Intermediate Logics Research Articles

Gödel–Dummett linear temporal logic

We investigate a version of linear temporal logic whose propositional fragment is Gödel–Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics: first a real-valued semantics, where statements have a degree of truth in the real unit interval, and second a ‘bi-relational’ semantics. We then show that these two semantics indeed define one and the same logic: the statements that are valid for the real-valued semantics are the same as those that are valid for the bi-relational semantics. This Gödel temporal logic does not have any form of the finite model property for these two semantics: there are non-valid statements that can only be falsified on an infinite model. However, by using the technical notion of a quasimodel, we show that every falsifiable statement is falsifiable on a finite quasimodel, yielding an algorithm for deciding if a statement is valid or not. Later, we strengthen this decidability result by giving an algorithm that uses only a polynomial amount of memory, proving that Gödel temporal logic is PSPACE-complete. We also provide a deductive calculus for Gödel temporal logic, and show this calculus to be sound and complete for the above-mentioned semantics, so that all (and only) the valid statements can be proved with this calculus.

Universal proof theory: Feasible admissibility in intuitionistic modal logics

We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities {□,◇} and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including CK, IK, their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments CK□, propositional lax logic and IPC. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than IPC has a constructive sequent calculus.

A model template for reachability-based containment checking of imprecise observations in timed automata

AbstractVerifying safety requirements by model checking becomes increasingly important for safety-critical applications. For the validity of such proof in practice, the model needs to capture the actual behavior of the real system, which could be tested by containment checks of real observation traces. Basic equivalence checks, however, are not applicable if the system is only partially or imprecisely observable, if the model abstracts from explicit states with symbolic semantics, or if the checks are not expressible in the logics supported by a model checker. In this article, we solve the problem of observation containment checking in timed automata via reachability checking on tester systems. We introduce the logic SRL (sequence reachability logic) to express observations as sequences of delayed reachability properties. Through SBLL (introduced by Aceto et al.) as intermediate logic, we synthesize a set of matcher model templates for partial and imprecise observations and further extend these templates for the case of limited state accessibility in a model. For the obtained matching traces, we define the back-transformation into the original model domain and formally prove the correctness of the transformation. We implemented the observation matching approach, and apply it to a set of 7 demo and 3 case study models with different levels of observability. The results show that all positive and negative observations are correctly classified, and that the most advanced matcher model instance still offers average run times between 0.1 and 1 s in all but 3 scenarios.

Bi-intermediate logics of trees and co-trees

A bi-Heyting algebra validates the Gödel-Dummett axiom (p→q)∨(q→p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension bi-GD of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice Λ(bi-GD) of extensions of bi-GD.We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of bi-GD. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of bi-GD. We introduce a sequence of co-trees, called the finite combs, and show that a logic in Λ(bi-GD) is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of bi-GD and consequently, a unique pre-locally tabular extension of bi-GD. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.

A Generic Solution to the Sorites Paradox

AbstractThis paper offers a generic revenge-proof solution to the Sorites paradox that is compatible with several philosophical approaches to vagueness, including epistemicism, supervaluationism, psychological contextualism and intuitionism. The solution is traditional in that it rejects the Sorites conditional and proposes a modally expressed weakened conditional instead. The modalities are defined by the first-order logic QS4M+FIN. (This logic is a modal companion to the intermediate logic QH+KF, which places the solution between intuitionistic and classical logic.) Borderlineness is introduced modally as usual. The solution is innovative in that its modal system brings out the semi-determinability of vagueness. Whether something is borderline and whether a predicate is vague or precise is only semi-determinable: higher-order vagueness is columnar. Finally, the solution is based entirely on two assumptions. (1) It rejects the Sorites conditional. (2) It maintains that if one specifies borderlineness in terms of the ‒suitably interpreted‒ modal logic QS4M+FIN, then one can explain why the Sorites appears paradoxical. From (1)+(2) it results that one can tell neither where exactly in a Sorites series the borderline zone starts and ends nor what its extension is. Accordingly, the solution is also called agnostic.

A Mixture of Negative-, Zero-, and Positive-Differential Transconductance Switching from Tellurium/Indium Gallium Zinc Oxide Heterostructures.

Conventional transistors have long emphasized signal modulation and amplification, often sidelining polarity considerations. However, the recent emergence of negative differential transconductance, characterized by a drain current decline during gate voltage sweeping, has illuminated an unconventional path in transistor technology. This phenomenon promises to simplify the implementation of ternary logic circuits and enhance energy efficiency, especially in multivalued logic applications. Our research has culminated in the development of a sophisticated mixed transconductance transistor (M-T device) founded on a precise Te and IGZO heterojunction. The M-T device exhibits a sequence of intriguing phenomena, zero differential transconductance (ZDT), positive differential transconductance (PDT), and negative differential transconductance (NDT) contingent on applied gate voltage. We clarify its operation using a three-segment equivalent circuit model and validate its viability with IGZO TFT, Te TFT, and Te/IGZO TFT components. In a concluding demonstration, the M-T device interconnected with Te TFT achieves a ternary inverter with an intermediate logic state. Remarkably, this configuration seamlessly transitions into a binary inverter when it is exposed to light.

ONE-VARIABLE FRAGMENTS OF FIRST-ORDER LOGICS

AbstractThe one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases—notably, the modal counterparts $\mathrm {S5}$ and $\mathrm {MIPC}$ of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively—but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically defined first-order logic—spanning families of intermediate, substructural, many-valued, and modal logics—to admit a certain natural axiomatization. More precisely, an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, using a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.

Algorithmic properties of modal and superintuitionistic logics of monadic predicates over finite Kripke frames

Abstract We show that the monadic fragment of the modal predicate logic of a single Kripke frame with finitely many possible worlds, but possibly infinite domains, is decidable. This holds true even for multimodal logics with equality, regardless of whether equality is interpreted as identity or as congruence. By the Gödel–Tarski translation, similar results follow for superintuitionistic predicate logics, with or without equality. Using these observations, we establish upper algorithmic bounds, which match the known lower bounds, for monadic fragments of some modal predicate logics. In particular, we prove that, if $L$ is a propositional modal logic contained in $\textbf{S5}$, $\textbf{GL.3}$ or $\textbf{Grz.3}$ and the class of finite Kripke frames validating $L$ is recursively enumerable, then the monadic fragment with equality of the predicate logic of finite Kripke frames validating $L$ is $\varPi ^{0}_{1}$-complete; this, in particular, holds if $L$ is one of the following propositional logics: $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{KTB}$, $\textbf{K4}$, $\textbf{K4.3}$, $\textbf{S4}$, $\textbf{S4.3}$, $\textbf{GL}$, $\textbf{Grz}$, $\textbf{K5}$, $\textbf{K45}$ and $\textbf{S5}$. We also prove that monadic fragments with equality of logics $\textbf{QAlt}^=_{n}$ and $\textbf{QTAlt}^=_{n}$ are decidable. The obtained results are easily extendable to the multimodal versions of the predicate logics we consider and to logics with the Barcan formula.

Merging Intuitionistic and De Morgan Logics

We introduce De Morgan Heyting logic for Heyting algebras with De Morgan negation (DH-algebras). The variety DH of all DH-algebras is congruence distributive. The lattice of all subvarieties of DH is distributive. We show the discrete dualities between De Morgan frames and DH-algebras. The Kripke completeness and finite approximability of some DH-logics are proven. Some conservativity of DH expansion of a Kripke complete superintuitionistic logic is shown by the construction of frame expansion. Finally, a cut-free terminating Gentzen sequent calculus for the DH-logic of De Morgan Boolean algebras is developed.

Efficient design approaches to model CNTFET-based Ternary Schmitt Trigger circuits

Digital computation using radix three offers the benefit of increased information density with less power consumption due to reduced interconnect complexity. Hence this brief presents two architectures for ternary Schmitt trigger designs that are implemented by combining shifting literals, ternary inverters and decoder equivalents using carbon nanotube technology. In the first proposed design, Schmitt trigger hysteresis curves are realized using shifting literals i.e. successor and predecessor cells. The second proposed design employs ternary inverters and a decoder equivalent of intermediate logic level for the implementation. The experimental analysis involves the simulations that are conducted using HSPICE and the standard Stanford CNTFET model. Simulation results confirm that the proposed ternary Schmitt trigger designs outperform in terms of power consumption and power delay product(PDP), showcasing the power reduction average of 75% and PDP reduction average of 79% respectively in comparison to recent counterparts. Moreover, Monte Carlo simulations are conducted to verify the robust operation of proposed designs and lesser deviations are observed in performance parameters towards process variations.

Uniform instruction set extensions for multiplications in contemporary and post-quantum cryptography

Hybrid key encapsulation is in the process of becoming the de-facto standard for integration of post-quantum cryptography (PQC). Supporting two cryptographic primitives is a challenging task for constrained embedded systems. Both contemporary cryptography based on elliptic curves or RSA and PQC based on lattices require costly multiplications. Recent works have shown how to implement lattice-based cryptography on big-integer coprocessors. We propose a novel hardware design that natively supports the multiplication of polynomials and big integers, integrate it into a RISC-V core, and extend the RISC-V ISA accordingly. We provide an implementation of Saber and X25519 to demonstrate that both lattice- and elliptic-curve-based cryptography benefits from our extension. Our implementation requires only intermediate logic overhead, while significantly outperforming optimized ARM Cortex M4 implementations, other hardware/software codesigns, and designs that rely on contemporary accelerators.

Nested sequents for intermediate logics: the case of Gödel-Dummett logics

We present nested sequent systems for propositional Gödel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these Gödel-Dummett logics, we introduce a new structural rule, called the linearity rule, which (bottom-up) operates by linearising branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking if data exists along certain paths within a nested sequent. Such rules require us to generalise our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalisation of the usual nested sequent formalism. Our calculi exhibit favourable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems.

Binary modal logic and unary modal logic

Abstract Standard unary modal logic and binary modal logic, i.e. modal logic with one binary operator, are shown to be definitional extensions of one another when an additional axiom $U$ is added to the basic axiomatization of the binary side. This is a strengthening of our previous results. It follows that all unary modal logics extending Classical Modal Logic, in other words all unary modal logics with a neighborhood semantics, can equivalently be seen as binary modal logics. This in particular applies to standard modal logics, which can be given simple natural axiomatizations in binary form. We illustrate this in the logic K. We call such logics binary expansions of the unary modal logics. There are many more such binary expansions than the ones given by the axiom $U$. We initiate an investigation of the properties of these expansions and in particular of the maximal binary expansions of a logic. Our results directly imply that all sub- and superintuitionistic logics with a standard modal companion also have binary modal companions. The latter also applies to the weak subintuitionistic logic WF of our previous papers. This logic doesn’t seem to have a unary modal companion.

Wright’s Strict Finitistic Logic in the Classical Metatheory: The Propositional Case

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional part of ‘strict finitistic logic’ from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model.

О трехзначных расширениях логики Клини

The paper is devoted to one of the most famous three-valued systems – Kleene's logic. The expressive capabilities of Kleene's logic and its three-valued expansions are described. We present two results. First, all possible three-valued expansions of Kleene's logic are found up to equivalence with respect to the mutual definability of connectives. It is shown that there are only twelve such expansions. This list includes both logics already known in the literature and completely new ones. For the found expansions, we describe the structure of the lattice ordered relative to the expressive power of its elements. Secondly, for Kleene's logic and its three-valued expansions we find how many extensions each of these logics has in the same language. Kleene's logic has only two proper extensions: the classical and the trivial ones. Generally, a three-valued logic in which Kleene's matrix is definable contains no more than three proper extensions: the classical one, the trivial one, and an intermediate logic, determined by the product of the the original logic's matrix and the matrix of classical logic in the same signature. Intermediate logics exist only for two types of three-valued expansions of Kleene's logic: in expansions equivalent to Łukasiewicz's logic, and in logics whose matrices contain both a bivalent submatrix, the universe of which consists of the classical truth values, and a submatrix, the universe of which consists the intermediate value alone. All three-valued expansions of Kleene's logic that do not preserve the classical values have only one extension of their own – the trivial one.

POLYHEDRAL COMPLETENESS OF INTERMEDIATE LOGICS: THE NERVE CRITERION

Abstract We investigate a recently devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper.

Hereditarily Structurally Complete Intermediate Logics: Citkin’s Theorem Via Duality

A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics.

On the proof complexity of logics of bounded branching

We investigate the proof complexity of extended Frege (EF) systems for basic transitive modal logics (K4, S4, GL, …) augmented with the bounded branching axioms BBk. First, we study feasibility of the disjunction property and more general extension rules in EF systems for these logics: we show that the corresponding decision problems reduce to total coNP search problems (or equivalently, disjoint NP pairs, in the binary case); more precisely, the decision problem for extension rules is equivalent to a certain special case of interpolation for the classical EF system. Next, we use this characterization to prove superpolynomial (or even exponential, with stronger hypotheses) separations between EF and substitution Frege (SF) systems for all transitive logics contained in S4.2GrzBB2 or GL.2BB2 under some assumptions weaker than PSPACE≠NP. We also prove analogous results for superintuitionistic logics: we characterize the decision complexity of multi-conclusion Visser's rules in EF systems for Gabbay–de Jongh logics Tk, and we show conditional separations between EF and SF for all intermediate logics contained in T2+KC.

Abductive Reasoning in Intuitionistic Propositional Logic via Theorem Synthesis

AbstractWith help of a compact Prolog-based theorem prover for Intuitionistic Propositional Logic, we synthesize minimal assumptions under which a given formula formula becomes a theorem. After applying our synthesis algorithm to cover basic abductive reasoning mechanisms, we synthesize conjunctions of literals that mimic rows of truth tables in classical or intermediate logics and we abduce conditional hypotheses that turn the theorems of classical or intermediate logics into theorems in intuitionistic logic. One step further, we generalize our abductive reasoning mechanism to synthesize more expressive sequent premises using a minimal set of canonical formulas, to which arbitrary formulas in the calculus can be reduced while preserving their provability. Organized as a self-contained literate Prolog program, the paper supports interactive exploration of its content and ensures full replicability of our results.

On intermediate justification logics

Abstract We study arbitrary intermediate propositional logics extended with a collection of axioms from (classical) justification logics. For these, we introduce various semantics by combining either Heyting algebras or Kripke frames with the usual semantic machinery used by Mkrtychev’s, Fitting’s or Lehmann and Studer’s models for classical justification logics. We prove unified completeness theorems for all intermediate justification logics and their corresponding semantics using a respective propositional completeness theorem of the underlying intermediate logic. Further, by a modification of a method of Fitting, we prove unified realization theorems for a large class of intermediate justification logics and accompanying intermediate modal logics.