While specifications and verifications of concurrent systems employ Linear Temporal Logic , it is increasingly likely that logical consequence in image will be used in the description of computations and parallel reasoning. Our paper considers logical consequence in the standard image with temporal operations image and image . The prime result is an algorithm recognizing consecutions admissible in image, so we prove that image is decidable w.r.t. admissible inference rules. As a consequence we obtain algorithms verifying the validity of consecutions (...) in image and solving the satisfiability problem. We start by a simple reduction of logical consecutions of image to equivalent ones in the reduced normal form . Then we apply a semantic technique based on image-Kripke structures with formula definable subsets. This yields necessary and sufficient conditions for a consecution to be not admissible in image. These conditions lead to an algorithm which recognizes consecutions admissible in image by verifying the validity of consecutions in special finite Kripke structures of size square polynomial in reduced normal forms of the consecutions. As a consequence, this also solves the satisfiability problem for image. (shrink)

Positive modal algebras are the$$\left\langle { \wedge, \vee,\diamondsuit,\square,0,1} \right\rangle $$-subreducts of modal algebras. We prove that the variety of positive S4-algebras is not locally finite. On the other hand, the free one-generated positive S4-algebra is shown to be finite. Moreover, we describe the bottom part of the lattice of varieties of positive S4-algebras. Building on this, we characterize structurally complete varieties of positive K4-algebras.

A quasivariety of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐generated algebra in can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of all satisfy the same existential positive sentences. We prove that if is PSC then it still has the JEP, and if it has the JEP and (...) its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if is relatively semisimple then it is generated by one ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics. (shrink)

The logics RŁ, RP, and RG have been obtained by expanding Łukasiewicz logic Ł, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in Ł, P, and G. Namely, RŁ is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules (...) in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of RG, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is Q-universal. (shrink)

We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known (...) results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 is countable and distributive and it forms a Heyting algebra. (shrink)

Positive logics are $\{ \wedge, \vee, \to \}$-fragments of intermediate logics. It is clear that the positive fragment of $Int$ is not structurally complete. We give a description of all hereditarily structurally complete positive logics, while the question whether there is a structurally complete positive logic which is not hereditarily structurally complete, remains open.

Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic \. A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system \ is structurally complete if any rule admissible for the logic defined by \ is derivable in \. It is known that any (...) logic can be defined by a structurally complete deductive system—its structural completion. The main goal of the paper is to study the following problem: given a superintuitionistic logic L, is the structural completion of L hereditarily structurally complete? It is shown that, on the one hand, there is continuum many of such logics, including \, and many of its standard extensions. On the other hand, there is continuum many superintutitionistic logics structural completion of which is not hereditarily structurally complete. It is observed that the class of hereditarily structurally complete superintuitionistic consequence relations does not have the smallest element and it contains continuum many members lacking the finite model property. The following statement is instrumental in obtaining negative results: if a Lindenbaum algebra of formulas on one variable is finite and has more than 15 elements, then a structural completion of such a logic is not hereditarily structurally complete. (shrink)

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. (shrink)