2 research outputs found
Modality Definition Synthesis for Epistemic Intuitionistic Logic via a Theorem Prover
We derive a Prolog theorem prover for an Intuitionistic Epistemic Logic by
starting from the sequent calculus {\bf G4IP} that we extend with operator
definitions providing an embedding in intuitionistic propositional logic ({\bf
IPC}). With help of a candidate definition formula generator, we discover
epistemic operators for which axioms and theorems of Artemov and Protopopescu's
{\em Intuitionistic Epistemic Logic} ({\bf IEL}) hold and formulas expected to
be non-theorems fail. We compare the embedding of {\bf IEL} in {\bf IPC} with a
similarly discovered successful embedding of Dosen's double negation modality,
judged inadequate as an epistemic operator. Finally, we discuss the failure of
the {\em necessitation rule} for an otherwise successful {\bf S4} embedding and
share our thoughts about the intuitions explaining these differences between
epistemic and alethic modalities in the context of the
Brouwer-Heyting-Kolmogorov semantics of intuitionistic reasoning and knowledge
acquisition. Keywords: epistemic intuitionistic logic, propositional
intuitionistic logic, Prolog-based theorem provers, automatic synthesis of
logic systems, definition formula generation algorithms, embedding of modal
logics into intuitionistic logic
Formula Transformers and Combinatorial Test Generators for Propositional Intuitionistic Theorem Provers
We develop combinatorial test generation algorithms for progressively more
powerful theorem provers, covering formula languages ranging from the
implicational fragment of intuitionistic logic to full intuitionistic
propositional logic. Our algorithms support exhaustive and random generators
for formulas of these logics.
To provide known-to-be-provable formulas, via the Curry-Howard
formulas-as-types correspondence, we use generators for typable lambda terms
and combinator expressions. Besides generators for several classes of formulas,
we design algorithms that restrict formula generation to canonical
representatives among equiprovable formulas and introduce program
transformations that reduce formulas to equivalent formulas of a simpler
structure. The same transformations, when applied in reverse, create harder
formulas that can catch soundness or incompleteness bugs.
To test the effectiveness of the testing framework itself, we describe use
cases for deriving lightweight theorem provers for several of these logics and
for finding bugs in known theorem provers. Our Prolog implementation available
at: https://github.com/ptarau/TypesAndProofs and a subset of formula generators
and theorem provers, implemented in Python is available at:
https://github.com/ptarau/PythonProvers.
Keywords: term and formula generation algorithms, Prolog-based theorem
provers, formulas-as-types, type inference and type inhabitation, combinatorial
testing, finding bugs in theorem provers.Comment: 32 page