Bunched Implications Research Articles

Semantical Analysis of the Logic of Bunched Implications

We give a novel approach to proving soundness and completeness for a logic (henceforth: the object-logic) that bypasses truth-in-a-model to work directly with validity. Instead of working with specific worlds in specific models, we reason with eigenworlds (i.e., generic representatives of worlds) in an arbitrary model. This reasoning is captured by a sequent calculus for a meta-logic (in this case, first-order classical logic) expressive enough to capture the semantics of the object-logic. Essentially, one has a calculus of validity for the object-logic. The method proceeds through the perspective of reductive logic (as opposed to the more traditional paradigm of deductive logic), using the space of reductions as a medium for showing the behavioural equivalence of reduction in the sequent calculus for the object-logic and in the validity calculus. Rather than study the technique in general, we illustrate it for the logic of Bunched Implications (BI), thus IPL and MILL (without negation) are also treated. Intuitively, BI is the free combination of intuitionistic propositional logic and multiplicative intuitionistic linear logic, which renders its meta-theory is quite complex. The literature on BI contains many similar, but ultimately different, algebraic structures and satisfaction relations that either capture only fragments of the logic (albeit large ones) or have complex clauses for certain connectives (e.g., Beth’s clause for disjunction instead of Kripke’s). It is this complexity that motivates us to use BI as a case-study for this approach to semantics.

A bunch of sessions: a propositions-as-sessions interpretation of bunched implications in channel-based concurrency

The emergence of propositions-as-sessions, a Curry-Howard correspondence between propositions of Linear Logic and session types for concurrent processes, has settled the logical foundations of message-passing concurrency. Central to this approach is the resource consumption paradigm heralded by Linear Logic. In this paper, we investigate a new point in the design space of session type systems for message-passing concurrent programs. We identify O’Hearn and Pym’s Logic of Bunched Implications (BI) as a fruitful basis for an interpretation of the logic as a concurrent programming language. This leads to a treatment of non-linear resources that is radically different from existing approaches based on Linear Logic. We introduce a new π-calculus with sessions, called πBI; its most salient feature is a construct called spawn, which expresses new forms of sharing that are induced by structural principles in BI. We illustrate the expressiveness of πBI and lay out its fundamental theory: type preservation, deadlock-freedom, and weak normalization results for well-typed processes; an operationally sound and complete typed encoding of an affine λ-calculus; and a non-interference result for access of resources.

A separation logic for negative dependence

Formal reasoning about hashing-based probabilistic data structures often requires reasoning about random variables where when one variable gets larger (such as the number of elements hashed into one bucket), the others tend to be smaller (like the number of elements hashed into the other buckets). This is an example of negative dependence , a generalization of probabilistic independence that has recently found interesting applications in algorithm design and machine learning. Despite the usefulness of negative dependence for the analyses of probabilistic data structures, existing verification methods cannot establish this property for randomized programs. To fill this gap, we design LINA, a probabilistic separation logic for reasoning about negative dependence. Following recent works on probabilistic separation logic using separating conjunction to reason about the probabilistic independence of random variables, we use separating conjunction to reason about negative dependence. Our assertion logic features two separating conjunctions, one for independence and one for negative dependence. We generalize the logic of bunched implications (BI) to support multiple separating conjunctions, and provide a sound and complete proof system. Notably, the semantics for separating conjunction relies on a non-deterministic , rather than partial, operation for combining resources. By drawing on closure properties for negative dependence, our program logic supports a Frame-like rule for negative dependence and monotone operations. We demonstrate how LINA can verify probabilistic properties of hash-based data structures and balls-into-bins processes.

A probabilistic separation logic

Probabilistic independence is a useful concept for describing the result of random sampling—a basic operation in all probabilistic languages—and for reasoning about groups of random variables. Nevertheless, existing verification methods handle independence poorly, if at all. We propose a probabili stic separation logic PSL, where separation models probabilistic independence. We first give a new, probabilistic model of the logic of bunched implications (BI). We then build a program logic based on these assertions, and prove soundness of the proof system. We demonstrate our logic by verifying information-theoretic security of cryptographic constructions for several well-known tasks, including private information retrieval, oblivious transfer, secure multi-party addition, and simple oblivious RAM. Our proofs reason purely in terms of high-level properties, like independence and uniformity.

Resource semantics

The Logic of Bunched Implications (BI) was introduced by O'Hearn and Pym. The original presentation of BI emphasised its role as a system for formal logic (broadly in the tradition of relevant logic) that has some interesting properties, combining a clean proof theory, including a categorical interpretation, with a simple truth-functional semantics. BI quickly found significant applications in program verification and program analysis, chiefly through a specific theory of BI that is commonly known as 'Separation Logic'. We survey the state of work in bunched logics - which, by now, is a quite large family of systems, including modal and epistemic logics and logics for layered graphs - in such a way as to organize the ideas into a coherent (semantic) picture with a strong interpretation in terms of resources. One such picture can be seen as deriving from an interpretation of BI's semantics in terms of resources, and this view provides a basis for a systematic interpretation of the family of bunched logics, including modal, epistemic, layered graph, and process-theoretic variants, in terms of resources. We explain the basic ideas of resource semantics, including comparisons with Linear Logic and ideas from economics and physics. We include discussions of BI's λ-calculus, of Separation Logic, and of an approach to distributed systems modelling based on resource semantics.

Semantics column

This quarter's Semantics column is devoted to a survey of BI-Logic - the logic of bunched implications - by David Pym. BI-Logics have a broad range of applications, including program verification. In particular they underpin Separation Logic which has attracted considerable attention both in the theory community and in industry. Notably, Steve Brookes and Peter O'Hearn were awarded the 2016 Gödel Prize for their work developing Concurrent Separation logic (CSL) (an accomplishment we were pleased to acknowledge in this series [Brookes and O'Hearn 2016]), and CSL has been successfully deployed at Facebook, and elsewhere. In fact, it was Peter who proposed that I invite David to write a column on BI-Logics, a suggestion I am happy to acknowledge.

A Stone-type Duality Theorem for Separation Logic Via its Underlying Bunched Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because — in addition to elegant abstraction — they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality theorems for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar Boolean BI, and concluding with Separation Logic. Our results encompass all the known existing algebraic approaches to Separation Logic and prove them sound with respect to the standard store-heap semantics. We additionally recover soundness and completeness theorems of the specific truth-functional models of these logics as presented in the literature. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualises the ‘resource semantics’ interpretation underpinning Separation Logic amongst them. As a consequence, theory from those fields — as well as algebraic and topological methods — can be applied to both Separation Logic and the systems of bunched logics it is built upon. Conversely, the notion of indexed resource frame (generalizing the standard model of Separation Logic) and its associated completeness proof can easily be adapted to other non-classical predicate logics.

A Substructural Modal Logic of Utility

Abstract We introduce a substructural modal logic of utility that can be used to reason aboutoptimality with respect to properties of states. Our notion of state is quite general, and is able to represent resource allocation problems in distributed systems. The underlying logic is a variant of the modal logic of bunched implications, and based on resource semantics, which is closely related to concurrent separation logic. We consider a labelled transition semantics and establish conditions under which Hennessy—Milner soundness and completeness hold. By considering notions of cost, strategy and utility, we are able to formulate characterizations of Pareto optimality, best responses, and Nash equilibrium within resource semantics. We also show that our logic is able to serve as a logic for a fully featured process algebra and explain the interaction between utility and the structure of processes.

Bunched sequential information

It is known that the logic BI of bunched implications is a logic of resources. Many studies have reported on the applications of BI to computer science. In this paper, an extension BIS of BI by adding a sequence modal operator is introduced and studied in order to formalize more fine-grained resource-sensitive reasoning. By the sequence modal operator of BIS, we can appropriately express “sequential information” in resource-sensitive reasoning. A Gentzen-type sequent calculus SBIS for BIS is introduced, and the cut-elimination and decidability theorems for SBIS are proved. An extension of the Grothendieck topological semantics for BI is introduced for BIS, and the completeness theorem with respect to this semantics is proved. The cut-elimination, decidability and completeness theorems for SBIS and BIS are proved using some theorems for embedding BIS into BI.

A modal separation logic for resource dynamics

The logic of Bunched implications (BI), and its boolean version (Boolean BI), are logics that allow us to express properties on resources and to provide logical frameworks for the so-called separation logics. In this paper we study a new modal separation logic that extends Boolean BI with two kinds of modalities, in order to deal with resources having dynamic properties (which depend on the current state of a system) and also to capture some resource evolutions or transformations. We show how we can model concurrent processes manipulating resources, and we provide a sound and complete tableau calculus, with a counter-model extraction method, for proving properties expressed in this logic.

Layered graph logic as an assertion language for access control policy models

We describe a uniform logical framework, based on a bunched logic that combines classical additives and very weak multiplicatives, for reasoning compositionally about access control policy models. We show how our approach takes account of the underlying system architecture, and so provides a way to identify and reason about how vulnerabilities may arise (and be removed) as a result of the architecture of the system. We consider, using frame rules, how local properties of access control policies are maintained as the system architecture evolves.

Single-succedent System Approach to Boolean BI

Abstract It is known that the logic BI of bunched implications is useful for describing shared mutable data structures and resource-aware reasoning. It has recently been clarified that some classical versions of BI are especially useful for describing shared mutable data structures. In this paper, a single-succedent Gentzen-type sequent calculus GcBI for a classical version (called Boolean BI) of an intuitionistic BI is introduced. Some theorems for embedding GcBI into its intuitionistic version GiBI, which are analogous to the Glivenko and Godel-Gentzen embedding theorems of classical logic into intuitionistic logic, are proved.

Temporal BI: Proof system, semantics and translations

The logic BI of bunched implications is a combination of intuitionistic logic and multiplicative intuitionistic linear logic. In this paper, a temporal extension tBI of BI is introduced and studied. A Gentzen-type sequent calculus GtBI for tBI is introduced, and the cut-elimination and decidability theorems for GtBI are proved using a theorem for syntactically embedding GtBI into a sequent calculus GBI for BI. A semantics for GtBI is introduced extending the Grothendieck topological semantics for BI, and the completeness theorem with respect to this semantics is proved using a theorem for semantically embedding tBI into BI. A semantics for GtBI without additive falsity constant is introduced extending the Kripke resource semantics for BI without additive falsity constant, and the completeness theorem with respect to this semantics is proved in a similar way. Moreover, an intuitionistic temporal linear logic, ITLL, is introduced as a Gentzen-type sequent calculus, and a theorem for embedding GtBI into ITLL is proved using a temporal extension of the Girard translation of intuitionistic logic into intuitionistic linear logic.

A theorem prover for Boolean BI

While separation logic is acknowledged as an enabling technology for large-scale program verification, most of the existing verification tools use only a fragment of separation logic that excludes separating implication. As the first step towards a verification tool using full separation logic, we develop a nested sequent calculus for Boolean BI (Bunched Implications), the underlying theory of separation logic, as well as a theorem prover based on it. A salient feature of our nested sequent calculus is that its sequent may have not only smaller child sequents but also multiple parent sequents, thus producing a graph structure of sequents instead of a tree structure. Our theorem prover is based on backward search in a refinement of the nested sequent calculus in which weakening and contraction are built into all the inference rules. We explain the details of designing our theorem prover and provide empirical evidence of its practicality.

Inconsistency-Tolerant Bunched Implications

It is known that logical systems with the property of paraconsistency can deal with inconsistency-tolerant and uncertainty reasoning more appropriately than systems which are non-paraconsistent. It is also known that the logic BI of bunched implications is useful for formalizing resource-sensitive reasoning. In this paper, a paraconsistent extension PBI of BI is studied. The logic PBI is thus intended to formalize an appropriate combination of inconsistency-tolerant reasoning and resource-sensitive reasoning. A Gentzen-type sequent calculus SPBI for PBI is introduced, and the cut-elimination and decidability theorems for SPBI are proved. An extension of the Grothendieck topological semantics for BI is introduced for PBI, and the completeness theorem with respect to this semantics is proved.

Bunched Logics Displayed

We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cut-elimination, and are sound and complete with respect to their standard presentations. We show how to constrain applications of display-equivalence in our calculi in such a way that an exhaustive proof search need be only finitely branching, and establish a full deduction theorem for the bunched logics with classical additives, BBI and CBI. We also show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic are very unlikely to exist.

Higher-order functional reactive programming in bounded space

Functional reactive programming (FRP) is an elegant and successful approach to programming reactive systems declaratively. The high levels of abstraction and expressivity that make FRP attractive as a programming model do, however, often lead to programs whose resource usage is excessive and hard to predict. In this paper, we address the problem of space leaks in discrete-time functional reactive programs. We present a functional reactive programming language that statically bounds the size of the dataflow graph a reactive program creates, while still permitting use of higher-order functions and higher-type streams such as streams of streams. We achieve this with a novel linear type theory that both controls allocation and ensures that all recursive definitions are well-founded. We also give a denotational semantics for our language by combining recent work on metric spaces for the interpretation of higher-order causal functions with length-space models of space-bounded computation. The resulting category is doubly closed and hence forms a model of the logic of bunched implications.

BI as an assertion language for mutable data structures

Reynolds has developed a logic for reasoning about mutable data structures in which the pre- and postconditions are written in an intuitionistic logic enriched with a spatial form of conjunction. We investigate the approach from the point of view of the logic BI of bunched implications of O'Hearn and Pym. We begin by giving a model in which the law of the excluded middle holds, thus showing that the approach is compatible with classical logic. The relationship between the intuitionistic and classical versions of the system is established by a translation, analogous to a translation from intuitionistic logic into the modal logic S4. We also consider the question of completeness of the axioms. BI's spatial implication is used to express weakest preconditions for object-component assignments, and an axiom for allocating a cons cell is shown to be complete under an interpretation of triples that allows a command to be applied to states with dangling pointers. We make this latter a feature, by incorporating an operation, and axiom, for disposing of memory. Finally, we describe a local character enjoyed by specifications in the logic, and show how this enables a class of frame axioms, which say what parts of the heap don't change, to be inferred automatically.

A Unified Display Proof Theory for Bunched Logic

We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cut-elimination, and are sound and complete with respect to their standard presentations. We show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic seem very unlikely to exist.

An Alternative Direct Simulation of Minsky Machines into Classical Bunched Logics via Group Semantics

Recently, Brotherston & Kanovich, and independently Larchey-Wendling & Galmiche, proved the undecidability of the bunched implication logic BBI. Moreover, Brotherston & Kanovich also proved the undecidability of the related logic CBI, as well as its neighbours. All of the above results are based on encodings of two-counter Minsky machines, but are derived using different techniques. Here, we show that the technique of Larchey-Wendling & Galmiche can also be extended, via group Kripke semantics, to prove the undecidability of CBI. Hence, we propose an alternative direct simulation of Minsky machines into both BBI and CBI. We identify a fragment called elementary Boolean BI (eBBI) which is common to the BBI/CBI families of logics and we show that the problem of Minsky machine acceptance can be encoded into eBBI. The soundness of the encoding is derived from the soundness of a goal directed sequent calculus designed for eBBI. The faithfulness of the encoding is obtained from a Kripke model based on the free commutative group Zn.