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Abstract Interpretation over Non-lattice Abstract Domains [chapter]

Graeme Gange, Jorge A. Navas, Peter Schachte, Harald Søndergaard, Peter J. Stuckey
2013 Lecture Notes in Computer Science  
Nonetheless, and for good reason, the literature on program analysis provides many examples of non-lattice domains, including non-convex numeric domains.  ...  Much of the power and elegance of that framework rests on the assumption that an abstract domain is a lattice.  ...  We now consider abstract interpretation over abstract domains that are only required to be bounded posets.  ... 
doi:10.1007/978-3-642-38856-9_3 fatcat:czcjdjiuajc53o5ofg6l63v35q

Combining abstract interpreters

Sumit Gulwani, Ashish Tiwari
2006 Proceedings of the 2006 ACM SIGPLAN conference on Programming language design and implementation - PLDI '06  
We present a methodology for automatically combining abstract interpreters over given lattices to construct an abstract interpreter for the combination of those lattices.  ...  We prove that our combination methodology yields the most precise abstract interpretation operators over the logical product of lattices when the individual lattices are over theories that are convex,  ...  The process of analyzing a program using abstract interpretation over a given abstract domain involves two main steps: designing the abstract interpretation operators for the abstract domain, and implementing  ... 
doi:10.1145/1133981.1134026 dblp:conf/pldi/GulwaniT06 fatcat:y5zkvl77dfasxosmfz4hucjjva

Combining abstract interpreters

Sumit Gulwani, Ashish Tiwari
2006 SIGPLAN notices  
We present a methodology for automatically combining abstract interpreters over given lattices to construct an abstract interpreter for the combination of those lattices.  ...  We prove that our combination methodology yields the most precise abstract interpretation operators over the logical product of lattices when the individual lattices are over theories that are convex,  ...  The process of analyzing a program using abstract interpretation over a given abstract domain involves two main steps: designing the abstract interpretation operators for the abstract domain, and implementing  ... 
doi:10.1145/1133255.1134026 fatcat:grdxypzkbncwtk2djze4feo23a

Abstract interpretation by dynamic partitioning

François Bourdoncle
1992 Journal of functional programming  
interesting abstract domains using data structures built over simpler domains.  ...  AbstractThe essential part of abstract interpretation is to build a machine-representable abstract domain expressing interesting properties about the possible states reached by a program at runtime.  ...  P snr ( )) the set of normalized subsets which have a non-redundant (resp. strongly non-redundant) domain.  ... 
doi:10.1017/s0956796800000496 fatcat:exidd7oy6zfeneloeshqcjxdwu

Abstract conflict driven learning

Vijay D'Silva, Leopold Haller, Daniel Kroening
2013 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '13  
Our generalisation provides a new method for static analyzers that operate over non-distributive lattices to reason about properties that require disjunction.  ...  We show that this algorithm can be generalised to solve the lattice-theoretic problem of determining if an additive transformer on a Boolean lattice is always bottom.  ...  We thank the abstract interpretation community, in particular Patrick and Radhia Cousot, for their support and encouragement of this work.  ... 
doi:10.1145/2429069.2429087 dblp:conf/popl/DSilvaHK13 fatcat:qyrtgnb655fbvee7pvyb463pra

Abstract Interpretation using a Language of Symbolic Approximation [article]

Matthieu Lemerre, Sébastien Bardin
2017 arXiv   pre-print
To solve these problems, we propose a new abstract interpretation framework, based on symbolic expressions used both as an abstraction of the program, and as the input analyzed by abstract domains.  ...  The traditional abstract domain framework for imperative programs suffers from several shortcomings; in particular it does not allow precise symbolic abstractions.  ...  [13] combines their symbolic abstract domain with constraint propagation over non-relational domains.  ... 
arXiv:1712.10058v1 fatcat:mkeooyor75fsdle62gffznuanq

Constraint-based reachability

Arnaud Gotlieb, Tristan Denmat, Nadjib Lazaar
2013 Electronic Proceedings in Theoretical Computer Science  
The keypoint of the approach is to interpret imperative constructions such as conditionals, loops, array and memory manipulations with the fundamental notion of constraint over a computational domain.  ...  This paper proposes an interpretation of classical filtering consistencies used in Constraint Programming as abstract domain computations, and shows how this approach can be used to produce a constraint  ...  an element of the abstract lattice that over-approximates the initial state.  ... 
doi:10.4204/eptcs.107.4 fatcat:2h3pzfzp3nginkn4dnkwp5u6ey

Compositional optimization of disjunctive abstract interpretations [chapter]

Roberto Giacobazzi, Francesco Ranzato
1996 Lecture Notes in Computer Science  
We show that the least disjunctive basis exists in most cases, and study its properties in relation with reduced product of abstract interpretations.  ...  We define the inverse operation for disjunctive completion, introducing the notion of least disjunctive basis for an abstract domain D: this is the most abstract domain inducing the same disjunctive completion  ...  The lattice of abstract interpretations.  ... 
doi:10.1007/3-540-61055-3_34 fatcat:ckulaamogzhubltw77l7esm72y

Computing Abstract Distances in Logic Programs [article]

Ignacio Casso, Jose F. Morales, Pedro Lopez-Garcia, Manuel V. Hermenegildo
2019 arXiv   pre-print
We survey and extend existing proposals for distances and metrics in lattices or abstract domains, and we propose metrics for some common domains used in logic program analysis, as well as extensions of  ...  However, choosing the abstract domain, widening, fixpoint, etc. that provides the best precision-cost trade-off remains an open problem.  ...  This is relevant because many abstract domains, such as those that are combinations of two different abstract domains, or non-relational domains which provide an abstract value from a lattice for each  ... 
arXiv:1907.13263v1 fatcat:47z53qq6azhkvphmslya4tg3zm

Abstract satisfaction

Vijay D'Silva, Leopold Haller, Daniel Kroening
2014 Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '14  
This article introduces an abstract interpretation framework that codifies the operations in SAT and SMT solvers in terms of lattices, transformers and fixed points.  ...  The lattices of tuples P(Val n ) represent the domains over which function symbols are interpreted.  ...  The lattices of environments P(Env Γ) represent the domains over which terms-in-context are interpreted.  ... 
doi:10.1145/2535838.2535868 dblp:conf/popl/DSilvaHK14 fatcat:hnhotzl7ajdhvpl5xmzdnwhkaq

A New Numerical Abstract Domain Based on Difference-Bound Matrices [article]

Antoine Miné
2007 arXiv   pre-print
This paper presents a new numerical abstract domain for static analysis by abstract interpretation.  ...  To illustrate the precision/cost tradeoff of this domain, we have implemented simple abstract interpreters for toy imperative and parallel languages which allowed us to prove some non-trivial algorithms  ...  V 0 -domains to the concrete lattice P({v 2 , . . . , v n } → I), following the abstract interpretation framework described in Cousot and Cousot's POPL'79 paper [5] .  ... 
arXiv:cs/0703073v2 fatcat:4myjr53eendupkdu3hodhwv3gy

Injecting Abstract Interpretations into Linear Cost Models

David Cachera, Arnaud Jobin
2010 Electronic Proceedings in Theoretical Computer Science  
We then present an abstraction technique inspired from abstract interpretation in order to effectively compute global cost information from the program.  ...  We show that our abstraction technique provides a correct approximation of the concrete cost computations.  ...  A is a complete lattice representing the abstract domain.  ... 
doi:10.4204/eptcs.28.5 fatcat:epfvpgtmy5getoi64efbfhtqya

A New Numerical Abstract Domain Based on Difference-Bound Matrices [chapter]

Antoine Miné
2001 Lecture Notes in Computer Science  
This paper presents a new numerical abstract domain for static analysis by abstract interpretation.  ...  To illustrate the precision/cost tradeoff of this domain, we have implemented simple abstract interpreters for toy imperative and parallel languages which allowed us to prove some non-trivial algorithms  ...  V 0 -domains to the concrete lattice P({v 2 , . . . , v n } → I), following the abstract interpretation framework described in Cousot and Cousot's POPL'79 paper [5] .  ... 
doi:10.1007/3-540-44978-7_10 fatcat:ms55tl7rdrbdvfgolzhw7i4ife

Deciding floating-point logic with abstract conflict driven clause learning

Martin Brain, Vijay D'Silva, Alberto Griggio, Leopold Haller, Daniel Kroening
2013 Formal methods in system design  
The core of our approach is a non-trivial, lattice-theoretic generalisation of the conflict-driven clause learning algorithm in modern SAT solvers to lattice-based abstractions.  ...  The generalisation of CDCL we propose is widely applicable and can be used to derive abstraction-based SMT solvers for other theories.  ...  The domain over A is called an abstract domain involving an abstract lattice and abstract transformers. A powerset domain is one in which the lattice is of the form ℘ (S).  ... 
doi:10.1007/s10703-013-0203-7 fatcat:fkn6fznwrrfzvahi3hlitjnywu

Abstract Interpretation as Automated Deduction

Vijay D'Silva, Caterina Urban
2016 Journal of automated reasoning  
This paper presents a new, logical perspective on abstract interpreters that perform reachability analysis using non-relational domains.  ...  We show that three components of an abstract interpreter, the lattice, transformers and iteration algorithm, represent a first-order, substructural theory, parametric deduction and abduction in that theory  ...  Our non-standard treatment of false is influenced by the way abstract domains reason about contradictions. We review the theory rules for our logics.  ... 
doi:10.1007/s10817-016-9382-4 fatcat:avm55dylavfohjycxobfuu25jy
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