cs.LO

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, pQLL, indexed by a hardness degree $p$, prove cut-elimination theorem for them, and show completeness for enriched residuated `soft' lattices. For $p = \infty$, pQLL reduces to MALL, with provability in pQLL converging to provability in MALL when $p \to \infty$.

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, pQLL, indexed by a hardness degree $p$, prove cut-elimination theorem for them, and show completeness for enriched residuated `soft' lattices. For $p = \infty$, pQLL reduces to MALL, with provability in pQLL converging to provability in MALL when $p \to \infty$.

Cubical type theories are designed around an abstract unit interval from which types of paths, used to represent equalities, are defined. Varying the operations available on this interval yields different type theories. A reversal is an involutive operator on the interval that swaps its two endpoints. We show that for cubical type theories with self-dual interval theories, such as the minimal theory of two endpoints or the theory of a bounded distributive lattice, the extension of the theory with a reversal that internalizes the duality is a conservative extension. The key tool is a "twist construction": the product of an interval and its dual is again an interval with a reversal given by swapping coordinates. Our conservativity result applies to "opaque" cubical type theories, without strict equations reducing the filling operator at concrete type formers or eliminators from higher inductive types at path constructors. Using the same twist construction, we also construct models of strict cubical type theory with reversals in categories of cubical sets without reversals. We thereby give the first model of a theory with reversals whose homotopy theory corresponds to that of topological spaces.

We show that a partial-correctness assertion about an iterative program is provable in Hoare Logic iffit is provable in standard second-order logic with comprehension restricted to first-order predicates. This equivalence was claimed twice in the past, both with faulty proofs, and seems to be the first foundational characterization of Hoare Logic.