Agenda. Specification models and their analysis. Agenda. Part I. Binary Decision Diagrams. Graph Theory: Some Definitions. Introduction to Petri Nets

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1 Agenda Specification models and their analysis Kai Lampka November 9, 29 Graph Theory: Some Definitions 2 Introduction to Petri Nets 3 Introduction to Computation Tree Logic and related model checking techniques 4 Introduction to Binary Decision Diagrams Specification models and their analysis: 36 Agenda Part I Binary Decision Diagramms Graph Theoretic Foundations 2 Petri Nets 3 Computation Tree Logic and related model checking techniques 4 Binary Decision Diagrams 5 Timed Automata and timed CTL

2 Binary Decision Diagrams: Zusammenfassung Binary Decision Tree Binäre Entscheidungsdiagramme sind bi-partite, ungewichtete, zyklenfreie Digraphen, in denen ein jeder inneren Knoten jeweils genau 2 Nachfolger hat, naemlich den -Nachfolger und de-nachfolger. Die Shannon-Expansion ordnet jedem Binären Entscheidungsdiagramm genau eine Boolesche Funktion zu. Da sich andersherum jede Boolesche Funktion durch und genau mit einem BDD darstellen lässt, sind BDDs kanonische Darstellung von Booleschen Funktionen. Ihre Verbindung zur Schaltalgebra ist somit evident. In den letzte Dekaden sind BDDs sehr gründlich erforscht worden und es existieren viele abgeleitete Formen, sowie effiziente Algorithmen zu ihrer Manipulierung. Letztendlich bilden BDDs und ihre verwandten Datentypen ein wichtiges Fundament im Very-large-scale integration (VLSI) Design und im Bereich des Model checkings. Da BDDs letztendlich eine Implementierung einer endlichen Booleschen Algebra darstellen spricht man in diesem Zusammenhang oft auch von symbolischen Verfahren, bspw. vom symbolischen Model checking. Specification models and their analysis: Binary Decision Diagramms 5 36 A Binary Decision Tree (BDT) is a bi-partite tree consisting of a set of inner nodes (NNT ) and a set of terminal nodes (NT ) with N := NNT NT. n x 6 2 Nodes are connected via - and -edges: NNT N and NNT N We read the tree from top to bottom, hence we can omitt the arrow heads Each inner node (circle) is asscoiated with a node label ni and a variable xj, e. g. var() = A dashed line leads to the -successor, the solid line to the -successor, e. g. child(n) = n3; child(n) = n2 Each terminal node is associated with a function value from B := {, }, e. g. value(t) = Specification models and their analysis: Binary Decision Diagramms 6 36 Variable Ordering Semantics For algorithmically working with BDDs it turns out that they should be ordered w.r.t. the variables of V. To do so one simply defines a total order V V and requires n NNT : n = child,(m) var(m) var(n) n x 6 3 What is the Boolean function represented by the BDT? What is the space complexity for representing Boolean functions with BDT? Shannon expansion for Boolean functions: f (x,... xn) = x f(x2,..., xn) + ( x) f(x2,..., xn) instead of the Boolean operators (,, ) we employ their arithmetic counterparts, e. g. x ( x), etc.. The recursion tree of a Shannon expansion is exactly what is represented by a BDT. Let BDT-node k be labelled with variable x. According to the Shannon expansion it represents the n-ary Boolean function f (x,..., xn). Its -successor represents than f(x2,..., xn) and its -successor represents function f(x2,..., xn). Function f(x,...,, xi+,..., xn) is denoted -cofactor of function f w. r. t. variable xi. Function f(x,...,, xi+,..., xn) is denoted -cofactor of function f w. r. t. variable xi. For the co-factors we also adapt the notation f x i :=b with b {, } A terminal node represents the -ary, constant or -function. Specification models and their analysis: Binary Decision Diagramms 7 36 Specification models and their analysis: Binary Decision Diagramms 8 36

3 Semantics According to the above discussion each BDT-node represents a Boolean function. We (recursively) define the equivalence relation on the set of BDT-nodes (N = NNT NT ) as follows: for two terminal BDT-nodes t, p NT : t p value(t) = value(p) Let node n represent function f n and let node k represent function f k : for two non-terminal BDT-nodes n, k NNT : Questio.: How can we decide if n k child(n) child(k) child(n) child(k) f n f k holds? equivalent, i. e., p t iff According to the above discussion each BDT-node represents a Boolean function. Questio.2: How does this effect the size of the obtained graphs? Specification models and their analysis: Binary Decision Diagramms 9 36 Specification models and their analysis: Binary Decision Diagramms 36 n x 6 3 n x 6 3 Specification models and their analysis: Binary Decision Diagramms 36 Specification models and their analysis: Binary Decision Diagramms 2 36

4 n x 6 3 n x 6 3 Specification models and their analysis: Binary Decision Diagramms 3 36 Specification models and their analysis: Binary Decision Diagramms 4 36 Specification models and their analysis: Binary Decision Diagramms 5 36 Specification models and their analysis: Binary Decision Diagramms 6 36

5 Specification models and their analysis: Binary Decision Diagramms 7 36 Specification models and their analysis: Binary Decision Diagramms 8 36 Specification models and their analysis: Binary Decision Diagramms 9 36 Specification models and their analysis: Binary Decision Diagramms 2 36

6 The reduction is applied on the fly, i.e., each allocated node is unique. Hence application of an a posteriori reduction not necessary. As we will see later, uniqueness of nodes is not only a key to memory efficiency but also to run-time efficiency w.r.t. the manipulation of Specification models and their analysis: Binary Decision Diagramms 2 36 Specification models and their analysis: Binary Decision Diagramms BDD: Multi-rooted BDDs Binary Decision Diagram: Uniqueness of BDD nodes allows one to share sub-graphs among different BDDs yielding multi-rooted BDDs: f f2 f3 f 4 child(n) = child(n) holds As shown by the example the Shannonexpansion yields, that such nodes can Questio.3: Can we do more, e. g. apply Shannon for function f3? Specification models and their analysis: Binary Decision Diagramms Specification models and their analysis: Binary Decision Diagramms 24 36

7 Binary Decision Diagram: Binary Decision Diagram: child(n) = child(n) holds child(n) = child(n) holds As shown by the example the Shannon- As shown by the example the Shannon- expansion yields, that such nodes can expansion yields, that such nodes can Specification models and their analysis: Binary Decision Diagramms Specification models and their analysis: Binary Decision Diagramms Binary Decision Diagram: Binary Decision Diagram: child(n) = child(n) holds child(n) = child(n) holds As shown by the example the Shannon- As shown by the example the Shannon- expansion yields, that such nodes can expansion yields, that such nodes can n 6 n 6 Specification models and their analysis: Binary Decision Diagramms Specification models and their analysis: Binary Decision Diagramms 28 36

8 Binary Decision Diagrams: Formal Definition Binary Decision Diagrams: Formal Definition A reduced ordered Binary Decision Diagram B < V, > is a 5-tuple {N, value, var, child, child} where V is a finite and non-empty set of boolean variables with the fixed ordering relation V V defined one. 2 N = NT NNT is a finite non-empty set of nodes, consisting of the set of terminal nodes NT and non-terminal nodes NNT, with NT NNT =. 3 The following functions are defined: the value-returning function value : NT B for each terminal node, 2 the variable-returning function var : NNT V for each non-terminal node, 3 the child node-returning functions child, child : NNT N for each non-terminal node, and 4 the root node-returning function getroot : B N. For the BDD to be ordered the following constraint must hold: u NNT : child(u) NNT : var(child(u)) var(u) child(u) NNT : var(child(u)) var(u). 2 A BDD is denoted reduced iff the following conditions apply: (a) Isomorphism rule: No isomorphic nodes; i.e. (i) Non-terminal case: n, m NNT : n m (var(n) var(m) child(n) child(m) child(n) child(m)) (ii) Terminal case: n, m NT : n m (value(n) value(m)) (b) Dnc-rule: No don t care nodes: u NNT : child(u) = child(u). Specification models and their analysis: Binary Decision Diagramms Specification models and their analysis: Binary Decision Diagramms 3 36 Binary Decision Diagrams: Canonicity BDD: Algorithmic manipulation Reduced ordered BDDs are (strongly) canonical representations for Boolean Functions, thus each Boolean function f produces its own BDD Bf. f g Bf Bg Questio.4: Why can equivalenz testing be done in constant time? Consider the following two Boolean functions: f := dab + ad c + abd + a c d g := a cb + cba + b a c + a bc Questio.5: Are f and g equivalent? Please justify by making use of BDDs For making use of BDDs in an algebraic framework it is neccessary to being capable of efficiently applying operators to them, s.t. the obtained BDD represents the resp. function. Hence any n-ary operator applicable to n Boolean functions should be applicable to their n BDD-based representations. In the following we consider -ary and 2-ary (binary) operators, s.t. f Negate(Bf ) f + g Plus(Bf, Bg ) f g Mult(Bf, Bg ) Excursio.: Proof of canonicity (on the black board) Specification models and their analysis: Binary Decision Diagramms 3 36 Specification models and their analysis: Binary Decision Diagramms 32 36

9 BDD-based algorithms: Negation BDD-based algorithms: Apply A binary operator op {+,,...} can be applied to BDDs by means of Bryant s Negate(node n) () IF n NT THEN () ELSE RETURN(makeTerminal( value(n))) (2) node t := Negate(child(n)) (3) node e := Negate(child(n)) (4) IF t = e THEN RETURN(t) (5) ELSE RETURN(makeNode(var(n), t, e)) END Apply algorithm. APPLY(op, node n, node m) () node e, t, res Reached terminal nodes, end of recursion () IF n, m N T THEN (2) int v := value(n) op value(m) (3) RETURN(makeTerminal(v)) Check op cache if result is already known (4) res := cachelookup(op, n, m) (5) IF res ɛ THEN RETURN(res) Depending on the node-labelling variables branch into recursion (6) IF var(n) = var(m) THEN (7) v := var(n) (8) e := APPLY(op, child (n), child (m)) (9) t := APPLY(op, child (n), child (m)) () ELSE IF var(n) var(m) THEN () v := var(n) (2) e := APPLY(op, child (n), m) (3) t := APPLY(op, child (n), m) (4) ELSE (5) v := var(m); (6) e := APPLY(op, n, child (m)) (7) t := APPLY(op, n, child (m)) Questio.6: Construct the recursion tree for Negate and the BDD depicted above Example.: Consider f := and f 2 :=. Please give the BDDs for f and f 2, construct the recursion tree for f f 2 and give the resulting BDD. Allocate new node, unique and non-dnc-node (8) IF t = e THEN RETURN(t) (9) ELSE (2) res := RETURN(makeNode(v, t, e)) Insert result into op cache and terminate recursion (9) cacheinsert(op, n, m, res) (2) RETURN(res) BDD: Specification models and their analysis: Binary Decision Diagramms Algorithmic manipulation Specification models and their analysis: Binary Decision Diagramms BDD-based approaches for the Verification of systems Besides the APPLY-algorithm, which is the most important one other algorithms have been developped. Let f be a n-ary Boolean function and let B f be its BDD, in the following we will employ the following operation and their resp. BDD-based implementations: f (x,..., xn) xi =b RESTRICT(B f, b) with b B Quantification: Existential quantification: xi : f ( x) f x i = f x i = ABSTRACT(B, xi, Mult) 2 Universal quantification: xi : f ( x) f x i = + f x i = ABSTRACT(B, xi, Plus) In total the so far discussed techniques gives us a framework for efficiently representing and manipulating Boolean functions. This is the basis for representing and verifying systems such as Symbolic analysis of switching functions Symbolic rechability set generation, especially in case of Petri nets Symbolic CTL model checking Relabeling: [x y]f B f {y x}, each occurence of variable x is replaced by variable y: (f x= (g(y) = y)) + (f x= (g(y) = y)) Example.2: BDDs Specification models and their analysis: Binary Decision Diagramms Specification models and their analysis: Binary Decision Diagramms 36 36