arXiv subjects

Viktor Kuncak

Publications and source records attributed to Viktor Kuncak.

26 records · Page 2Linked to original sources

arXiv2004

On Spatial Conjunction as Second-Order Logic

Spatial conjunction is a powerful construct for reasoning about dynamically allocated data structures, as well as concurrent, distributed and mobile computation. While researchers have identified many uses of spatial conjunction, its precise expressive power compared to traditional logical constructs was not previously known. In this paper we establish the expressive power of spatial conjunction. We construct an embedding from first-order logic with spatial conjunction into second-order logic, and more surprisingly, an embedding from full second order logic into first-order logic with spatial conjunction. These embeddings show that the satisfiability of formulas in first-order logic with spatial conjunction is equivalent to the satisfiability of formulas in second-order logic. These results explain the great expressive power of spatial conjunction and can be used to show that adding unrestricted spatial conjunction to a decidable logic leads to an undecidable logic. As one example, we show that adding unrestricted spatial conjunction to two-variable logic leads to undecidability. On the side of decidability, the embedding into second-order logic immediately implies the decidability of first-order logic with a form of spatial conjunction over trees. The embedding into spatial conjunction also has useful consequences: because a restricted form of spatial conjunction in two-variable logic preserves decidability, we obtain that a correspondingly restricted form of second-order quantification in two-variable logic is decidable. The resulting language generalizes the first-order theory of boolean algebra over sets and is useful in reasoning about the contents of data structures in object-oriented languages.

cs.LO↗

arXiv2004

The First-Order Theory of Sets with Cardinality Constraints is Decidable

We show that the decidability of the first-order theory of the language that combines Boolean algebras of sets of uninterpreted elements with Presburger arithmetic operations. We thereby disprove a recent conjecture that this theory is undecidable. Our language allows relating the cardinalities of sets to the values of integer variables, and can distinguish finite and infinite sets. We use quantifier elimination to show the decidability and obtain an elementary upper bound on the complexity. Precise program analyses can use our decidability result to verify representation invariants of data structures that use an integer field to represent the number of stored elements.

cs.LO↗

arXiv2004

On computing the fixpoint of a set of boolean equations

This paper presents a method for computing a least fixpoint of a system of equations over booleans. The resulting computation can be significantly shorter than the result of iteratively evaluating the entire system until a fixpoint is reached.

cs.PL↗

arXiv2004

Roles Are Really Great!

We present a new role system for specifying changing referencing relationships of heap objects. The role of an object depends, in large part, on its aliasing relationships with other objects, with the role of each object changing as its aliasing relationships change. Roles therefore capture important object and data structure properties and provide useful information about how the actions of the program interact with these properties. Our role system enables the programmer to specify the legal aliasing relationships that define the set of roles that objects may play, the roles of procedure parameters and object fields, and the role changes that procedures perform while manipulating objects. We present an interprocedural, compositional, and context-sensitive role analysis algorithm that verifies that a program respects the role constraints.

cs.PL↗

arXiv2004

Typestate Checking and Regular Graph Constraints

We introduce regular graph constraints and explore their decidability properties. The motivation for regular graph constraints is 1) type checking of changing types of objects in the presence of linked data structures, 2) shape analysis techniques, and 3) generalization of similar constraints over trees and grids. We define a subclass of graphs called heaps as an abstraction of the data structures that a program constructs during its execution. We prove that determining the validity of implication for regular graph constraints over the class of heaps is undecidable. We show undecidability by exhibiting a characterization of certain "corresponder graphs" in terms of presence and absence of homomorphisms to a finite number of fixed graphs. The undecidability of implication of regular graph constraints implies that there is no algorithm that will verify that procedure preconditions are met or that the invariants are maintained when these properties are expressed in any specification language at least as expressive as regular graph constraints.

cs.PL↗

arXiv2004

On the Theory of Structural Subtyping

We show that the first-order theory of structural subtyping of non-recursive types is decidable. Let $Σ$ be a language consisting of function symbols (representing type constructors) and $C$ a decidable structure in the relational language $L$ containing a binary relation $\leq$. $C$ represents primitive types; $\leq$ represents a subtype ordering. We introduce the notion of $Σ$-term-power of $C$, which generalizes the structure arising in structural subtyping. The domain of the $Σ$-term-power of $C$ is the set of $Σ$-terms over the set of elements of $C$. We show that the decidability of the first-order theory of $C$ implies the decidability of the first-order theory of the $Σ$-term-power of $C$. Our decision procedure makes use of quantifier elimination for term algebras and Feferman-Vaught theorem. Our result implies the decidability of the first-order theory of structural subtyping of non-recursive types.

cs.LO↗

arXiv2004

On Role Logic

We present role logic, a notation for describing properties of relational structures in shape analysis, databases, and knowledge bases. We construct role logic using the ideas of de Bruijn's notation for lambda calculus, an encoding of first-order logic in lambda calculus, and a simple rule for implicit arguments of unary and binary predicates. The unrestricted version of role logic has the expressive power of first-order logic with transitive closure. Using a syntactic restriction on role logic formulas, we identify a natural fragment RL^2 of role logic. We show that the RL^2 fragment has the same expressive power as two-variable logic with counting C^2 and is therefore decidable. We present a translation of an imperative language into the decidable fragment RL^2, which allows compositional verification of programs that manipulate relational structures. In addition, we show how RL^2 encodes boolean shape analysis constraints and an expressive description logic.

cs.PL↗

arXiv2004

On Generalized Records and Spatial Conjunction in Role Logic

We have previously introduced role logic as a notation for describing properties of relational structures in shape analysis, databases and knowledge bases. A natural fragment of role logic corresponds to two-variable logic with counting and is therefore decidable. We show how to use role logic to describe open and closed records, as well the dual of records, inverse records. We observe that the spatial conjunction operation of separation logic naturally models record concatenation. Moreover, we show how to eliminate the spatial conjunction of formulas of quantifier depth one in first-order logic with counting. As a result, allowing spatial conjunction of formulas of quantifier depth one preserves the decidability of two-variable logic with counting. This result applies to two-variable role logic fragment as well. The resulting logic smoothly integrates type system and predicate calculus notation and can be viewed as a natural generalization of the notation for constraints arising in role analysis and similar shape analysis approaches.

cs.PL↗