This short note is written in celebration of David Schmidt's sixtieth
birthday. He has now been active in the program analysis research community for
over thirty years and we have enjoyed many interactions with him. His work on
characterising simulations between Kripke structures using Galois connections
was particularly influential in our own work on using probabilistic abstract
interpretation to study Larsen and Skou's notion of probabilistic bisimulation.
We briefly review this work and discuss some recent applications of these ideas
in a variety of different application areas.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455

Hankin, Chris

Electronic Proceedings in Theoretical Computer Science

English

arXiv

This short note is written in celebration of David Schmidt's sixtieth birthday. He has now been active in the program analysis research community for over thirty years and we have enjoyed many interactions with him. His work on characterising simulations between Kripke structures using Galois connections was particularly influential in our own work on using probabilistic abstract interpretation to study Larsen and Skou's notion of probabilistic bisimulation. We briefly review this work and discuss some recent applications of these ideas in a variety of different application areas

Chris Hankin

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A. Banerjee, O. Danvy, K.-G. Doh, J. Hatcliff (Eds):David A. Schmidt’s 60th Birthday FestschriftEPTCS 129, 2013, pp. 337–340, doi:10.4204/EPTCS.129.20A short note on Simulation and AbstractionChris HankinInstitute for Security Science and TechnologyImperial College London, UKc.hankin@imperial.ac.ukThis short note is written in celebration of David Schmidt’ssixtieth birthday. He has now been activein the program analysis research community for over thirty years and we have enjoyed many inter-actions with him. His work on characterising simulations between Kripke structures using Galoisconnections was particularly influential in our own work on using probabilistic abstract interpreta-tion to study Larsen and Skou’s notion of probabilistic bisimulation. We briefly review this work anddiscuss some recent applications of these ideas in a varietyof different application areas.1 IntroductionSince his earliest contributions on state transition machines for lambda calculus expressions [13], DavidSchmidt has been at the forefront of research in programmingla uage theory, particularly programanalysis.His work on program analysis started through his collaborati n with Neil Jones. No doubt partlyinspired by Patrick and Radhia Cousot’s work on abstract interpretation [2], he has made a study ofvarious aspects of Galois connections. An early contribution was [7]. A later example, which wasinfluential in our own work, was [15] where he shows how to characterise simulation relations usingGalois Connections.The early work of our group was also inspired by the Cousots [1]. Over the last fourteen years,we have been working on the analysis of probabilistic and quantitative programming languages andsystems [12]. This has led to a framework calledprobabilistic abstract interpretation(PAI). Rather thanusing lattices and Galois Connections, PAI uses Hilbert Spaces and Moore-Penrose Pseudo Inverses.Inspired by [15], we have used PAI to characterise probabilistic bisimulation [8, 10]; we demonstratedthe probabilistc analogue of Dave’s earlier results which amounted to characterising Larsen and Skou’sprobabilistic bisimulation [5] using Moore-Penrose Pseudo Inverses. A major feature of our approach isthat it becomes natural to introduce a notion of approximatebisimulation – this has proved to be veryuseful in studies of language-based security [9].The author first met David Schmidt nearly thirty years ago. Hevisited Imperial College whilstdeveloping the work in [7], his visit coincided with our own work on the ideas in [1]. We found a lotto talk about and it was the start of many further interactions. I addition to his scientific contributions,he has always given time to developing text books; his [14] was used to educate several generations ofImperial students in the principles of programming language design. The author has also worked withDave on many programme committees; perhaps the high point was when they were co-General Chairsof POPL in 2001. Dave has a had a long and successful career andwe wish him many more years. Theauthor would like to add his congratulations to Dave on this important milestone.338 Simulation and Abstraction2 Simulation and Galois ConnectionsIn [15], Dave addresses,inter alia, the question of characterising simulation relations betwe n Kripkestructures using Galois connections.Recall that we define(L,α ,γ ,M) to be aGalois connectionbetween the complete lattices(L,⊑) and(M,⊑) if and only ifα : L → M andγ : M → L are monotone functionsthat satisfy:γ ◦α ⊒ idLα ◦ γ ⊑ idMFor Kripke structuresC = 〈ΣC,→C,IC〉 andA= 〈ΣA,→A,IA〉, a binary relationR ⊆ ΣC×ΣA is asimulationof C by A (C⊳R A), if for everyc∈ ΣC, a∈ ΣA:if c R a andc→C c′ then∃a′ ∈ ΣA[a→A a′ andc′ R a′]In the framework studied in [15], the concrete Kripke structures are often infinite state structuresrepresenting programs and the abstract Kripke structures are ome program analysis. In this setting,we should abstract sets of states of the concrete structure to a single state in the abstract structure.Given a Galois connection,(α : P(ΣC)→ ΣA,γ : ΣA → P(ΣC)), we can construct the relation,R(α ,γ) ⊆P(ΣC)×ΣA:SR(α ,γ)a if and only if α(S)⊑A aThis can be shown to be the suitable basis for a simulation relation.Whilst [15] achieves much more than described here, it was the e ideas that inspired our own workon probabilistic bisimulation (originally introduced by Larsen and Skou [5]) for reactive systems (alsocalled fully probabilistic systems):Definition 1 A probabilistic bisimulationis an equivalence relation∼b on states of a probabilistic tran-sition system satisfying for all actions a∈ A:p∼b q and p→a π ⇔ q→a ρ andπ ∼b ρ .whereπ andρ are distributions of states..3 Probabilistic Bisimulation and Moore-Penrose Pseudo InversesIn [10, 8] we introduced an approximate version of bisimulation and confinement where the approxima-tion can be used as a measureε for the information leakage of the system under analysis. Werepresentedprobabilistic transition systems by linear operators, i.e. by their transition matricesM . In the caseof probabilistic programs and systems these matricesM are the usual well known stochastic matriceswhich are the generators of the corresponding Markov chains(for details see [10, 8]).We then showed that two systemsM1 andM2 are bisimilar if there exist simplified, or abstracted,versions ofM1 andM2, represented by matricesM#1 andM#2, such thatM#1 = M#2. In the probabilisticChris Hankin 339abstract interpretation setting that we use, a bounded linear operator and its Moore-Penrose PseudoInverse are the analogue of the adjoined pair of monotonic fun tions in a Galois insertion. The abstractsystems are obtained bylumpingstates, i.e. by identifying each concrete statesi with a classCj of stateswhich are all behavioural equivalent to each other.Concretely, we compute this vian×m matricesK (wheren is the number of concrete states andmthe number of abstract classes) withK ij = 1 iff si ∈Cj and 0 otherwise. We refer to such matrices whichhave exactly one entry 1 in each row while all other entries ar0 asclassification matrices, and denotethe set of all classification matrices byK . The abstract systems are then given byM#i = K†i M iK i withK i some classification matrix and † constructing the so calledMoore-Penrose pseudo-inverse– in thecase of classification matricesK† can be constructed as the row-normalised transpose ofK .The problem of showing that two systemsM1 andM2 are behaviourally equivalent, i.e. are (proba-bilistically) bisimilar, is now translated into finding twoclassification matricesK i ∈ K such thatM#1 = K†1M1K1 = K†2M2K2 = M#2.In case that two systems are not bisimilar we can still define aquantity ε which describes how(non-)bisimilar the two systems are. Thisε is formally defined in terms of the norm of a linear operatorrepresenting the partition induced by the ‘minimal’ bisimulation on the set of the states of a given system,i.e. the one minimising the observational difference betwen the system’s components (see again [10]for further details, in particular regarding labeled probabilistic transition systems):Definition 2 Let M1 and M2 be the matrix representations of two probabilistic transition systems. Wesay thatM1 andM2 are ε-bisimilar, denoted byM1 ∼εb M2, iffinfK1,K2∈K‖K†1M1K1−K†2M2K2‖= εwhere‖.‖ denotes an appropriate norm, e.g. the supremum norm‖.‖∞.In [10] we show that, whenε = 0 this gives the standard notion of probabilistic bisimulation.4 ConclusionThis short note has sketched some early work by David which forms part of his deep study of the useof Galois connections and relations in program analysis. Our own work on characterising probabilisticbisimulation using probabilistic abstract interpretation has found a number of applications, including:• the detection and removal of timing channels in probabilistic transition systems [11] – we study aconcept called probabilistic time bisimilarity and use it to detect timing channels;• the detection of sub-communities in social media [6] – we evaluate a number of algorithms in-cluding one using the notion of stability from [3] which effectively lumps nodes together if theirmutual interactions are “stronger” than interactions outside the group; and• the abstraction of stochastic and Bayesian games to providedecision support in cyber security [4] –where we hope to apply probabilistic abstract interpretation directly to the underlying probabilistictransition systems in the games, thereby developing a princi led way of reducing the state spacesto achieve tractability of game solutions.We look forward to discussing some of this work with David in the future but, in the meantime, reiterateour best wishes on this important anniversary.340 Simulation and Abstraction5 AcknowledgementsMuch of the work discussed above was done in collaboration with Alessandra Di Pierro and HerbertWiklicky. More recently, I have enjoyed working on the application of these ideas to other areas withErwan Le Martelot and Pasquale Malacaria.References[1] Geoffrey L. Burn, Chris Hankin & Samson Abramsky (1986):Strictness Analysis for Higher-Order Func-tions. Sci. Comput. Program.7(3), pp. 249–278, doi:10.1016/0167-6423(86)90010-9.[2] Patrick Cousot & Radhia Cousot (1977):Abstract Interpretation: A Unified Lattice Model for StaticAnalysis of Programs by Construction or Approximation of Fixpoints. In: POPL, ACM, pp. 238–252,doi:10.1145/512950.512973.[3] Jean-Charles Delvenne, Sophia Yaliraki & Mauricio Barahona (2010):Stability of graph communities acrosstime scales. Proc. Nat. Acad. Sci.107(29), pp. 12755–12760, doi:10.1073/pnas.0903215107.[4] Chris Hankin & Pasquale Malacaria (2013):Payoffs, Intensionality and Abstraction in Games. In: Compu-tation, Logic, Games, and Quantum Foundations - The Many Facets of Samson Abramsky, Lecture Notes inComputer Science7860, Springer, doi:10.1007/978-3-642-38164-5-6.[5] Kim Guldstrand Larsen & Arne Skou (1989):Bisimulation Through Probabilistic Testing. In: POPL, ACM,pp. 344–352, doi:10.1145/75277.75307.[6] Erwan Le Martelot & Chris Hankin (2013):Fast Multi-Scale Detection of Relevant Communities in Large-Scale Networks. Computer Journal, doi:10.1093/comjnl/bxt002.[7] Austin Melton, David A. Schmidt & George E. Strecker (1986): Galois Connections and ComputerScience Applications. In: CTCS, Lecture Notes in Computer Science240, Springer, pp. 299–312,doi:10.1007/3-540-17162-2-130.[8] Alessandra Di Pierro, Chris Hankin & Herbert Wiklicky (2003): Quantitative Relations and ApproximateProcess Equivalences. In: CONCUR, Lecture Notes in Computer Science2761, Springer, pp. 498–512,doi:10.1007/978-3-540-45187-7-33.[9] Alessandra Di Pierro, Chris Hankin & Herbert Wiklicky (2004): Approximate Non-interference. Journal ofComputer Security12(1), pp. 37–82.[10] Alessandra Di Pierro, Chris Hankin & Herbert Wiklicky (2005):Measuring the confinement of probabilisticsystems. Theor. Comput. Sci.340(1), pp. 3–56, doi:10.1016/j.tcs.2005.03.002.[11] Alessandra Di Pierro, Chris Hankin & Herbert Wiklicky (2011): Probabilistic timing covert channels: toclose or not to close?Int. J. Inf. Sec.10(2), pp. 83–106, doi:10.1007/s10207-010-0107-0.[12] Alessandra Di Pierro & Herbert Wiklicky (2000):Concurrent constraint programming: towards probabilisticabstract interpretation. In: PPDP, ACM, pp. 127–138, doi:10.1145/351268.351284.[13] David A. Schmidt (1980):State transition machines for lambda calculus expressions. In: Semantics-Directed Compiler Generation, Lecture Notes in Computer Science94, Springer, pp. 415–440,doi:10.1007/3-540-10250-7-32.[14] David A. Schmidt (1986):Denotational semantics: a methodology for language development. Allyn andBacon.[15] David A. Schmidt (1999):Binary relations for abstraction and refinement. In: Workshop on Refinement andAbstraction.

A short note on Simulation and Abstraction

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A short note on Simulation and Abstraction

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