Quantified Underapproximation via Labeled Bunches

We present a framework for inductive definitions in the logic of bunched implications, BI, and formulate two sequent calculus proof systems for inductive reasoning in this framework. The first proof system adopts a traditional approach to inductive proof, extending the usual sequent calculus for predicate BI with explicit induction rules for the inductively defined predicates. The second system allows an alternative mode of reasoning with inductive definitions by cyclic proof. In this system, the induction rules are replaced by simple case-split rules, and the proof structures are cyclic graphs formed by identifying some sequent occurrences in a derivation tree. Because such proof structures are not sound in general, we demand that cyclic proofs must additionally satisfy a global trace condition that ensures soundness. We illustrate our inductive definition framework and proof systems with simple examples which indicate that, in our setting, cyclic proof may enjoy certain advantages over the traditional induction approach.

The Impact of Internal and External Resources, and Strategic Actions in Business Networks on Firm Performance in the Software Industry

Program sensitivity measures the distance between the outputs of a program when run on two related inputs. This notion, which plays a key role in areas such as data privacy and optimization, has been the focus of several program analysis techniques introduced in recent years. Among the most successful ones, we can highlight type systems inspired by linear logic, as pioneered by Reed and Pierce in the Fuzz programming language. In Fuzz, each type is equipped with its own distance, and sensitivity analysis boils down to type checking. In particular, Fuzz features two product types, corresponding to two different notions of distance: the tensor product combines the distances of each component by adding them, while the with product takes their maximum.In this work, we show that these products can be generalized to arbitrary $$L^p$$ L p distances, metrics that are often used in privacy and optimization. The original Fuzz products, tensor and with, correspond to the special cases $$L^1$$ L 1 and $$L^\infty $$ L ∞ . To ease the handling of such products, we extend the Fuzz type system with bunches—as in the logic of bunched implications—where the distances of different groups of variables can be combined using different $$L^p$$ L p distances. We show that our extension can be used to reason about quantitative properties of probabilistic programs.

Experience‐oriented problems can be solved based on previous cases or an expert's experience, but cannot be solved by conventional algorithms. Effectively managing knowledge acquisition and problem‐solving tasks is crucial in the construction domain in which experience accumulation is at a retarded pace. While Expert Systems are extensively employed to solve complex problems in the construction domain, the Case‐Based Reasoning technique provides a valuable alternative when previous cases can be properly modeled. However, use of a single technique alone may not yield optimum solutions. This paper presents a hybrid model, capable of integrating Case‐Based Reasoning and Expert System techniques, for knowledge acquisition and problem‐solving of experience‐oriented problems. The proposed model is implemented and demonstrated by estimating the slurry wall's duration at the project planning stage. Test results indicate that solutions generated by the proposed hybrid model are better than those generated by using a single technique.

Most systems based on separation logic consider only restricted forms of implication or non-separating conjunction, as full support for these connectives requires a non-trivial notion of variable context, inherited from the logic of bunched implications (BI). We show that in an expressive type theory such as Coq, one can avoid the intricacies of BI, and support full separation logic very efficiently, using the native structuring primitives of the type theory. Our proposal uses reflection to enable equational reasoning about heaps, and Hoare triples with binary postconditions to further facilitate it. We apply these ideas to Hoare Type Theory, to obtain a new proof technique for verification of higher-order imperative programs that is general, extendable, and supports very short proofs, even without significant use of automation by tactics. We demonstrate the usability of the technique by verifying the fast congruence closure algorithm of Nieuwenhuis and Oliveras, employed in the state-of-the-art Barcelogic SAT solver.

Most systems based on separation logic consider only restricted forms of implication or non-separating conjunction, as full support for these connectives requires a non-trivial notion of variable context, inherited from the logic of bunched implications (BI). We show that in an expressive type theory such as Coq, one can avoid the intricacies of BI, and support full separation logic very efficiently, using the native structuring primitives of the type theory.

The λΛ‐calculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proof‐objects of a structural variation, with Dereliction, of a fragment of BI, the logic of bunched implications. As such, it is also closely related to linear logic. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intuitionistic quantifiers. The λΛ‐calculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we study the categorical semantics of the λΛ‐calculus, gives a theory of ‘Kripke resource models’, i.e. monoid‐indexed sets of functorial Kripke models, in which the monoid gives an account of resource consumption. A class of concrete, set‐theoretic models is given by the category of families of sets parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.

We describe a polymorphic extension of the substructural lambda calculus αλ associated with the logic of bunched implications. This extension is particularly novel in that both variables and type variables are treated substructurally, being maintained through a system of zoned, bunched contexts. Polymorphic universal quantifiers are introduced in both additive and multiplicative forms, and then metatheoretic properties, including subject-reduction and normalization, are established. A sound interpretation in a class of indexed category models is defined and the construction of a generic model is outlined, yielding completeness. A concrete realization of the categorical models is given using pairs of partial equivalence relations on the natural numbers. Polymorphic existential quantifiers are presented, together with some metatheory. Finally, potential applications to closures and memory-management are discussed.

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We show how to extend O'Hearn and Pym's logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a non-conservative extension of (propositional) Boolean BI that includes multiplicative versions of falsity, negation and disjunction. We give an algebraic semantics for CBI that leads us naturally to consider resource models of CBI in which every resource has a unique dual. We then give a cut-eliminating proof system for CBI, based on Belnap's display logic, and demonstrate soundness and completeness of this proof system with respect to our semantics.

The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource that is rich enough, for example, to form the logical basis for ‘pointer logic’ and ‘separation logic’ semantics for programs that manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels for BI's tableaux to solve the resource-distribution problem, the labels being the elements of resource models. For BI with inconsistency, , the challenge consists in dealing with BI's Grothendieck topological models within such a proof-search method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from so-called dependency graphs. Then, from these results, we can define a new resource semantics of BI, based on partially defined monoids, and prove that this semantics is complete. Such a semantics, based on partiality, is closely related to the semantics of BI's (intuitionistic) pointer and separation logics. Returning to the tableaux calculus, we propose a new version with liberalised rules for which the countermodels are closely related to the topological Kripke semantics of BI. As consequences of the relationships between semantics of BI and resource tableaux, we prove two new strong results for propositional BI: its decidability and the finite model property with respect to topological semantics.

The logic of bunched implications (BI), introduced by O’Hearn and Pym, is a substructural logic which freely combines additive and multiplicative implications. Boolean BI (BBI) denotes BI with classical interpretation of additives and its model is the commutative monoid. We show that when the monoid is finitely generated and propositions are recursively defined, or the monoid is infinitely generated and propositions are restricted to generator propositions, the model checking problem is undecidable. In the case of finitely related monoid and generator propositions, the model checking problem is EXPSPACE-complete.KeywordsModel CheckWord ProblemCommutative SemigroupCongruence ClassGenerator PropositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a connection-based characterization of propositional BI (logic of bunched implications), a logic combining linear and intuitionistic connectives. This logic, with its sharing interpretation, has been recently used to reason about mutable data structures and needs proof search methods. Our connection-based characterization for BI is based on standard notions but involves, in a specific way, labels and constraints in order to capture the interactions between connectives during the proof-search. As BI is conservative w.r.t. intuitionistic logic and multiplicative intuitionistic linear logic, we deduce, by some restrictions, new connection-based characterizations and methods for both logics.

We show how to extend O'Hearn and Pym's logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a non-conservative extension of (propositional) Boolean BI that includes multiplicative versions of falsity, negation and disjunction. We give an algebraic semantics for CBI that leads us naturally to consider resource models of CBI in which every resource has a unique dual. We then give a cut-eliminating proof system for CBI, based on Belnap's display logic, and demonstrate soundness and completeness of this proof system with respect to our semantics.

Pseudo-distributive Laws