Formal Specification and Verification

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1 Formal Specification and Verification Stefan Ratschan Katedra číslicového návrhu Fakulta informačních technologíı České vysoké učení technické v Praze Stefan Ratschan (FIT ČVUT) PI-PSC / 19

2 History of System Design build try Stefan Ratschan (FIT ČVUT) PI-PSC / 19

3 History of System Design build model test model Intel Pentium FDIV bug (1994) implement model Stefan Ratschan (FIT ČVUT) PI-PSC / 19

4 History of System Design build model prove correctness implement model Stefan Ratschan (FIT ČVUT) PI-PSC / 19

5 Example Modulo 8 counter: State: S. = B 3 v 0 = v 0 v 1 = v 0 v 1 v 2 = (v 0 v 1 ) v 2 Stefan Ratschan (FIT ČVUT) PI-PSC / 19

6 Example: Digital Circuit Transition relation: T S S T ((v 0, v 1, v 2 ), (v 0, v 1, v 2 )). = ((v 0, v 1, v 2 ), (v 0, v 1, v 2)) (v 0 v 0) (v 1 v 0 v 1 ) (v 2 (v 0 v 1 ) v 2 ) Non-determinism Stefan Ratschan (FIT ČVUT) PI-PSC / 19

7 Formal Specification Transition System: Set of states S Set I S of initial states Transition relation R S S For example: S = B n, I, R given by formulas in propositional logic S is program state (program counter + ranges of variables) I, R given by computer program S = {1,..., n} R n, R given by clock checks and resets: timed systems/automata S = R n, I given by (in)equalities, R given by ordinary differential equations (models of physical systems) S = {1,..., n}, I, R given by (in)equalities, ordinary differential equations, and discrete rules: hybrid (dynamical) systems (models of embedded systems) Stefan Ratschan (FIT ČVUT) PI-PSC / 19

8 Formal Specification How to specify whether a given system behaves correctly? First we have to specify properties of single states: state properties Example: error(s): state s is an error state zero(s): state s represents zero Now we can extend this to one behavior of system: A path in a transition system (S, I, R) is an infinite sequence of states s 0 s 1 s 2... s.t. s 0 I, for all i {0, 1,... }, (s i, s i+1 ) R. How to specify whether a given path shows correct/incorrect behavior? Stefan Ratschan (FIT ČVUT) PI-PSC / 19

9 Properties on Paths π State property g holds on first element of path: π = g : g(π(0)) State property holds on next element of path: π = Xg : π 1 = g Train eventually reaches full speed π = Fg : there is k 0 s.t. π k = g (F: in the future ) Number of motor rotations always stays in safe area π = Gg : for all k 0, π k = g (G: globally ) The train eventually stops and until then the doors remain closed π = guh : there is i s.t. π i = h and for all j < i, π j = g (U: until ) As long as the plane does not reach full height the fasten seat belts sign is on π = grh : for all j 0, if for all i < j, not π i = g then π j = h (R: release ) Stefan Ratschan (FIT ČVUT) PI-PSC / 19

10 Combining Operators If the elevator is called, it will eventually show up (g Fh). The train will never move with open doors G (g h) So: Boolean combinations (,, ). Combining temporal operators. For example: FGg: Eventually property g will hold forever. GFg: Always eventually g will hold. Result: Linear Temporal Logic (LTL) Syntax: every state property is an LTL formula If g and h are LTL formulas then also Xg, guh, grh, Fg, Gg, and guh, g, g h, g h, are LTL formulas. Stefan Ratschan (FIT ČVUT) PI-PSC / 19

11 Semantics For a path π and LTL formulas g, h, π = g iff g is a state property and g(π(0)) π = Xg iff π 1 = g π = Fg iff there is k 0 s.t. π k = g π = Gg iff for all k 0, π k = g π = guh iff there is i s.t. π i = h and for all j < i, π j = g. π = grh iff for all j 0, if for all i < j, not π i = g then π j = h π = g iff not π = g π = g h iff π = g and π = h π = g h iff π = g or π = h Stefan Ratschan (FIT ČVUT) PI-PSC / 19

12 Specification of System We already can specify correctness of one path. Correctness of system: = g iff for all paths π of, π = g Industrial Usage: PSL (property specification language) LTL + regular expressions Stefan Ratschan (FIT ČVUT) PI-PSC / 19

13 Formal Verification Problem: Even finite transition systems may have paths of infinite length! Checking G ok: Check prefixes of paths of length 1, 2,... : bounded model checking BMC(n) In Boolean case, individual checks can be done efficiently by SAT (in practice, not in theory) Especially: Bugs can often be found fast Bounded model checking algorithms for other types of systems: Software: CBMC [Clarke et al., 2004] Hybrid Systems: isat [Fränzle and Herde, 2007, Fränzle et al., 2007]... Stefan Ratschan (FIT ČVUT) PI-PSC / 19

14 Full LTL? Unbounded Case? Full LTL [Biere et al., 2003] For every LTL formula φ, for all n, = φ implies BMC(φ, n) Opposite direction? Observation: = Gs iff BMC(Gs, S ) If system does not fulfill Gs then it has an error path of length at most S Theorem for all finite transition system, for all LTL formula φ there is a bound n s.t. for all n n, BMC(φ, n ) iff = φ But: bound may be huge! Stefan Ratschan (FIT ČVUT) PI-PSC / 19

15 Unbounded Model Checking If we want to prove correctness over unbounded time, or we search for a bug that shows up after long time. Reach set computation: let R be the set of initial states add reachable state reachable from R until no more new reachable states If for all x R, ok(x), then = G ok. For full LTL (and other temporal logics), more complicated [Clarke et al., 1999]. Stefan Ratschan (FIT ČVUT) PI-PSC / 19

16 Unbounded Model Checking for Specific System Types In Boolean case, sets can be represented by (reduced ordered) binary decision diagrams (BDD) BDDs provide a unique representation for Boolean formulas Hence can be used equivalence checking (i.e., specification is another circuit) Other systems: Software: Huge field of software model checking [Jhala and Majumdar, 2009] Timed automata: Uppaal ( Hybrid systems: HyTech [Henzinger et al., 1997] PHAver [Frehse, 2008] HSolver [Ratschan and She, 2007] Stefan Ratschan (FIT ČVUT) PI-PSC / 19

17 Literature I Armin Biere, Alessandro Cimatti, Edmund M. Clarke, Ofer Strichman, and Yunshan Zhu. Bounded model checking. volume 58 of Advances in Computers, pages Elsevier, doi: DOI: /S (03) Edmund Clarke, Daniel Kroening, and Flavio Lerda. A tool for checking ANSI-C programs. In Kurt Jensen and Andreas Podelski, editors, Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2004), volume 2988 of Lecture Notes in Computer Science, pages Springer, ISBN X. Edmund M. Clarke, Orna Grumberg, and Doron A. Peled. Model Checking. MIT Press, Stefan Ratschan (FIT ČVUT) PI-PSC / 19

18 Literature II M. Fränzle, C. Herde, S. Ratschan, T. Schubert, and T. Teige. Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT Journal on Satisfiability, Boolean Modeling and Computation, Special Issue on SAT/CP Integration, 1: , Martin Fränzle and Christian Herde. Hysat: An efficient proof engine for bounded model checking of hybrid systems. Formal Methods in System Design, 30(3): , Goran Frehse. Phaver: algorithmic verification of hybrid systems past hytech. International Journal on Software Tools for Technology Transfer (STTT), 10(3): , doi: /s x. Thomas A. Henzinger, Pei-Hsin Ho, and Howard Wong-Toi. HYTECH: a model checker for hybrid systems. International Journal on Software Tools for Technology Transfer (STTT), 1: , Stefan Ratschan (FIT ČVUT) PI-PSC / 19

19 Literature III Ranjit Jhala and Rupak Majumdar. Software model checking. ACM Comput. Surv., 41(4):1 54, ISSN doi: Stefan Ratschan and Zhikun She. Safety verification of hybrid systems by constraint propagation based abstraction refinement. ACM Transactions in Embedded Computing Systems, 6(1), Stefan Ratschan (FIT ČVUT) PI-PSC / 19

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