Gödel modal logic defined over (crisp) equivalence relations. Although the latter modal logics in general lack the finite model property with respect to their frame semantics, an alternative semantics is defined that has this property and used to establish co-NP-completeness results for the one-variable fragments of the corresponding intermediate logics both with and without constant domains.

THE LATTICE OF SUPER-BELNAP LOGICS

AbstractWe study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety.

Interpolation for intermediate logics via injective nested sequents

AbstractWe introduce a novel, semantically inspired method of constructing nested sequent calculi for propositional intermediate logics. Applying recently developed methods for proving Craig interpolation to these nested sequent calculi, we obtain constructive proofs of the interpolation property for most non-trivial interpolable intermediate logics, as well as Lyndon interpolation for Gödel logic. Finally, we provide a prototype implementation combining proof search and countermodel construction.

Two pretabular linear extensions of relevance logic R

Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett’s famous super-intuitionistic logic LC and the well-known semi-relevance logic RM (= R-Mingle). In this paper, we investigate Anderson and Belnap’s relevance logic R with the extra axiom: (p → q) ∨ (q → p), which we name LR, and which is (much) weaker than RM, and so is not pretabular. This means that over LR, there may be some pretabular extensions other than RM, two of which this paper presents and with which it provides axiomatizations, characteristic algebras, and Routley-Meyer semantics.

Proof complexity of substructural logics

In this paper, we investigate the proof complexity of a wide range of substructural systems. For any proof system P at least as strong as Full Lambek calculus, FL, and polynomially simulated by the extended Frege system for some superintuitionistic logic of infinite branching, we present an exponential lower bound on the proof lengths. More precisely, we will provide a sequence of P-provable formulas {An}n=1∞ such that the length of the shortest P-proof for An is exponential in the length of An. The lower bound also extends to the number of proof lines (proof lengths) in any Frege system (extended Frege system) for a logic between FL and any superintuitionistic logic of infinite branching. As an example, Hilbert-style proof systems for any finitely axiomatizable extension of FL that are weaker than the intuitionistic logic, in particular the usual Hilbert-style proof systems for the logics FLS for the set of structural rules S⊆{e,i,o,c}, fall in this category. We will also prove a similar result for the proof systems and logics extending Visser's basic propositional calculus BPC and its logic BPC, respectively. Finally, in the classical substructural setting, we will establish an exponential lower bound on the number of proof lines in any proof system polynomially simulated by the cut-free version of CFLew.

Inquisitive Heyting Algebras

In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures (prime elements represent declarative propositions, non-prime elements represent questions, join is a question-forming operation) and provide several alternative characterizations of these algebras. For instance, it is shown that a Heyting algebra is inquisitive if and only if its prime filters and filters generated by sets of prime elements coincide and prime elements are closed under relative pseudocomplement. We prove that the weakest inquisitive superintuitionistic logic is sound with respect to a Heyting algebra iff the algebra is what we call a homomorphic p-image of some inquisitive Heyting algebra. It is also shown that a logic is inquisitive iff its Lindenbaum–Tarski algebra is an inquisitive Heyting algebra.

Algorithmic properties of first-order superintuitionistic logics of finite Kripke frames in restricted languages

Abstract We consider the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate superintuitionistic logics of finite Kripke frames on the logics’ algorithmic properties. By a finite frame we mean a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that positive fragments of many predicate superintuitionistic logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter; this applies to the logics of finite frames of the predicate counterparts of propositional logics lying between the intuitionistic logic and the logic of the weak law of the excluded middle.

An Extension of the Stable Semantics via Lukasiewicz Logic

Logic Programming and fuzzy logic are active areas of research, and their scopes in terms of applications are growing fast. Fuzzy logic is a branch of many-valued logic based on the paradigm of inference under vagueness. In this work we recall some of the interplay between three 3-valued logics that are relevant in these areas: The Lukasiewicz logic, the intermediate logic G3 and the paraconsistent logic G3′, and we present a contribution to the area of answer sets that consists in extending a definition of stable model based on proof theory in logic G3, to a more general definition that can be based on any of the more expressive logics G3′ or Lukasiewicz. Finally we present and explore a new 4-valued logic that bears relation to G3 and to Lukasiewicz 4-valued logic.

Distinctive Field-Effect Transistors and Ternary Inverters Using Cross-Type WSe2/MoS2 Heterojunctions Treated with Polymer Acid.

The electrical and optical characteristics of two-dimensional (2D) transition-metal dichalcogenides (TMDCs) can be improved by surface modification. In this study, distinctive field-effect transistors (FETs) were realized by forming cross-type 2D WSe2/MoS2 p-n heterojunctions through surface treatment using poly(methyl methacrylate-co-methacrylic acid) (PMMA-co-PMAA). The FETs were applied to new ternary inverters as multivalued logic circuits (MVLCs). Laser confocal microscope photoluminescence spectroscopy indicated the generation of trions in the WSe2 and MoS2 layers, and the intensity decreased after PMMA-co-PMAA treatment. For the cross-type WSe2/MoS2 p-n heterojunction FETs subjected to PMMA-co-PMAA treatment, the channel current and the region of anti-ambipolar transistor characteristics increased considerably, and ternary inverter characteristics with three stable logic states, "1", "1/2", and "0", were realized. Interestingly, the intermediate logic state 1/2, which results from the negative differential transconductance characteristics, was realized by the turn-on of all component FETs, as the current of the FETs increased after PMMA-co-PMAA treatment. The electron-rich carboxyl acid moieties in PMMA-co-PMAA can undergo coordination with the metal Mo or W atoms present in the Se or S vacancies, respectively, resulting in the modulation of charge density. These features yielded distinctive FETs and ternary inverters for MVLCs using cross-type WSe2/MoS2 heterojunctions.

Epimorphism surjectivity in varieties of Heyting algebras

It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by the Rieger-Nishimura lattice, and all of its (locally finite) subvarieties that contain the original counter-example K. It is known that, whenever a variety of Heyting algebras has finite depth, then it has surjective epimorphisms. In contrast, we show that for every integer n⩾2, the variety of all Heyting algebras of width at most n has a non-surjective epimorphism. Within the so-called Kuznetsov-Gerčiu variety (i.e., the variety generated by finite linear sums of one-generated Heyting algebras), we describe exactly the subvarieties that have surjective epimorphisms. This yields new positive examples, and an alternative proof of epimorphism surjectivity for all varieties of Gödel algebras. The results settle natural questions about Beth-style definability for a range of intermediate logics.

Model Checking MITL Formulae on Timed Automata

Timed Automata (TA) is de facto a standard modelling formalism to represent systems when the interest is the analysis of their behaviour as time progresses. This modelling formalism is mostly used for checking whether the behaviours of a system satisfy a set of properties of interest. Even if efficient model-checkers for Timed Automata exist, these tools are not easily configurable. First, they are not designed to easily allow adding new Timed Automata constructs, such as new synchronization mechanisms or communication procedures, but they assume a fixed set of Timed Automata constructs. Second, they usually do not support the Metric Interval Temporal Logic (MITL) and rely on a precise semantics for the logic in which the property of interest is specified, which cannot be easily modified and customized. Finally, they do not easily allow using different solvers that may speed up verification in different contexts. This article presents a novel technique to perform model checking of Metric Interval Temporal Logic (MITL) properties on TA. The technique relies on the translation of both the TA and the MITL formula into an intermediate Constraint LTL over clocks (CLTLoc) formula, which is verified through an available decision procedure. The technique is flexible, since the intermediate logic allows the encoding of new semantics as well as new TA constructs, by just adding new CLTLoc formulae. Furthermore, our technique is not bound to a specific solver as the intermediate CLTLoc formula can be verified using different procedures.

A Generalized Proof-Theoretic Approach to Logical Argumentation Based on Hypersequents

In this paper we introduce hypersequent-based frameworks for the modelling of defeasible reasoning by means of logic-based argumentation and the induced entailment relations. These structures are an extension of sequent-based argumentation frameworks, in which arguments and the attack relations among them are expressed not only by Gentzen-style sequents, but by more general expressions, called hypersequents. This generalization allows us to overcome some of the known weaknesses of logical argumentation frameworks and to prove several desirable properties of the entailments that are induced by the extended (hypersequent-based) frameworks. It also allows us to incorporate as the deductive base of our formalism some well-known logics (like the intermediate logic LC, the modal logic S5, and the relevance logic RM), which lack cut-free sequent calculi, and so are not adequate for standard sequent-based argumentation. We show that hypersequent-based argumentation yields robust defeasible variants of these logics, with many desirable properties.

On the concurrent computational content of intermediate logics

We provide a proofs-as-concurrent-programs interpretation for a large class of intermediate logics that can be formalized by cut-free hypersequent calculi. Obtained by adding classical disjunctive tautologies to intuitionistic logic, these logics are used to type concurrent λ-calculi by Curry–Howard correspondence; each of the calculi features a specific communication mechanism, enhanced expressive power when compared to the λ-calculus, and implements forms of code mobility. We thus confirm Avron's 1991 thesis that intermediate logics formalizable by hypersequent calculi can serve as basis for concurrent λ-calculi.