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31
Full functional verification of linked data structures
 In ACM Conf. Programming Language Design and Implementation (PLDI
, 2008
"... We present the first verification of full functional correctness for a range of linked data structure implementations, including mutable lists, trees, graphs, and hash tables. Specifically, we present the use of the Jahob verification system to verify formal specifications, written in classical high ..."
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Cited by 79 (17 self)
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We present the first verification of full functional correctness for a range of linked data structure implementations, including mutable lists, trees, graphs, and hash tables. Specifically, we present the use of the Jahob verification system to verify formal specifications, written in classical higherorder logic, that completely capture the desired behavior of the Java data structure implementations (with the exception of properties involving execution time and/or memory consumption). Given that the desired correctness properties include intractable constructs such as quantifiers, transitive closure, and lambda abstraction, it is a challenge to successfully prove the generated verification conditions. Our Jahob verification system uses integrated reasoning to split each verification condition into a conjunction of simpler subformulas, then apply a diverse collection of specialized decision procedures,
Towards efficient satisfiability checking for boolean algebra with presburger arithmetic
 In CADE21
, 2007
"... 1 Introduction This paper considers the satisfiability problem for a logic that allows reasoning about sets and their cardinalities. We call this logic quantifierfree Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivationfor QFBAPA is proving the validity of formulas arisi ..."
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Cited by 28 (17 self)
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1 Introduction This paper considers the satisfiability problem for a logic that allows reasoning about sets and their cardinalities. We call this logic quantifierfree Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Our motivationfor QFBAPA is proving the validity of formulas arising from program verification [12,13,14], but
Decision procedures for algebraic data types with abstractions
 IN 37TH ACM SIGACTSIGPLAN SYMPOSIUM ON PRINCIPLES OF PROGRAMMING LANGUAGES (POPL), 2010. DECISION PROCEDURES FOR ORDERED COLLECTIONS 15 SHE75. SAHARON SHELAH. THE MONADIC THEORY OF ORDER. THA ANNALS OF MATHEMATICS OF MATHEMATICS
, 2010
"... We describe a family of decision procedures that extend the decision procedure for quantifierfree constraints on recursive algebraic data types (term algebras) to support recursive abstraction functions. Our abstraction functions are catamorphisms (term algebra homomorphisms) mapping algebraic data ..."
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Cited by 23 (10 self)
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We describe a family of decision procedures that extend the decision procedure for quantifierfree constraints on recursive algebraic data types (term algebras) to support recursive abstraction functions. Our abstraction functions are catamorphisms (term algebra homomorphisms) mapping algebraic data type values into values in other decidable theories (e.g. sets, multisets, lists, integers, booleans). Each instance of our decision procedure family is sound; we identify a widely applicable manytoone condition on abstraction functions that implies the completeness. Complete instances of our decision procedure include the following correctness statements: 1) a functional data structure implementation satisfies a recursively specified invariant, 2) such data structure conforms to a contract given in terms of sets, multisets, lists, sizes, or heights, 3) a transformation of a formula (or lambda term) abstract syntax tree changes the set of free variables in the specified way.
Using firstorder theorem provers in the Jahob data structure verification system
 In Byron Cook and Andreas Podelski, editors, Verification, Model Checking, and Abstract Interpretation, LNCS 4349
, 2007
"... Abstract. This paper presents our integration of efficient resolutionbased theorem provers into the Jahob data structure verification system. Our experimental results show that this approach enables Jahob to automatically verify the correctness of a range of complex dynamically instantiable data st ..."
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Cited by 21 (1 self)
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Abstract. This paper presents our integration of efficient resolutionbased theorem provers into the Jahob data structure verification system. Our experimental results show that this approach enables Jahob to automatically verify the correctness of a range of complex dynamically instantiable data structures, including data structures such as hash tables and search trees, without the need for interactive theorem proving or techniques tailored to individual data structures. Our primary technical results include: (1) a translation from higherorder logic to firstorder logic that enables the application of resolutionbased theorem provers and (2) a proof that eliminating type (sort) information in formulas is both sound and complete, even in the presence of a generic equality operator. Moreover, our experimental results show that the elimination of this type information dramatically decreases the time required to prove the resulting formulas. These techniques enabled us to verify complex correctness properties of Java programs such as a mutable set implemented as an imperative linked list, a finite map implemented as a functional ordered tree, a hash table with a mutable array, and a simple library system example that uses these container data structures. Our system verifies (in a matter of minutes) that data structure operations correctly update the finite map, that they preserve data structure invariants (such as ordering of elements, membership in appropriate hash table buckets, or relationships between sets and relations), and that there are no runtime errors such as null dereferences or array out of bounds accesses. 1
An integrated proof language for imperative programs
 In PLDI’09
"... We present an integrated proof language for guiding the actions of multiple reasoning systems as they work together to prove complex correctness properties of imperative programs. The language operates in the context of a program verification system that uses multiple reasoning systems to discharge ..."
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Cited by 18 (3 self)
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We present an integrated proof language for guiding the actions of multiple reasoning systems as they work together to prove complex correctness properties of imperative programs. The language operates in the context of a program verification system that uses multiple reasoning systems to discharge generated proof obligations. It is designed to 1) enable developers to resolve key choice points in complex program correctness proofs, thereby enabling automated reasoning systems to successfully prove the desired correctness properties; 2) allow developers to identify key lemmas for the reasoning systems to prove, thereby guiding the reasoning systems to find an effective proof decomposition; 3) enable multiple reasoning systems to work together productively to prove a single correctness property by providing a mechanism that developers can use to divide the property into lemmas, each of which is suitable for
Decidable logics combining heap structures and data
 IN: POPL’11
, 2011
"... We define a new logic, STRAND, that allows reasoning with heapmanipulating programs using deductive verification and SMT solvers. STRAND logic (“STRucture ANd Data ” logic) formulas express constraints involving heap structures and the data they contain; they are defined over a class of pointerstru ..."
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Cited by 17 (7 self)
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We define a new logic, STRAND, that allows reasoning with heapmanipulating programs using deductive verification and SMT solvers. STRAND logic (“STRucture ANd Data ” logic) formulas express constraints involving heap structures and the data they contain; they are defined over a class of pointerstructures R defined using MSOdefined relations over trees, and are of the form ∃⃗x∀⃗yϕ(⃗x, ⃗y), where ϕ is a monadic secondorder logic (MSO) formula with additional quantification that combines structural constraints as well as dataconstraints, but where the dataconstraints are only allowed to refer to ⃗x and ⃗y. The salient aspects of the logic are: (a) the logic is powerful, allowing existential and universal quantification over the nodes, and complex combinations of data and structural constraints; (b) checking Hoaretriples for linear blocks of statements with preconditions and postconditions expressed as Boolean combinations of existential and universal STRAND formulas reduces to satisfiability of a STRAND formula; (c) there are powerful decidable fragments of STRAND, one semantically defined and one syntactically defined, where the decision procedure works by combining the theory of MSO over trees and the quantifierfree theory of the underlying datalogic. We demonstrate the effectiveness and practicality of the logic by checking verification conditions generated in proving properties of several heapmanipulating programs, using a tool that combines an MSO decision procedure over trees (MONA) with an SMT solver for integer constraints (Z3).
Decision Procedures for Multisets with Cardinality Constraints
"... Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of ..."
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Cited by 11 (7 self)
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Abstract. Applications in software verification and interactive theorem proving often involve reasoning about sets of objects. Cardinality constraints on such collections also arise in these applications. Multisets arise in these applications for analogous reasons as sets: abstracting the content of linked data structure with duplicate elements leads to multisets. Interactive theorem provers such as Isabelle specify theories of multisets and prove a number of theorems about them to enable their use in interactive verification. However, the decidability and complexity of constraints on multisets is much less understood than for constraints on sets. The first contribution of this paper is a polynomialspace algorithm for deciding expressive quantifierfree constraints on multisets with cardinality operators. Our decision procedure reduces in polynomial time constraints on multisets to constraints in an extension of quantifierfree Presburger arithmetic with certain “unbounded sum ” expressions. We prove bounds on solutions of resulting constraints and describe a polynomialspace decision procedure for these constraints. The second contribution of this paper is a proof that adding quantifiers to a constraint language containing subset and cardinality operators yields undecidable constraints. The result follows by reduction from Hilbert’s 10th problem. 1
Verifying complex properties using symbolic shape analysis
 In Workshop on heap abstraction and verification
, 2007
"... One of the main challenges in the verification of software systems is the analysis of unbounded data structures with dynamic memory allocation, such as linked data structures and arrays. We describe Bohne, a new analysis for verifying data structures. Bohne verifies data structure operations and sho ..."
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Cited by 11 (6 self)
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One of the main challenges in the verification of software systems is the analysis of unbounded data structures with dynamic memory allocation, such as linked data structures and arrays. We describe Bohne, a new analysis for verifying data structures. Bohne verifies data structure operations and shows that 1) the operations preserve data structure invariants and 2) the operations satisfy their specifications expressed in terms of changes to the set of objects stored in the data structure. During the analysis, Bohne infers loop invariants in the form of disjunctions of universally quantified Boolean combinations of formulas, represented as sets of binary decision diagrams. To synthesize loop invariants of this form, Bohne uses a combination of decision procedures for Monadic SecondOrder Logic over trees, SMTLIB decision procedures (currently CVC Lite), and an automated reasoner within the Isabelle interactive theorem prover. This architecture shows that synthesized loop invariants can serve as a useful communication mechanism between different decision procedures. In addition, Bohne uses field constraint analysis, a combination mechanism that enables the use of uninterpreted function symbols
On Combining Theories with Shared Set Operations
"... Abstract. We explore the problem of automated reasoning about the nondisjoint combination of theories that share set variables and operations. We prove a combination theorem and apply it to show the decidability of the satisfiability problem for a class of formulas obtained by applying propositional ..."
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Cited by 10 (5 self)
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Abstract. We explore the problem of automated reasoning about the nondisjoint combination of theories that share set variables and operations. We prove a combination theorem and apply it to show the decidability of the satisfiability problem for a class of formulas obtained by applying propositional operations to quantified formulas belonging to several expressive decidable logics. 1
On Linear Arithmetic with Stars
"... Abstract. We consider an extension of integer linear arithmetic with a star operator that takes closure under vector addition of the set of solutions of linear arithmetic subformula. We show that the satisfiability problem for this language is in NP (and therefore NPcomplete). Our proof uses a gene ..."
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Cited by 8 (6 self)
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Abstract. We consider an extension of integer linear arithmetic with a star operator that takes closure under vector addition of the set of solutions of linear arithmetic subformula. We show that the satisfiability problem for this language is in NP (and therefore NPcomplete). Our proof uses a generalization of a recent result on sparse solutions of integer linear programming problems. We present two consequences of our result. The first one is an optimal decision procedure for a logic of sets, multisets, and cardinalities that has applications in verification, interactive theorem proving, and description logics. The second is NPcompleteness of the reachability problem for a class of “homogeneous ” transition systems whose transitions are defined using integer linear arithmetic formulas. 1