Constructive Many-One Reduction from the Halting Problem to Semi-Unification

Semi-unification is the combination of first-order unification and first-order matching. The undecidability of semi-unification has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s by Turing reduction from Turing machine immortality (existence of a diverging configuration). The particular Turing reduction is intricate, uses non-computational principles, and involves various intermediate models of computation. The present work gives a constructive many-one reduction from the Turing machine halting problem to semi-unification. This establishes RE-completeness of semi-unification under many-one reductions. Computability of the reduction function, constructivity of the argument, and correctness of the argument is witnessed by an axiom-free mechanization in the Coq proof assistant. Arguably, this serves as comprehensive, precise, and surveyable evidence for the result at hand. The mechanization is incorporated into the existing, well-maintained Coq library of undecidability proofs. Notably, a variant of Hooper's argument for the undecidability of Turing machine immortality is part of the mechanization.Comment: CSL 2022 - LMCS special issu

Distributivity and Additivity: The Case of the Romanian Distributive Marker "câte"

The paper explores the semantic import of Romanian nominal distributive marker “câte” and identifies two distinct contributions to the interpretation of the DP hosting it: a licensing condition excluding non-distributive interpretations and an additivity implicature requiring that the distributive share should be monotonic with respect to the sortal key. The latter property can only be tested when the marker modifies measure phrases. For this purpose, an experiment was conducted in order to test the hypothesis that the association between “câte” and non-monotonic measure phrases correlates with a significantly lower acceptability as opposed to “câte” and monotonic measure phrases

Dieudonné-type theorems for lattice group-valued kk-triangular set functions

summary:Some versions of Dieudonné-type convergence and uniform boundedness theorems are proved, for $k$-triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case. Furthermore, we pose some open problems

Logic and Truth: Some Logics without Theorems

Two types of logical consequence are compared: one, with respect to matrix and designated elements and the other with respect to ordering in a suitable algebraic structure. Particular emphasis is laid on algebraic structures in which there is no top-element relative to the ordering. The significance of this special condition is discussed. Sequent calculi for a number of such structures are developed. As a consequence it is re-established that the notion of truth as such, not to speak of tautologies, is inessential in order to define validity of an argument

On the isomorphism problem of concept algebras

Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on {\it concepts}. They have been introduced to capture the equational theory of concept algebras \cite{Wi00}. They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in \cite{Kw04}, is whether complete {\wdl}s are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem \ref{T:main}). We also provide a new proof of a well known result due to M.H. Stone \cite{St36}, saying that {\em each Boolean algebra is a field of sets} (Corollary \ref{C:Stone}). Before these, we prove that the boundedness condition on the initial definition of {\wdl}s (Definition \ref{D:wdl}) is superfluous (Theorem \ref{T:wcl}, see also \cite{Kw09}).Comment: 15 page