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Title: Algorithmic Algebraic Model Checking II


1
Algorithmic Algebraic Model Checking II
Decidability of Semi-Algebraic Model Checking and
its applications to Systems Biology
  • Venkatesh Mysore
  • Courant Institute Of Mathematical Sciences, NYU
  • Carla Piazza
  • University of Udine, Italy
  • Bud Mishra
  • Courant Institute Of Mathematical Sciences, NYU
  • School of Medicine, NYU

2
Quick Outline
  • Semi-Algebraic Hybrid Automata Yet another
    hybrid systems subclass ??! WHY WHY WHY ?
  • Real Quantifier Elimination (Yes, the algorithm
    that Prof. Manna abhors, and gave a - -
    complexity grade)
  • Timed Computation Tree Logic (TCTL) to decide or
    not to decide ? That is the question
  • Blum-Shube-Smale Complexity What really are Real
    Turing Machines ?
  • Algorithmic Algebraic Model Checking (AAMC)
    over-ambitious hyperbole ?

3
Simple Goals of This Talk
  • Jargon ! Jargon ! Everywhere,
  • ..and not a chance to Think !
  • Fancy phrases where they come from, and why you
    dont really need them
  • Will I waste your 30 minutes ? NO !
  • Simple general ideas you can use
  • I have the difficult task of addressing a very
    diverse audience from the wise men of Weizmann
    to the dainty doctors-to-be

4
Motivation Model Checking In Biology
  • Systems Biology Model, simulate and analyze
    biochemical systems to test hypotheses, validate
    predictions and suggest experiments

5
The Model-Checking ProblemVerify Temporal
Properties Of A Reactive System
  • Step 1 Formally encode the behavior of the
    system
  • Step 2 Formally encode the properties of
    interest
  • Step 3 Automate the process of checking if the
    formal model of the system satisfies the formally
    encoded properties
  • Step 4 Conclude that the original system
    satisfies original properties (proof /
    counter-example)

6
Task Verify temporal properties of a reactive
system
Step 1 Formally encode the behavior of the
system as a semi-algebraic hybrid automaton
7
Hybrid Systems
  • Let H (Z,V,E,Init,Inv,Flow,Jump) be a hybrid
    automaton of dimension k
  • States have invariants and initial values
  • Transitions have jumps (guards and resets)

8
Continuous Transition
  • Invariant should hold at every point (except
    end-point) along the flow-evolution curve
  • Flow(v)(r, s, t, h) is an algebraic relation
    between the continuous state r at time t and the
    continuous state s after h time units in the
    discrete state v

9
Discrete Transition
  • Guard condition satisfied before the transition
  • Reset condition determines the values after the
    transition
  • Discrete state transitions take zero time

10
Transition Relation Trace
  • Transition relation expression connecting the
    possible values of the system variables before
    and after a zero-time discrete step or a
    continuous evolution for any time period t gt 0
  • Trace sequence of admissible locations

11
Trace
Total time h1 h2 h3 h4
s1
s2
s3
s4
h1
h2
h3
h4
time
12
Hybrid Systems For Biochemical Modeling
  • Chemical Kinetics The kinetic mass-action
    equations for the time variation of the
    concentrations of the interacting species of
    biochemicals can be written down in the form of a
    system of ordinary differential equations
  • The discrete states of the hybrid system can then
    be used to describe regimes of system behavior
    which are qualitatively different in terms of
    which species and reactions predominate

13
Subclasses Of Hybrid Systems
  • Timed Automaton - a discrete transition system
    where the only continuous variable allowed is the
    clock
  • Multirate Automaton - a discrete transition
    system where there can be many continuous
    variables with a constant flow
  • Rectangular Automaton is a discrete transition
    system where the flows are allowed to vary within
    a range
  • Linear Systems - The reachability problem for
    sub-classes of linear hybrid systems have been
    proved
  • O-Minimal Systems - restricted jump condition
    the new continuous state cannot depend on the old
    state, and the system is assumed to be
    time-invariant

14
Semi-Algebraic Hybrid Systems
  • Restriction The expressions for invariant,
    initial, guard and reset are restricted to be
    boolean combinations of polynomial equations and
    inequalities
  • Motivation The quantified expressions
    corresponding to the translation of the temporal
    logic queries become amenable to quantifier
    elimination (and other techniques from real
    algebraic geometry)

15
Semi-Algebraic Set
  • Every quantifier-free formula composed of
    polynomial equations and inequalities, and
    Boolean connectives defines a semi-algebraic set.

16
Flow Expression Accuracy
  • Case 1 Closed-form solution is a polynomial
  • Case 2 Differential equation is a polynomial

17
Approximate Symbolic Integration Euler Forward
Discretization
  • First order Taylor polynomial
  • Approximate as a straight line with slope equal
    to first derivative
  • If r represents the vector of variables of the
    hybrid system at time t in discrete state v, the
    approximate value of r(t h) is given by

Leonhard Euler (1707-1783)
  • Improved Two-Way Euler f(t) 1/2h f(th)
    f(t-h)

18
Approximate Symbolic Integration Taylor Series
  • Differential flow equations discretized using
    Taylor polynomials
  • Degree of the Taylor polynomial influences the
    complexity of formulæ and the number of steps
    needed to get a sufficient precision
  • Error Control Upper bound time spent in one
    step of continuous evolution

Brook Taylor (1685-1731)
In other words
19
Approximate Symbolic Integration Second Order
Runge-Kutta
  • At time to, find k1 - the derivative of y(t)
  • Find an initial value for y(toh) using the
    Euler formula
  • From y(toh) estimate k2 - the derivative of
    y(t) at toh
  • Get a new value for y'(toh) based on the
    average of the values of k1 and k2

Carl Runge (1856-1927)
M. W. Kutta (1867-1944)
20
Summary Of Flow Constraint
  • Accurate Solution
  • If there exists an accurate closed-form
    semi-algebraic formula connecting y(t) and y(th)
    valid for all y,t and h
  • The solution of the differential equation must be
    polynomial
  • Lafferriere et al.s work shows that in some
    cases exponential and trigonometric solutions can
    be expressed as semi-algebraic sets
  • Approximate Solution
  • If the differential equation is polynomial
  • Approximate Symbolic Integration techniques Eg.
    Euler, Runge-Kutta or Taylor Series
  • Upper bound continuous time step to control error

21
Task verify temporal properties of a reactive
system
  • Step 1. Formally encode the behavior of the
    system as a semi-algebraic automaton
  • Step 2. Formally encode the properties of
    interest in TCTL

22
Linear Temporal Logic (LTL)
  • Interpreted over sequential natural models for
    which LTL is expressively complete
  • We do not explicitly talk about the different
    paths the system can evolve through
  • A property is sequence-valid on a Kripke
    structure if it is valid in all natural models
    which are generated from it
  • Temporal Operators Next, Eventually, Henceforth,
    Until, and past-counterparts
  • We can have a arbitrary number of such temporal
    operators preceding a property allowing us to
    capture very complex temporal properties along
    the path

Amir Pnueli
Zohar Manna
23
Computation Tree Logic (CTL)
  • Branching Time temporal logic interpreted over
    an execution tree where branching denotes
    non-deterministic actions
  • A property is tree-valid in a Kripke model if it
    is valid in the root of the unique maximal tree
    generated from it
  • Second order logic as we explicitly quantify over
    two modes the path and the time
  • Each time we talk about a temporal property, we
    also specify whether it is true on all possible
    paths or whether it is true on atleast one path -
    Path quantifiers
  • A for all future paths
  • E for some future path

Ed Clarke
EA Emerson
24
Continuous-Time Logics
  • Linear Time
  • Metric Temporal Logic (MTL)
  • Timed Propositional Temporal Logic (TPTL)
  • Real-Time Temporal Logic (RTTL)
  • Explicit-Clock Temporal Logic (ECTL)
  • Metric Interval Temporal Logic (MITL)
  • Branching time
  • Real-Time Computation Tree Logic (RTCTL)
  • Timed Computation Tree Logic (TCTL)
  • TCTL the most used branching time temporal
    logic for real-time systems (Farn Wang, 2004)

25
TCTL Syntax And Semantics
Rajeev Alur
David Dill
26
TCTL One-Step Until
  • q can be reached within one step of the hybrid
    system and p holds until that point in the
    transition
  • p continuously holds until some intermediate
    point immediately followed by q being true
  • p or q holding all along that one step of the
    hybrid system and q being true at the end of the
    one-step evolution
  • Discrete time model-checking next state
    operator X
  • Continuous-mode single-step until operator

Tom Henzinger
27
TCTL Model Checking
  • Only Until requires computation
  • Until Iterative computation of one-step Until
  • Least fixpoint computation

28
Task verify temporal properties of a reactive
system
  • Step 1. Formally encode the behavior of the
    system as a semi-algebraic hybrid automaton
  • Step 2. Formally encode the properties of
    interest in TCTL
  • Step 3. Automate the process of checking if the
    formal model of the system satisfies the formally
    encoded properties using quantifier elimination

29
Single-Step Until For Semi-Algebraic Hybrid
Systems
  • p or q holds on discrete step
  • q must be true after jump
  • p or q holds on continuous step
  • Every intermediate point must satisfy p or q
  • q must be true at the end of the evolution
  • Can be simplified into p must hold at every
    intermediate point with q holding at the end

30
One-Step Until
State v
State u
q
ltv,rgt
ltu,sgt
p or q
h
q
ltv,sgt
time
31
Semi-Algebraic Sets Are Amenable To Quantifier
Elimination
  • Recall Semi-Algebraic Set
  • Every quantifier-free formula composed of
    polynomial inequalities and Boolean connectives
    defines a semialgebraic set
  • 1930s Tarski proved Quantifier Elimination is
    possible for quantified semi-algebraic sets but
    his algorithm was too slow

Alfred Tarski 1902-1983
32
Quantifier Elimination
  • 1973 Collins discovered new method
    cylindrical algebraic decomposition (CAD)
  • Doubly exponential in number of variables
  • Polynomial in number and degree of polynomials,
    number of atomic formulae
  • Hoon Hong implemented the system Quantifier
    Elimination by Partial Cylindrical Algebraic
    Decomposition (Qepcad)
  • Input (Ex) x2 b x c 0
  • Output b2 - 4 c gt 0

Hoon Hong
33
Quantifier Elimination Suffering from a
Complexity Complex
  • Tarskis almost impractical algorithm
  • Collins cylindrical algebraic decomposition
    (CAD) algorithm - double-exponential dependence
    on the number of variables
  • Collins doctoral student Hong implemented the
    first quantifier elimination software Qepcad
  • Alternative CAD-based methods Grigoriev, Renegar
    and Heintz that are doubly exponential in the
    number of quantifier alternations
  • Weispfennings work on cubic quantified variables
  • Implemented on Reduce as Redlog and Risa/Asir
    1
  • Complexity independent of the number of free
    variables
  • New quantifier elimination approaches Basu,
    Pollack and Roy

34
Semi-Decidability Of TCTL
  • Global time variable
  • Allows interpretation of the TCTL operators
    freeze (z.X) and subscripted until (Ua)
  • Initial value 0, flow 1 in all discrete states
    and never reset
  • While one-step until is decidable, the fixpoint
    is not guaranteed to converge
  • So TCTL is semi-decidable
  • Existential segment and negation of Universal
    Segment
  • Subscripted operators are decidable in non-zeno
    systems

35
Lets squeeze in a yawn and a stretch before
continuing Also a good place to check how many
minutes we have left
36
General Undecidability Of Reachability
  • Classical theory of computation and complexity
    analysis centered around the binary Turing
    machine is not sufficient to fully characterize
    problems involving real-valued mathematics
  • Blum-Cucker-Shub-Smale proposed the more general
    real Turing machine that has exact rational
    operations and comparison of real numbers
    built-in as atomic operations represented as
    maps

37
Relation To Semi-Algebraic Sets
38
Undecidability Of The Mandelbrot Set
  • The Mandelbrot set is not decidable over R. This
    follows from the fact that the Mandelbrot set
    cannot be the countable union of semi-algebraic
    sets over R as its boundary has complex
    mathematical properties
  • Complement of

Benoit Mandelbrot (1924-)
39
Mandelbrot Hybrid Automaton
Let
Invariant False Flows Null
Then
Reachability Query
40
Implementation Tolque
  • Implemented in C / C
  • Accepts the hybrid system specification, with
    flow equation already approximated by user
  • Accepts existential until (EU) TCTL query
  • At each iteration, it computes the p one-step
    until q formula and calls the quantifier-eliminat
    ion software Qepcad

Abstract Interpretation Of Tolque In Action
41
Tolque Limitations
  • The most severe limitation of Tolque comes from
    the computational complexity of the cylindrical
    algebraic decomposition algorithm -
    double-exponential dependence on the number of
    variables
  • The Qepcad implementation has the additional
    disadvantage of not supporting real numbers
  • Degrees of the resulting polynomials increasing
    at each time-step quickly leading to Qepcad
    choking on the query

42
So, what have we learnt ?
  • Quantifier elimination is gooooooooood !
  • Not just YES or NO answers, but actual ranges and
    constraints on parameters that need to hold for a
    certain property to be true can be solved
  • Its a good idea to see what is possible, and
    then see what sub-problem you can solve more
    efficiently good perspective
  • Can you express the problem that YOU are studying
    as a quantifier elimination problem ? Show
    something to be decidable for a change ?

43
Algebraic Model Checking
  • Mats Jirstrand - Qepcad for the problems of
    stationarizable sets, range of controllable
    output, following a curve and reachability
  • Hirokazu Anai and Martin Fraenzle - independently
    suggested the use of quantifier elimination for
    the verification of polynomial (semi-algebraic)
    hybrid systems
  • Anai and Weispfenning - expounded the use of
    quantifier elimination for the reachability
    analysis of continuous systems with parametric
    inhomogenous linear differential equations
  • Fraenzle - proved that progress, safety, state
    recurrence and reachability are semi-decidable
    using quantifier elimination developed proof
    engines for bounded model checking
  • Lafferiere et al. - a quantifier-elimination-centr
    ic method for symbolic reachability computation
    of linear vector fields

44
Algebraic Model Checking
  • Ratschan and She - constraint propagation based
    abstraction refinement for verification of hybrid
    systems
  • Carbonell Tiwari and Sankaranarayanan et al. -
    schemes for generating invariants for hybrid
    systems
  • Becker et al.s integration of bounded model
    checking and inductive verification
  • Lanotte and Schettinis - monotonic hybrid
    systems
  • Lanotte and Tini - approximating each formula in
    any (non-polynomial) hybrid system definition
    with its Taylor polynomial (of some degree k) is
    an over-approximation

45
AAMC I The Case of Biochemical Systems and their
Reachability Analysis Carla Piazza, Marco
Antoniotti, Venkatesh Mysore, Alberto Policriti,
Franz Winkler and Bud Mishra, Computer Aided
Verification (CAV), 2005
  • Introduced Semi-Algebraic Hybrid Automata
  • Characterized the widest range of automata that
    admit sound albeit expensive mathematical
    techniques, as opposed to focusing on a very
    narrow class of systems that often prematurely
    sacrifices generalizability for the sake of
    efficiency
  • Bounded reachability problem shown to be solvable
    using real algebraic techniques like Taylor
    series approximation and quantifier elimination
  • Suitability for Systems Biology
  • Found sufficiently powerful in analyzing such
    systems as the Delta-Notch protein interaction
    example

46
AAMC II Decidability of Semi-Algebraic Model
Checking and its Applications to Systems Biology
Venkatesh Mysore, Carla Piazza and Bud Mishra,
International Symposium on Automated Technology
for Verification and Analysis (ATVA), 2005
  • Solved the algebraic model-checking problem over
    the dense time logic TCTL demonstrated in
    Tolque
  • Exploited algebraic bounded reachability
    algorithm of AAMC-I, Franzles ideas for
    polynomial hybrid systems, and Henzinger et al.s
    characterization of the Until operator as a
    fix-point expression involving the one-step-until
    operator
  • The ability to perform an entirely symbolic
    analysis of arbitrary polynomial hybrid systems
    over a full temporal logic, limited only by
    computational power, distinguishes our approach
    from the other methods in literature
  • Proved that reachability is undecidable even in
    Blum et al.s real Turing machine
    (finite-dimensional machine over a field)
    formalism

47
AAMC III Approximate Methods Venkatesh Mysore
and Bud Mishra, Verification of Infinite State
Systems (Infinity), 2005
  • Made existing ideas applicable to semi-algebraic
    hybrid systems, by using quantifier elimination
    in place of the original efficient-but-restrictive
    computational method
  • Bisimulation Partitioning
  • Polytopes
  • Rectangular Grids
  • Time Discretization
  • Obtained new optimizations and techniques
  • Identified well-behaved subclasses

48
Future Theoretical Work
  • Algebraic Enhancements
  • Groebner basis Characteristic Sets
  • Characterizing recursive paths and invariants
  • Conditions for convergence of the fixpoint
    expressions
  • Extension of Cousots widening technique to
    semi-algebraic sets
  • Estimation of expected number of iterations
  • Analysis of perturbed and robust systems
  • Discretization Of Space Time
  • Depart from the continuous infinite space and
    time
  • Approximate with rectangular grids, ellipsoids
    and polyhedra
  • Use polynomials to identify connected components
    relevant to query abstraction over each
    component
  • Chaotic dynamical systems and decidability
  • Flux balance analysis and topology

49
Future Software Development
  • Tolque being integrated with Simpathica (in Lisp)
  • Different integration-discretization schemes and
    the continuous / discrete modes of operation
  • Extend repertoire of temporal logic operators
  • Extension to real-time LTL
  • Integrate other quantifier elimination tools like
    Approximate Quantified Constraint Solving (AQCS)
    and Redlog
  • Translate and break down the TL query into small
    quantifier elimination problems, and then use
    heuristics to decide which quantifier elimination
    tool to call for each sub-query
  • Eventually, our own symbolic algebra system will
    work hand in hand with the quantifier
    elimination, Groebner basis and characteristic
    set tools to simplify and systematically simplify
    the formulæ at each fixpoint iteration

50
Acknowledgements
Dr. Carla Piazza
For a full list of references, please see
AAMC-I (CAV05) , AAMC-II (ATVA05) AAMC-III
(Infinity05)
Dr. Amir Pnueli
Dr. Bud Mishra
51
Thank You
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