Unification in description logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive description logic $\mathcal{EL}$ is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using $\mathcal{EL}$. On the other hand, unification in $\mathcal{EL}$ has been shown to be NP-complete and, thus, of considerably lower complexity than unification in other description logics of similarly restricted expressive power.

However, $\mathcal{EL}$ allows the use of the top concept ($\top$), which represents the whole interpretation domain, whereas the large medical ontology SNOMED CT makes no use of this feature. Surprisingly, removing the top concept from $\mathcal{EL}$ makes the unification problem considerably harder. More precisely, we will show that unification in $\mathcal{EL}$ without the top concept is PSPACE-complete. In addition to the decision problem, we also consider the problem of actually computing ${\mathcal{EL}}^{-\top }$-unifiers.

In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive lattices.

A term $\mathit{td}(p,q,r)$ is called a ternary deductive (TD) term for a variety of algebras $\mathcal{V}$ if the identity $\mathit{td}(p,p,r)\approx r$ holds in $\mathcal{V}$ and $(\mathsf{c},\mathsf{d})\in \theta (\mathsf{a},\mathsf{b})$ yields $\mathit{td}(\mathsf{a},\mathsf{b},\mathsf{c})\approx \mathit{td}(\mathsf{a},\mathsf{b},\mathsf{d})$ for any $\mathcal{A}\in \mathcal{V}$ and any principal congruence $\theta$ on $\mathcal{A}$. A connective $f(p_1,\dots ,p_n)$ is called $\mathit{td}$-distributive if $\mathit{td}(p,q,f(r_1,\dots ,r_n\left)\right)\approx$ $f\left(\mathit{td}\right(p,q,r_1),\dots ,\mathit{td}(p,q,r_n\left)\right)$. If $\mathsf{L}$ is a propositional logic and $\mathcal{V}$ is a corresponding variety (algebraic semantic) that has a TD term $\mathit{td}$, then any admissible in $\mathsf{L}$ rule, the premises of which contain only $\mathit{td}$-distributive operations, is derivable, and the substitution $r\mapsto \mathit{td}(p,q,r)$ is a projective $\mathsf{L}$-unifier for any formula containing only $\mathit{td}$-distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal to prove structural completeness of the implication fragment of intuitionistic propositional logic.

We characterize all finitary consequence relations over $\mathbf{S4.3}$, both syntactically, by exhibiting so-called (admissible) 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 $\mathbf{S4}$ has projective unification if and only if $L$ contains $\mathbf{S4.3}$. 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 $\mathbf{S4.3}$ (the lattice of quasivarieties of $\mathbf{S4.3}$-algebras) is countable and distributive and it forms a Heyting algebra.

We show the admissibility for $\mathsf{BCI}$ of a rule form of the characteristic implicational axiom of abelian logic, this rule taking us from $(\alpha \to \beta )\to \beta$ to $\alpha$. This is done in Section 8, with surrounding sections exploring the admissibility and derivability of various related rules in several extensions of $\mathsf{BCI}$.

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.