The papers selected for this Special Issue address many of these topics, with foci on problems and topics, such as: How to model cut-free arithmetic for interfaces with existing proof-engines; The design of algorithms for provability or unprovability in many implicational logics; A constructive approach, leading to proved-correct procedures, or various search problems; The efficient management of linear contexts in linear logic programming; For non-trivial applications, the adaptation of a well-known procedure to increase the automation of proof-search in the Calculus of Constructions; The use of proof-nets as an alternative tool for automated deduction in linear logic; Reasoning about specifications of computation with an intuitionistic logic that includes definitions and induction; Characterization of cases in which classical provability and intuitionistic provability coincide and their relationships with uniform provability; Proof-search in intuitionistic logic viewed as a pertubation on proof-search in classical logic via a type-theoretic, and hence semantic, calculus.