Abstract. The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of -modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP , one can factor out the complexity of encoding specific features of logical systems which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and sub-structural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincaré Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP . We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP : strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments. Moreover, we provide a canonical presentation of LFP , based on a suitable extension of the notion of βη-long normal form, allowing for smooth formulations of adequacy statements.

In this paper, we introduce the probabilistic justification logic PJ, a logic in which we can reason about the probability of justification statements. We present its syntax and semantics, and establish a strong completeness theorem. Moreover, we investigate the relationship between PJ and the logic of uncertain justifications.

We present a framework for trustworthy symbolic execution of JavaScripts programs, whose aim is to assist developers in the testing of their code: the developer writes symbolic tests for which the framework provides concrete counter-models. We create the framework following a new, general methodology for designing compositional program analyses for dynamic languages. We prove that the underlying symbolic execution is sound and does not generate false positives. We establish additional trust by using the theory to precisely guide the implementation and by thorough testing. We apply our framework to whole-program symbolic testing of real-world JavaScript libraries and compositional debugging of separation logic specifications of JavaScript programs.

We extend the constructive dependent type theory of the Logical Framework LF with monadic, dependent type constructors indexed with predicates over judgements, called Locks. These monads capture various possible proof attitudes in establishing the judgment of the object logic encoded by an LF type. Standard examples are factoring-out the verification of a constraint or delegating it to an external oracle, or supplying some non-apodictic epistemic evidence, or simply discarding the proof witness of a precondition deeming it irrelevant. This new framework, called Lax Logical Framework, LLFP , is a conservative extension of LF, and hence it is the appropriate metalanguage for dealing formally with side-conditions in rules or external evidence in logical systems. LLFP arises once the monadic nature of the lock type-constructor, L P M,σ [•], introduced by the authors in a series of papers, together with Marina Lenisa, is fully exploited. The nature of the lock monads permits to utilize the very Lock destructor, U P M,σ [•], in place of Moggi's monadic letT , thus simplifying the equational theory. The rules for U P M,σ [•] permit also the removal of the monad once the constraint is satisfied. We derive the meta-theory of LLFP by a novel indirect method based on the encoding of LLFP in LF. We discuss encodings in LLFP of call-by-value λ-calculi, Hoare's Logic, and Fitch-Prawitz Naive Set Theory.