Abstract
In the context of intuitionistic sequent calculus, naturality means permutation-freeness (the terminology is essentially due to Mints). We study naturality in the context of the lambda-calculus with generalized applications and its multiary extension, to cover, under the Curry-Howard correspondence, proof systems ranging from natural deduction (with and without general elimination rules) to a fragment of sequent calculus with an iterable left-introduction rule, and which can still be recognized as a call-by-name lambda-calculus. In this context, naturality consists of a certain restricted use of generalized applications. We consider the further restriction obtained by the combination of naturality with normality w.r.t. the commutative conversion engendered by generalized applications. This combination sheds light on the interpretation of naturality as a vectorization mechanism, allowing a multitude of different ways of structuring lambda-terms, and the structuring of a multitude of interesting fragments of the systems under study. We also consider a relaxation of naturality, called weak naturality: this not only brings similar structural benefits, but also suggests a new weak system of natural deduction with generalized applications which is exempt from commutative conversions. In the end, we use all of this evidence as a stepping stone to propose a computational interpretation of generalized application (whether multiary or not, and without any restriction): it includes, alongside the argument(s) for the function, a general list - a new, very general, vectorization mechanism, that structures the continuation of the computation.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Abstract
With the incresasing importance of program verification, an issue that has been receiving more attention is the certification of verification tools, addressing the vernacular question: Who verifies the verifier?. In this paper we approach this meta-verification problem by focusing on a fundamental component of program verifiers: the Verification Conditions Generator (VCGen), responsible for producing a set of proof obligations from a program and a specification. The semantic foundations of VCGens lie in program logics, such as Hoare logic, Dynamic logic, or Separation logic, and related predicate transformers. Dynamic logic is the basis of the KeY system, one of the foremost deductive verifiers, whose logic makes use of the notion of update, which is quite intricate to formalize. In this paper we derive systematically, based on a KeY-style dynamic logic, a correct-by-construction VCGen for a toy programming language. Our workflow covers the entire process, from the logic to the VCGen. It is implemented in the Why3 tool, which is itself a program verifier. We prove the soundness and (an appropriate notion of) completeness of the logic, then define a VCGen for our language and establish its soundness. Dynamic logic is one of a variety of research topics that our dear friend and colleague Luis Soares Barbosa has, over the years, initiated and promoted at the University of Minho. It is a pleasure for us to dedicate this work to him on the occasion of his 60th birthday.

Abstract
This paper presents a minimal model of the functioning of program verification and property checking tools based on (i) the encoding of loops as non-iterating programs, either conservatively, making use of invariants and assume/assert commands, or in a bounded way; and (ii) the use of an intermediate single-assignment (SA) form. The model captures the basic workflow of tools like Boogie, Why3, or CBMC, building on a clear distinction between operational and axiomatic semantics. This allows us to consider separately the soundness of program annotation, loop encoding, translation into SA form, and VC generation, as well as appropriate notions of completeness for each of these processes. To the best of our knowledge, this is the first formalization of a bounded model checking of software technique, including soundness and completeness proofs using Hoare logic; we also give the first completeness proof of a deductive verification technique based on a conservative encoding of invariant-annotated loops with assume/assert in SA form, as well as the first soundness proof based on a program logic. © 2019 IEEE.

Abstract
In a world where many human lives depend on the correct behavior of software systems, program verification assumes a crucial role. Many verification tools rely on an algorithm that generates verification conditions (VCs) from code annotated with properties to be checked. In this paper, we revisit two major methods that are widely used to produce VCs: predicate transformers (used mostly by deductive verification tools) and the conditional normal form transformation (used in bounded model checking of software). We identify three different aspects in which the methods differ (logical encoding of control flow, use of contexts, and semantics of asserts), and show that, since they are orthogonal, they can be freely combined. This results in six new hybrid verification condition generators (VCGens), which together with the fundamental methods constitute what we call the VCGen cube. We consider two optimizations implemented in major program verification tools and show that each of them can in fact be applied to an entire face of the cube, resulting in optimized versions of the six hybrid VCGens. Finally, we compare all VCGens empirically using a number of benchmarks. Although the results do not indicate absolute superiority of any given method, they do allow us to identify interesting patterns. © 2018 IEEE.

Abstract
This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a ?-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry- Howard interpretation (a kind of formal vector notation for -terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs. © José Espírito Santo, Maria João Frade, and Luís Pinto; licensed under Creative Commons License CC-BY 22nd International Conference on Types for Proofs and Programs (TYPES 2016).