1 / 9

Example: Control Flow Graphs

process Flowgraph = {. x : int, pc: {c 0 , c 1 , c 2 , c 3 , c 4 , c 5 } init pc = c 0 update pc = c 0 → x := 1 ; pc := c 1 pc = c 1 → pc := c 2 pc = c 2 ∩ x ≤ 100 → pc := c 3 pc = c 2 ∩ x > 100 → pc := c 5 pc = c 3 → x := x + 1 ; pc := c 4 pc = c 4 → pc := c 2. c 0.

kaemon
Download Presentation

Example: Control Flow Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. process Flowgraph = { • x : int, pc: {c0, c1, c2, c3, c4, c5} • init pc = c0 • update • pc = c0→ x := 1 ; pc := c1 • pc = c1→ pc := c2 • pc = c2∩ x ≤ 100 → pc := c3 • pc = c2 ∩ x > 100 → pc := c5 • pc = c3→ x := x + 1 ; pc := c4 • pc = c4→ pc := c2 c0 x := 1 c1 c2 c4 c5 x ≤ 100 false c3 true x := x + 1 } Example: Control Flow Graphs Control Flow Graph Unity – like Process -6-

  2. Example: Mutual Exclusion P = m : cobegin P0 || P1coend m’ P0 :: l0 : while true do { nc0 : wait (turn = 0) cr0 : turn := 1 } l0’ : P = P0 || P1 P0 = { • turn : {0,1} , pc0 : { nc0, cr0} • init pc0 = nc0 • update • pc0 = nc0 ∩ turn = 0 → pc0 := cr0 • pc0 = nc0 ∩ turn = 1 → pc0 := nc0 • pc0 = cr0→ turn := 1 ; pc0 := nc0 } P1 :: l1 : while true do { nc1 : wait (turn = 1) cr1 : turn := 0 } l1’ : P1 = { • turn : {0,1} , pc1 : { nc1, cr1} • init pc1 = nc1 • update • pc1 = nc1 ∩ turn = 1 → pc1 := cr1 • pc1 = nc1 ∩ turn = 0 → pc1 := nc1 • pc1 = cr1→ turn := 0 ; pc1 := nc1 } Pseudo - code -7-

  3. Example: Mutual Exclusion Expanded process P = { • turn : {0,1} , pc0 : { nc0, cr0} , pc1 : { nc1, cr1} • init pc0 = nc0∩ pc1 = nc1 • update • pc0 = nc0 ∩ turn = 0 → pc0 := cr0 • pc0 = nc0 ∩ turn = 1 → pc0 := nc0 • pc0 = cr0→ turn := 1 ; pc0 := nc0 • pc1 = nc1 ∩ turn = 1 → pc1 := cr1 • pc1 = nc1 ∩ turn = 0 → pc1 := nc1 • pc1 = cr1 → turn := 0 ; pc1 := nc1 } -8-

  4. Q = int x arcs I = pc = c0 = { (x, c0) | x Є int } R = pc0 = c0 ∩ x’ = 1 ∩ pc’ := c1U pc = c1 ∩ pc’ := c2U pc = c2∩ x ≤ 100 ∩ pc’ := c3U pc = c2 ∩ x > 100 ∩ pc’ := c5U pc = c3 ∩ x’ = x + 1 ∩ pc’ := c4U pc = c4∩ pc’ := c2 ^ ^ Example: Transition Relation Unity – like Process Semantics type arcs = {c0, c1, c2, c3, c4, c5} process Flowgraph = { • x : int, pc: arcs • init pc = c0 • update • pc = c0→ x := 1 ; pc := c1 • pc = c1→ pc := c2 • pc = c2∩ x ≤ 100 → pc := c3 • pc = c2 ∩ x > 100 → pc := c5 • pc = c3→ x := x + 1 ; pc := c4 • pc = c4→ pc := c2 } -9-

  5. post [ pc = 2 ∩ x ≤ 100 → pc := c3 ](φ) = pc’, x’ . pc’ = 2 ∩ x’ ≤ 100 ∩ pc = c3 ∩ φ(pc’, x’) pre [ pc = 2 ∩ x ≤ 100 → pc := c3 ](φ) = pc’, x’ . pc = 2 ∩ x ≤ 100 ∩ pc’ = 3 ═>φ(pc’, x’) = pc = 2 ∩ x ≤ 100 => φ[3/ pc’, x/ x’] E ~ A Example: Predicate Transformers -10-

  6. SA = (QA,vTiA, I A) is an abstraction of S iff (1) Iγ(IA) (2) i, q Є QA . Post [ Ti ](γ(q)) γ(TiA(q)) A ΤiA γ γ Τi Abstract Semantics of Programs Abstract State Graphs: S = (Q,vTi, I) state graph of the program QAa lattice of abstract states α : P(Q) → QAα(γ(q)) = q γ : QA→ P(Q) φ ═>γ(α(φ)) A Galois connection given γ→ α(φ) = ∏ {q Є QA | φ => γ(q)} Consequence: every concrete execution is represented by at least one abstract -11-

  7. Galois connection: γ(expA(B1, .., Bl)) = expA[φ/ B ] α(φ)= Λ {expA(B1, .., Bl )| φ═> expA[φ/ B ] } -not easily computed═> use apper approx (monomial) α’(φ)= Λ {(Bi | φ═> φi } l i = 1 Abstract State Lattice Main idea: Given {φ1, .. , φl} predicates on vars of concrete P. B1, .., Bl boolean variables (all concrete states that satisfy φi) Abstract states: set of predicates overB1, .., Bl(QA). Monomial on B1, .. , Bl: conjunction of Bi’s and ¬Bi’s (at most once) Theorem: The set M of monomials on B1 .. Bl forms: - complete lattice and (α ; γ) Galois connection -12-

  8. false if expA[φ/ B ] ═> ¬qi(3.0) Bj if post[Ti ](expA[φ/ B ]) ═> φj(3.1) Λ ¬Bj ifpost[Ti ](expA[φ/ B ]) ═> ¬φj(3.2) true otherwise(3.3) l i = 1 Ti A (expA) = otherwise Abstract Transitions Main idea:Ti A (expA) = α(post[Ti ](γ(expA))) Ti A Problem: difficult to compute. α( post[Ti ]( γ(expA))) expA γ α Ti Ti A (expA) = α’(post[Ti ](γ(expA))) post[Ti ](γ(expA)) γ(expA) Use monomial approximation α’ of α. Has following form: By using (2.1) and (2.2) : (3.1)BjifexpA[φ/ B ] Λ qj ═> φj [ assi(x)/ x ] -13-

  9. Abstract Successors Computing abstract successors: check validity of implications (3) - use a decision procedure (DP) - automatic theorem prover implementing such DPs. → for all i Є [1, .. , l ]. prove ( 3.0 | 3.1 | 3.2 ) Imposibility of proving (3): - post[Ti ](expA[φ/ B ]) intersects bothφj and ¬φj - abstract expA too big → cut in smaller pieces - because of monomials - abstract state space is too rough. - applied proof strategy is not powerful enough Abstract initial state: IA = α’(init) - in most practical cases, defines one value for most variables. - can be computed by evaluating φi -14-

More Related