Computations in fragments of intuitionistic propositional logic

Abstract

This article ~s a report on research m progress into the structure of fimte diagrams of mtuinomst~c proposmonal logic with the aid of automated reasoning systems for larger calculanons. A/ragment of a proposmonal logic is the set of formulae built up from a fimte number of proposinonal variables by means of a number of connectives of the logic, among which possibly non-standard ones like ~ or ~ which are stud~ed here. The dtagram of that fragment ~s the set of eqmvalence classes of its formulae parnally ordered by the derivability relation. N.G. de Brmjn's concept of exact model has been used to construct subdlagrams of the [p, q, A, ~, ~]-fragment.

Modality Definition Synthesis for Epistemic Intuitionistic Logic via a Theorem Prover

“…As the IPC fragment with two variables, implication and negation has exactly 518 equivalence classes of formulas [15,16], one would expect the construction deriving "*" from "#" to reach a fixpoint. We can use our prover to find out when that happens.…”

Algebraic semantics for the ‐fragment of and its properties

“…Note that the cardinality of the free equivalential algebras with zero has been already computed in these three cases in . In the figures below, each dot denotes a labelled element of the frame, and a solid circle or ellipse shows an equivalence class in the frame.…”

Algebraic semantics for the (↔, ¬¬)‐fragment of IPC

“…The fragments of intuitionistic logic with double negation were also studied, e.g., in [2,10,14] as well as in [5], where ¬¬ was treated as a modal operator. Our construction works for some of these fragments, e.g., for (→, ¬¬) and (∧, →, ¬¬), where as the equivalent algebraic semantics one can take Hilbert algebras with regularization (HI r ) and Brouwerian semilattices with regularization (B r ), respectively, i.e., Hilbert algebras or Brouwerian semilattices endowed with a retraction r into the set of regular elements that is simultaneously a closure operator with respect to the natural order.…”