Differentiable Logics are deployed in neuro-symbolic learning tasks as a way of embedding logical constraints in the training objective of neural networks. A differentiable logic consists of a syntax to write logical properties and a semantics to interpret them as real-valued functions to be folded in the loss function. A defining trade-off of the field is that between logical properties of the connectives, and analytic concerns for the semantics, with both aspects being relevant in applications. At one extreme we find fuzzy logics, that have well-established algebraic and proof-theoretic foundations, and at the other ad-hoc differentiable logics like Fischer's DL2, conceived for deep learning applications. However, no satisfactory foundation has emerged yet. We propose a resolution to this long-standing tension via a novel logic, QLL, with foundational ambitions. Our design is driven by naturality---the idea that, since logical constraints are translated to losses, the semantics of the connectives should be pertinent operations used in ML practice (that is, sum and log-sum-exp) on additive quantities (like logits). We then judge the result on two aspects: logical adequacy---that they satisfy most of the standard logical laws of Linear Logic; and empirical effectiveness---test-time performance (as measured by adversarial attacks) is well-correlated to the actual verification of the logical constraints (as measured by off-the-shelf neural network verifier), which makes QLL stand out among SoTA techniques.

Real-valued logics have seen a renewed interest in verification for probabilistic and quantitative systems, in particular machine learning models, where they can be used to directly integrate specifications in the training objective. To do so effectively one has to strike a balance between the logical properties of the connectives and their semantics. A major hurdle in this sense is to give ''soft'' (i.e. differentiable) semantics to additive connectives---in linear and fuzzy logics, additives are necessarily ''hard'' lattice operations.
In this paper, we solve this problem by combining an accurate analysis of the properties of sum and product on the reals with a significant revision of sequent calculus. We introduce 'quantitative sequent calculi', which simultaneously generalize hypersequent calculi of fuzzy logics and deep inference, and in which validity of a proof and provability of a sequent are real-valued quantities. We present a family of calculi, \(p\)QLL, indexed by a hardness degree \(p\), prove cut-elimination theorem for them, and show completeness for enriched residuated 'soft' lattices. For \(p = \infty \), \(p\)QLL reduces to MALL, with provability in \(p\)QLL converging to provability in MALL when \(p \to \infty \).

Assume-guarantee reasoning is a technique for compositional model checking in which system specifications are checked under certain assumptions on system parameters or inputs, and provide guarantees on observations of system state. We present a categorical framework for assume-guarantee reasoning for safety problems by viewing systems as lenses, following our earlier work on the compositionality of generalized Moore machines. Generalized Moore machines include ordinary Moore machines, partially observable Markov (decision) processes, and systems of parameterized ODEs (control systems); our framework gives assume-guarantee reasoning specially adapted to each of these cases. In particular, we give a novel formulation of assume-guarantee reasoning for (local) input-to-state stability ((L)ISS) Lyapunov functions on systems of parameterized ODEs.
Our framework is categorically natural and straightforwardly compositional. A flavor of generalized Moore machine is determined by a tangency: a fibration with a section. We show that symmetric monoidal loose right modules of assume-guarantee certified generalized Moore machines over symmetric monoidal double categories of certified wiring diagrams can be constructed 2-functorially from fibrations internal to the 2-category of tangencies.

Talk at the internal seminar for the ARIA 'Safeguarded AI' programme on propositional quantitative linear logic.