6 research outputs found
Towards an Intelligent Tutor for Mathematical Proofs
Computer-supported learning is an increasingly important form of study since
it allows for independent learning and individualized instruction. In this
paper, we discuss a novel approach to developing an intelligent tutoring system
for teaching textbook-style mathematical proofs. We characterize the
particularities of the domain and discuss common ITS design models. Our
approach is motivated by phenomena found in a corpus of tutorial dialogs that
were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor
for textbook-style mathematical proofs can be built on top of an adapted
assertion-level proof assistant by reusing representations and proof search
strategies originally developed for automated and interactive theorem proving.
The resulting prototype was successfully evaluated on a corpus of tutorial
dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453
A Relational Logic for Higher-Order Programs
Relational program verification is a variant of program verification where
one can reason about two programs and as a special case about two executions of
a single program on different inputs. Relational program verification can be
used for reasoning about a broad range of properties, including equivalence and
refinement, and specialized notions such as continuity, information flow
security or relative cost. In a higher-order setting, relational program
verification can be achieved using relational refinement type systems, a form
of refinement types where assertions have a relational interpretation.
Relational refinement type systems excel at relating structurally equivalent
terms but provide limited support for relating terms with very different
structures.
We present a logic, called Relational Higher Order Logic (RHOL), for proving
relational properties of a simply typed -calculus with inductive types
and recursive definitions. RHOL retains the type-directed flavour of relational
refinement type systems but achieves greater expressivity through rules which
simultaneously reason about the two terms as well as rules which only
contemplate one of the two terms. We show that RHOL has strong foundations, by
proving an equivalence with higher-order logic (HOL), and leverage this
equivalence to derive key meta-theoretical properties: subject reduction,
admissibility of a transitivity rule and set-theoretical soundness. Moreover,
we define sound embeddings for several existing relational type systems such as
relational refinement types and type systems for dependency analysis and
relative cost, and we verify examples that were out of reach of prior work.Comment: Submitted to ICFP 201
Toward Structured Proofs for Dynamic Logics
We present Kaisar, a structured interactive proof language for differential
dynamic logic (dL), for safety-critical cyber-physical systems (CPS). The
defining feature of Kaisar is *nominal terms*, which simplify CPS proofs by
making the frequently needed historical references to past program states
first-class. To support nominals, we extend the notion of structured proof with
a first-class notion of *structured symbolic execution* of CPS models. We
implement Kaisar in the theorem prover KeYmaera X and reproduce an example on
the safe operation of a parachute and a case study on ground robot control. We
show how nominals simplify common CPS reasoning tasks when combined with other
features of structured proof. We develop an extensive metatheory for Kaisar. In
addition to soundness and completeness, we show a formal specification for
Kaisar's nominals and relate Kaisar to a nominal variant of dL