Two Approaches to Inte terprocedural l Data ta Flow w Analys - PowerPoint PPT Presentation

CS updat ate operat ation on ∘  ∘∶ Γ × 𝐹 ∗ → Γ updates call strings along edges 𝑛𝑏𝑗𝑜 𝑞 𝑞 = 〈𝑑 1 〉 𝜇 ∘ 𝑑 1 , 𝑠 𝑞 𝑑 1 𝑠 call 𝑞 1 𝑞 , 𝑑 2 = 〈𝑑 1 〉 𝑑 1 ∘ 𝑠 1. Interproc. 𝑞 = 〈𝑑 1 , 𝑑 2 〉 𝑑 1 ∘ 𝑑 2 , 𝑠 2. Call 𝑑 2 call 𝑞 1 3. Return 𝑜 2 𝑑 1 𝑑 2 ∘ 𝑓 𝑞 , 𝑜 2 = 𝑑 1 𝑑 1 ∘ 𝑜 2 ,𝑓 𝑞 = 〈𝑑 1 〉 𝑜 1 𝑓 𝑞 return 𝑑 1 ∘ 𝑓 𝑞 , 𝑜 1 = 𝜇 12.06.2010 Nikolai Knopp 29

CS updat ate operat ation on ∘  ∘∶ Γ × 𝐹 ∗ → Γ updates call strings along edges 𝑛𝑏𝑗𝑜 𝑞 𝑞 = 〈𝑑 1 〉 𝜇 ∘ 𝑑 1 , 𝑠 𝑞 𝑑 1 𝑠 call 𝑞 1 𝑞 , 𝑑 2 = 〈𝑑 1 〉 𝑑 1 ∘ 𝑠 1. Interproc. 𝑞 = 〈𝑑 1 , 𝑑 2 〉 𝑑 1 ∘ 𝑑 2 , 𝑠 2. Call 𝑑 2 call 𝑞 1 3. Return 𝑜 2 4. invalid 𝑑 1 𝑑 2 ∘ 𝑓 𝑞 , 𝑜 2 = 𝑑 1 𝑑 1 ∘ 𝑓 𝑞 , 𝑜 2 undef. 𝑑 1 ∘ 𝑜 2 ,𝑓 𝑞 = 〈𝑑 1 〉 𝑜 1 𝑓 𝑞 return 𝑑 1 ∘ 𝑓 𝑞 , 𝑜 1 = 𝜇 𝑑 1 𝑑 2 ∘ 𝑓 𝑞 , 𝑜 1 undef. 12.06.2010 Nikolai Knopp 30

CS updat ate operat ation on ∘  ∘∶ Γ × 𝐹 ∗ → Γ updates call strings along edges 𝑛𝑏𝑗𝑜 𝑞 𝑞 = 〈𝑑 1 〉 𝜇 ∘ 𝑑 1 , 𝑠 𝑞 𝑑 1 𝑠 call 𝑞 1 𝑞 undef. 𝑑 1 ∘ 𝑑 1 , 𝑠 𝑞 , 𝑑 2 = 〈𝑑 1 〉 𝑑 1 ∘ 𝑠 1. Interproc. 𝑞 = 〈𝑑 1 , 𝑑 2 〉 𝑑 1 ∘ 𝑑 2 , 𝑠 2. Call 𝑑 2 call 𝑞 1 3. Return 𝑜 2 4. invalid 𝑑 1 𝑑 2 ∘ 𝑓 𝑞 , 𝑜 2 = 𝑑 1 𝑑 1 ∘ 𝑓 𝑞 , 𝑜 2 undef. 𝑑 1 ∘ 𝑜 2 ,𝑓 𝑞 = 〈𝑑 1 〉 𝑜 1 𝑓 𝑞 return 𝑑 1 ∘ 𝑓 𝑞 , 𝑜 1 = 𝜇 𝑑 1 𝑑 2 ∘ 𝑓 𝑞 , 𝑜 1 undef. 12.06.2010 Nikolai Knopp 31

∘ consi sist stent with 𝐽𝑊𝑄  𝛿 ′ ∘ 𝑛, 𝑜 = 𝛿 only defined iff: 1. (By definition) A path 𝑟 ′ ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑛 exists with 𝐷𝑁 𝑟 ′ = 𝛿 ′ 2. (Lemma) 𝑟 = 𝑟 ′ ||(𝑛,𝑜) lies in 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 and 𝐷𝑁 𝑟 = 𝛿 ⇒ “ ∘ only updates and yields call strings for interprocedurally valid paths” 12.06.2010 Nikolai Knopp 32

Edge effect cts s doma main 𝐺 ∗  𝐺 ∗ ≔ 𝑀 ∗ → 𝑀 ∗ embeds edge effects from 𝐺 into 𝑀 ∗ domain  For 𝑛, 𝑜 ∈ 𝐹 ∗ and data 𝜊 = . . , 𝛿 ′ , 𝑤 ′ , . . ∈ 𝑀 ∗ at 𝑛 , data at 𝑜 is 𝑔 𝑛,𝑜 ∗ 𝜊 = . . , 𝛿, 𝑤 ,. .  update each 𝛿 ′ to reflect 𝑛, 𝑜 taken: 𝛿 = 𝛿 ′ ∘ 𝑛,𝑜  propagate effect 𝑔 𝑛,𝑜 ∈ 𝑀 applied to each 𝑤 ′ : 𝑤 = 𝑔 𝑛,𝑜 𝑤 ′

Edge effect cts s 𝑔 𝑛,𝑜 ∗ Defini nition: n: Let 𝑛, 𝑜 ∈ 𝐹 ∗ , 𝑔 𝑛,𝑜 ∈ 𝐺, 𝜊 ∈ 𝑀 ∗ . 𝑑 1 , 1 , 𝜊 a := a - 1 𝑑 1 𝑑 2 ,⊤ 𝑛 𝑔 𝑛,𝑑 2 call p 𝑑 2 12.06.2010 Nikolai Knopp 34

Edge effect cts s 𝑔 𝑛,𝑜 ∗ Defini nition: n: Let 𝑛, 𝑜 ∈ 𝐹 ∗ , 𝑔 𝑛,𝑜 ∈ 𝐺, 𝜊 ∈ 𝑀 ∗ . 𝑑 1 , 1 , 𝜊 a := a - 1 𝑑 1 𝑑 2 ,⊤ 𝑛 𝑔 𝑛,𝑑 2 𝑑 1 , ⊤ , 𝜊 ∗ 𝑔 𝑛,𝑑 2 call p 𝑑 1 𝑑 2 ,⊤ 𝑑 2 12.06.2010 Nikolai Knopp 35

Edge effect cts s 𝑔 𝑛,𝑜 ∗ Defini nition: n: Let 𝑛, 𝑜 ∈ 𝐹 ∗ , 𝑔 𝑛,𝑜 ∈ 𝐺, 𝜊 ∈ 𝑀 ∗ . 𝜊 𝑑 1 𝑑 1 , 1 , 𝜊 a := a - 1 𝑑 1 𝑑 2 ,⊤ 𝑛 𝑔 𝑛,𝑑 2 ∗ 𝑑 1 , ⊤ , 𝑔 𝑛,𝑑 2 𝜊 𝑑 1 𝜊 ∗ 𝑔 𝑛,𝑑 2 call p 𝑑 1 𝑑 2 ,⊤ 𝑑 2 if ∃𝛿 ′ with ∀ 𝛿 ∈ Γ 𝛿 = 𝛿 ′ ∘ 𝑛, 𝑜 𝜊 𝛿′ 𝑔 𝑛,𝑜 𝜊 𝛿 ∗ 𝑔 𝑛,𝑜 ≔ ⊥ else 12.06.2010 Nikolai Knopp 36

Edge effect cts s doma main 𝐺 ∗ Define 𝐺 ∗ ⊆ 𝑀 ∗ → 𝑀 ∗ such that 𝐺 ∗ smallest set containing 1. | 𝑛, 𝑜 ∈ 𝐹 ∗ ∪ 𝑗𝑒 𝑀 ∗ ∗ 𝑔 𝑛,𝑜 𝐺 ∗ closed under functional composition and ⋀ 2.

Lemm mma: a: Prop operties s of 𝐺 ∗ monotone monotone distributive in 𝑀 ⇒ 𝐺 ∗ distributive in 𝑀 ∗ 𝐺 1. 𝐺 distributive in 𝑀 ⇒ 𝐺 ∗ continuo nuous us in 𝑀 ∗ : 2. For an infinite collection 𝜊 𝑙 𝑙≥1 ⊆ 𝑀 ∗ and 𝑛, 𝑜 ∈ 𝐹 ∗ : ∗ ∗ 𝑔 𝑛,𝑜 𝜊 𝑙 = 𝑔 𝑛,𝑜 𝜊 𝑙 𝑙 𝑙 12.06.2010 Nikolai Knopp 38

Dataf aflow equat ations ons 𝑦 𝑜 ∗  The dataflow problem for 𝐺 ∗ ,𝑀 ∗ : ∗  𝑦 𝑠 𝑛𝑏𝑗𝑜 = 𝜇, ⊤ ∗ = ⋀ 𝑜 ∈ 𝑂 ∗ − 𝑠 𝑛𝑏𝑗𝑜 ∗ ∗  𝑦 𝑜 𝑔 𝑛,𝑜 𝑦 𝑛 𝑛,𝑜 ∈𝐹 ∗  Claim: ∃ maximum fixed-point solution ( MFP MFP CS CS ) for 𝑜∈𝑂 ∗ ∗ 𝑦 𝑜 12.06.2010 Nikolai Knopp 40

Proo oof: Exist stence ce of max. FP solution on  Iteratively solving the equations yields decreasing ( 𝐺 ∗ monotone) chains ∗ 𝑗 ≥ 𝑦 𝑜 ∗ 𝑗+1 ∀𝑜 ∈ 𝑂 ∗ . 𝑦 𝑜 Then ∀𝛿 ∈ Γ ∗ 𝑗 𝛿 ≥ 𝑦 𝑜 ∗ 𝑗+1 𝛿 𝑦 𝑜 ∗ 𝛿 (exists as a decreasing chain in 𝑀 with limit lim𝑦 𝑜 𝑀 bounded). ∗ 𝑗 possibly infinite but well-defined ∗ ≔ lim  𝑦 𝑜 𝑗→∞ 𝑦 𝑜 𝑜∈𝑂 ∗ is maximal FP solution to DF problem. ∗ ⇒ 𝑦 𝑜 12.06.2010 Nikolai Knopp 41

Maximum mum FP solution on of (𝑀 ∗ , 𝐺 ∗ )  Initialise 𝑦 𝑜 𝑜∈𝑂 ∗ with ∗ ∗ 0  𝑦 𝑠 𝑛𝑏𝑗𝑜 = 𝜇, ⊤ ∗ 0 = ⊥ ∗ 𝑜 ∈ 𝑂 ∗ − 𝑠 𝑛𝑏𝑗𝑜  𝑦 𝑜  Result if Γ  finite te: Any iterative algorithm terminates and reaches MFP.  infini nite te: Iteration may diverge, intermediate steps unsafe approx. of MFP 12.06.2010 Nikolai Knopp 42

Exam ample main p 𝑞 if a=0 T 𝑠 read a, b F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 call p 𝑑 2 call p 12.06.2010 Nikolai Knopp 43

Exam ample main p 𝑞 if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 call p 𝑑 2 call p 12.06.2010 Nikolai Knopp 44

Exam ample main p 𝑞 if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 𝑑 2 call p 12.06.2010 Nikolai Knopp 45

Exam ample main p 𝑞 〈𝑑 1 〉,1 if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 𝑑 2 call p 12.06.2010 Nikolai Knopp 46

Exam ample main p 𝑞 〈𝑑 1 〉,1 if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 ∗ 𝑜 2 〈𝑑 1 〉,1 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 𝑑 2 call p 12.06.2010 Nikolai Knopp 47

Exam ample main p 𝑞 〈𝑑 1 〉,1 if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 ∗ 𝑜 2 〈𝑑 1 〉,1 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 𝑑 2 𝑑 1 ,⊤ call p 12.06.2010 Nikolai Knopp 48

Exam ample main p 〈𝑑 1 〉, 1 , 𝑞 𝑑 1 𝑑 2 , ⊤ if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 ∗ 𝑜 2 〈𝑑 1 〉,1 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 𝑑 2 𝑑 1 ,⊤ call p 12.06.2010 Nikolai Knopp 49

Exam ample main p 〈𝑑 1 〉, 1 , 𝑞 𝑑 1 𝑑 2 , ⊤ if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉, 1 , ∗ 𝑑 1 𝑑 2 , ⊤ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 𝑑 2 𝑑 1 ,⊤ call p 12.06.2010 Nikolai Knopp 50

Exam ample main p 〈𝑑 1 〉, 1 , 𝑞 𝑑 1 𝑑 2 , ⊤ if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉, 1 , ∗ 𝑑 1 𝑑 2 , ⊤ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉, 1 , 𝑑 2 𝑑 1 𝑑 2 , ⊤ call p 12.06.2010 Nikolai Knopp 51

Exam ample main p 〈𝑑 1 〉,1 , 𝑞 𝑑 1 𝑑 2 ,⊤ , if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑑 1 𝑑 2 𝑑 2 , ⊤ 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉, 1 , ∗ 𝑑 1 𝑑 2 , ⊤ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉, 1 , 𝑑 2 𝑑 1 𝑑 2 , ⊤ call p 12.06.2010 Nikolai Knopp 52

Exam ample main p 〈𝑑 1 〉,1 , 𝑞 𝑑 1 𝑑 2 ,⊤ , if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑑 1 𝑑 2 𝑑 2 , ⊤ 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉, 1 , ∗ 𝑑 1 𝑑 2 , ⊤ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉, 1 , 𝑑 2 𝑑 1 𝑑 2 , ⊤ call p ... 12.06.2010 Nikolai Knopp 53

Chap apter summ mmar ary Call string ng approach: ch:  New dataflow problem on top of (𝑀, 𝐺)  Data “tagged” with propagation history  Interprocedural flow explicit  Existence of MFP CS  No functional compositions  Only iteratively computable if Γ finite 12.06.2010 Nikolai Knopp 54

Struct cture 1. Definition of a new DF problem (𝑀 ∗ , 𝐺 ∗ ) n to 𝑴 ∗ , 𝑮 ∗ ≡ MOP solution 2. Proof: Solut ution 2. 3. Feasibility and precise variants 4. Approximative solution 12.06.2010 Nikolai Knopp 55

MOP P solution on and MFP CS CS  This chapter: Show that MOP solution 𝑧 𝑜 = 𝑔 𝑟 : 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 ,𝑜 ⊤ can be acquired from MFP CS .  Chapter outline: 1. Define MOP CS 2. Show MOP CS = MFP CS ′ ∈ 𝑀 from MFP CS 3. Extract results 𝑦 𝑜 ′ = MOP 4. Using MOP CS and MFP CS , show 𝑦 𝑜 12.06.2010 Nikolai Knopp 56

MOP P solution on for call strings s (MOP CS CS )  Propagation of data along paths in 𝐻 ∗ For 𝑟 = 𝑠 𝑛𝑏𝑗𝑜 , 𝑡 1 , … ,𝑡 𝑙 ,𝑜 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ : ∗ ≔ 𝑔 𝑡 𝑙 ,𝑜 ∗ ∗ ∗ ∗ 𝑔 ∘ 𝑔 𝑡 𝑙−1 ,𝑡 𝑙 ∘ ⋯∘ 𝑔 𝑡 1 ,𝑡 2 ∘ 𝑔 𝑟 (𝑠 𝑛𝑏𝑗𝑜 ,𝑡 1 )  Call string MOP solution ( MOP CS CS ) ∗ ≔ ⋀ 𝑔 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∀𝑜 ∈ 𝑂 ∗ : 𝑧 𝑜 ∗ ∶ 𝑟 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 𝑟 𝑛𝑏𝑗𝑜 𝑛𝑏𝑗𝑜 𝑠 ∗ ∗ … 𝑔 𝑔 𝑟 1 𝑟 2 𝑜 12.06.2010 Nikolai Knopp 57

MOP P solution on for call strings s (MOP CS CS )  Propagation of data along paths in 𝐻 ∗ For 𝑟 = 𝑠 𝑛𝑏𝑗𝑜 , 𝑡 1 , … ,𝑡 𝑙 ,𝑜 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ : ∗ ≔ 𝑔 𝑡 𝑙 ,𝑜 ∗ ∗ ∗ ∗ 𝑔 ∘ 𝑔 𝑡 𝑙−1 ,𝑡 𝑙 ∘ ⋯∘ 𝑔 𝑡 1 ,𝑡 2 ∘ 𝑔 𝑟 (𝑠 𝑛𝑏𝑗𝑜 ,𝑡 1 )  Call string MOP solution ( MOP CS CS ) ∗ ≔ ⋀ 𝑔 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∀𝑜 ∈ 𝑂 ∗ : 𝑧 𝑜 ∗ ∶ 𝑟 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 𝑟 𝑞𝑏𝑢ℎ 𝐻 ∗ ⊇ 𝐽𝑊𝑄 𝑛𝑏𝑗𝑜 𝑛𝑏𝑗𝑜 𝑠 ∗ ∗ … ? 𝑔 𝑔 𝑟 1 𝑟 2 𝑜 12.06.2010 Nikolai Knopp 58

Valid – pat ath-only prop opag agat ation on Lemma: : Let 𝑜 ∈ 𝑂 ∗ ,𝑟 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ 𝑠 𝑛𝑏𝑗𝑜 ,𝑜 , 𝛿 ∈ Γ . Then 𝑛𝑏𝑗𝑜 , 𝑜 and 𝐷𝑁 𝑟 = γ 𝑟 ∈ 𝐽𝑊𝑄 𝑠 1. ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 𝛿 defined ∗ ⇔ 𝑔 𝑟 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∗ 𝑔 𝛿 = 𝑔 𝑟 ⊤ 2. 𝑟 ⇒ “MOP CS only contains paths in 𝐽𝑊𝑄 and for each such 𝑞 yields the same result as 𝑀, 𝐺 ” 12.06.2010 Nikolai Knopp 59

Valid – pat ath-only prop opag agat ation on (Proo oof) By induc uction on 𝑚 𝑟 for 𝑟 = (𝑠 𝑛𝑏𝑗𝑜 , 𝑡 1 ,… ,𝑡 𝑙 , 𝑜) :  𝒎 𝒓 = 𝟏 : Let 𝑜 ∈ 𝑂 ∗ . 𝑛𝑏𝑗𝑜 ⇒ 𝐷𝑁 𝑟 = 𝜇 . 𝑟 = 𝑠 𝑛𝑏𝑗𝑜 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 ,𝑠 ∗ 𝑠 only def. for 𝜇 with ⊤ = 𝑔 𝑟 ⊤ . 𝑔 𝑛𝑏𝑗𝑜 = 𝜇, ⊤ 𝑟  IH: Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 then 𝑔 𝑞 ⊤ . ∗ 𝜇 = 𝑔 𝑟 12.06.2010 Nikolai Knopp 60

Valid – pat ath-only prop opag agat ation on (Proo oof)  IH: Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 then 𝑔 𝑞 ⊤ ∗ 𝜇 = 𝑔 𝑟  IS: 𝑚 𝑟 = 𝑙 . 𝑛𝑏𝑗𝑜 𝑠 ... 𝑡 𝑙−1 𝑟 𝑡 𝑙 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 𝛿 ∗ ∗ 𝑜 ≝ 𝑔 𝑡 𝑙 ,𝑜 𝑦 𝑡 𝑙 𝑔 𝑟 12.06.2010 Nikolai Knopp 61

Valid – pat ath-only prop opag agat ation on (Proo oof)  IH: Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 then 𝑔 𝑞 ⊤ ∗ 𝜇 = 𝑔 𝑟  IS: 𝑚 𝑟 = 𝑙 . 𝑛𝑏𝑗𝑜 𝑠 ... 𝛿 ′ = 𝐷𝑁 𝑟 ′ 𝛿 = 𝛿 ′ ∘ 𝑛, 𝑜 , 𝑟 ′ 𝑡 𝑙−1 𝑟 𝛿 ′ ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 𝑡 𝑙 ∗ 𝑔 𝑟 ′ ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 𝛿 ∗ ∗ 𝑜 ≝ 𝑔 𝑡 𝑙 ,𝑜 𝑦 𝑡 𝑙 𝑔 𝑟 12.06.2010 Nikolai Knopp 62

Valid – pat ath-only prop opag agat ation on (Proo oof)  IH: Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 then 𝑔 𝑞 ⊤ ∗ 𝜇 = 𝑔 𝑟  IS: 𝑚 𝑟 = 𝑙 . 𝑛𝑏𝑗𝑜 𝑠 ... 𝛿 ′ = 𝐷𝑁 𝑟 ′ 𝛿 = 𝛿 ′ ∘ 𝑛, 𝑜 , 𝑟 ′ 𝑡 𝑙−1 𝑟 𝛿 ′ ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 = 𝑱𝑰 𝑔 𝑟 ′ ⊤ 𝑡 𝑙 ∗ 𝑔 𝑟 ′ ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 𝛿 ∗ ∗ 𝑜 ≝ 𝑔 𝑡 𝑙 ,𝑜 𝑦 𝑡 𝑙 𝑔 𝑟 12.06.2010 Nikolai Knopp 63

Valid – pat ath-only prop opag agat ation on (Proo oof)  IH: Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 then 𝑔 𝑞 ⊤ ∗ 𝜇 = 𝑔 𝑟  IS: 𝑚 𝑟 = 𝑙 . 𝑛𝑏𝑗𝑜 𝑠 ... 𝛿 ′ = 𝐷𝑁 𝑟 ′ 𝛿 = 𝛿 ′ ∘ 𝑛, 𝑜 , 𝑟 ′ 𝑡 𝑙−1 𝑟 𝛿 ′ ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 = 𝑱𝑰 𝑔 𝑟 ′ ⊤ 𝑡 𝑙 ∗ 𝑔 𝑟 ′ ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 𝑟 ⊤ 𝛿 ∗ 𝑜 = 𝑔 𝑡 𝑙 ,𝑜 𝑔 𝑟 ′ ⊤ ≝ 𝑔 𝑔 ∎ 𝑟 12.06.2010 Nikolai Knopp 64

∗ = 𝑧 𝑜 MFP CS = MOP CS CS : : 𝑦 𝑜 ∗ ∗ = 𝑧 𝑜 Theorem: (𝑀, 𝐺) distributive ⇒ ∀𝑜 ∈ 𝑂 ∗ : 𝑦 𝑜 ∗ Proof: ∗ : Let 𝑟 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ = 𝑠 𝑛𝑏𝑗𝑜 ,𝑡 1 , … , 𝑡 𝑙 , 𝑜 . ∗ ≤ 𝒛 𝒐  𝒚 𝒐 𝑛𝑏𝑗𝑜 𝑠 : 𝑟 ∗ ≤ 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑡 1 ∗ 𝑢 𝑡 1 ∗ 𝑦 𝑡 1 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∗ ≤ 𝑔 𝑡 1 ,𝑡 2 ∗ ∗ 𝑡 2 𝑦 𝑡 2 𝑦 𝑡 1 ... ∗ ∗ ≤ 𝑔 𝑡 𝑙 ,𝑜 ∗ 𝑜 𝑦 𝑜 𝑦 𝑡 𝑙 12.06.2010 Nikolai Knopp 65

∗ = 𝑧 𝑜 MFP CS = MOP CS CS : : 𝑦 𝑜 ∗ ∗ = 𝑧 𝑜 Theorem: (𝑀, 𝐺) distributive ⇒ ∀𝑜 ∈ 𝑂 ∗ : 𝑦 𝑜 ∗ Proof: ∗ : Let 𝑟 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ = 𝑠 𝑛𝑏𝑗𝑜 ,𝑡 1 , … , 𝑡 𝑙 , 𝑜 . ∗ ≤ 𝒛 𝒐  𝒚 𝒐 𝑛𝑏𝑗𝑜 𝑠 : 𝑟 ∗ ≤ 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑡 1 ∗ 𝑢 𝑡 1 ∗ 𝑦 𝑡 1 𝑦 𝑠 𝑛𝑏𝑗𝑜 By monotonicity ∗ ≤ 𝑔 𝑡 1 ,𝑡 2 ∗ ∗ 𝑡 2 𝑦 𝑡 2 𝑦 𝑡 1 ∗ ≤ 𝑔 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∗ ⇒ 𝑦 𝑜 ... 𝑟 ∗ 𝑁𝑃𝑄 𝐷𝑇 = ⋀ 𝑔 ∗ 𝑟 𝑟 ∗ ≤ 𝑔 𝑡 𝑙 ,𝑜 ∗ 𝑜 𝑦 𝑜 𝑦 𝑡 𝑙 ∗ ≤ ⋀ 𝑔 ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∗ ⇒ 𝑦 𝑜 𝑟 𝑟 12.06.2010 Nikolai Knopp 66

∗ = 𝑧 𝑜 MFP CS = MOP CS CS : : 𝑦 𝑜 ∗ ∗ ≥ 𝒛 𝒐 ∗ :  𝒚 𝒐 ∗ 𝑗 ≥ 𝑧 𝑜 By induction on 𝑗: ∀𝑗 ≥ 0, 𝑜 ∈ 𝑂 ∗ : 𝑦 𝑜 ∗  𝒋 = 𝟏 : ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∗(0) ∗ ∗ 𝑜 = 𝑠 𝑛𝑏𝑗𝑜 ∶ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ≝ 𝜇, ⊤ = 𝑔 ≥ 𝑧 𝑠 𝑛𝑏𝑗𝑜 𝑟 0 ∗ 0 ≝ ⊥ ∗ ≥ 𝑧 𝑜 ∗ 𝑜 ≠ 𝑠 𝑛𝑏𝑗𝑜 ∶ 𝑦 𝑜  IH: ∗ 𝑗 ≥ 𝑧 ∗ ∀𝑗 ≤ 𝑙, 𝑜 ∈ 𝑂 ∗ : 𝑦 𝑜 𝑜  IS: Cont.+Dist. of 𝑔 𝑛,𝑜 ∗ IH ∗ 𝑙+1 ≝ ∗ ∗ 𝑙 ∗ ∗ ∗ 𝑦 𝑜 𝑔 𝑛,𝑜 𝑦 𝑛 𝑔 𝑛,𝑜 𝑧 𝑛 ≥ 𝑧 𝑜 ≥ 𝑛,𝑜 ∈𝐹 ∗ 𝑛,𝑜 ∈𝐹 ∗ ∎ 12.06.2010 Nikolai Knopp 67

Resu sult for 𝑀, 𝐺 from om MFP CS CS ′ ∈ 𝑀 as follows: Define 𝑦 𝑜 ′ ≔ 𝑦 𝑜 ∗ 𝛿 𝑦 𝑜 𝛿∈Γ ′ = 𝑦 𝑜 ∗ 𝜇 ∧ 𝑦 𝑜 ∗ 𝑦 𝑜 𝑑 1 ∧ 𝑑 1 ,1 , ∗ 𝑦 𝑜 𝑑 1 𝑑 2 ∧ ⋯ 𝑑 1 𝑑 2 , ⊤ , ∗ 𝑜 = 𝑦 𝑜 a := a - 1 = ⊥ ∧ 1 ∧ ⊤ ∧ ⋯ 𝑑 1 𝑑 2 𝑑 2 ,⊤ , = 1 ∧ ⊤ ∧ ⋯ = ⊤ … ′ is the MOP solution in (𝑀, 𝐺) Claim: 𝑦 𝑜 12.06.2010 Nikolai Knopp 68

MFP CS CS ≡ MOP ′ = 𝑧 𝑜 Theorem: For each 𝑜 ∈ 𝑂 ∗ : 𝑦 𝑜 Proof: ′ ′ Def 𝑦 𝑜 ∗ (𝛿) 𝑦 𝑜 = 𝑦 𝑜 𝛿∈Γ MFP CS = ∗ 𝑦 𝑠 𝑛𝑏𝑗𝑜 ∗ = 𝑔 𝛿 |𝑟 ∈ 𝑞𝑏𝑢ℎ 𝐻 ∗ (𝑠 𝑛𝑏𝑗𝑜 ,𝑜) MOP CS 𝑟 𝛿∈Γ Only-valid-path = 𝑔 𝑟 ⊤ |𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 sth. 𝐷𝑁 𝑟 = 𝛿 propag. Lemma 𝛿∈Γ 𝐷𝑁 −1 Γ ∩ 𝐽𝑊𝑄 = 𝑔 𝑟 ⊤ |𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 = 𝐽𝑊𝑄 = 𝑧 𝑜 ∎ 12.06.2010 Nikolai Knopp 69

Chap apter summ mmar ary  Shown:  Get MOP solution from MFP CS ′ straightforward and without ∗ → 𝑦 𝑜  Conversion 𝑦 𝑜 functional composition ⇒ Call string approach is actually useful  But: Computing MFP CS still problematic 12.06.2010 Nikolai Knopp 70

Struct cture 1. Definition of a new DF problem (𝑀 ∗ , 𝐺 ∗ ) 2. Proof: Solution to 𝑀 ∗ ,𝐺 ∗ ≡ MOP solution 3. Feasibility and preci cise variant nts 3. 4. Approximative solution 12.06.2010 Nikolai Knopp 71

Motivat ation  Observation: 𝑀 finite ⇒ functional approach converges, CS approach not necessarily  Idea: Ensure termination of CS approach by limiting to finite CS subset Γ 0 ⊆ Γ  Show: Can be done for all (𝑀, 𝐺) with finite 𝑀 without losing precision 12.06.2010 Nikolai Knopp 72

Redefinitions ons using Γ 𝑝  Γ 0 is finite subset of Γ fulfilling: If 𝛿 ∈ Γ 0 and 𝛿 ′ initial subtuple of 𝛿 ⇒ 𝛿 ′ ∈ Γ 0  𝐽𝑊𝑄 ′ ≔ 𝑟 ∈ 𝐽𝑊𝑄 | ∀ prefix 𝑟 ′ of 𝑟: 𝐷𝑁 𝑟 ′ ∈ Γ 0  Let ∘ 0 only act in Γ 0  discards 𝛿 ′ entirely iff 𝛿 ′ ∘ 0 𝑛,𝑜 ∉ Γ 0  ∘ 0 is consistent with 𝐽𝑊𝑄 ′ 12.06.2010 Nikolai Knopp 73

Definitions: ons: DF anal alysis s with Γ 0 ∗ with 𝑀 0 ∗ ,𝐺 0 ∗ ≔ Γ 0 → 𝑀 finite Now consider 𝑀 0  Dataflow equations now iteratively solvable ∗ 𝛿 ( MFP CS0 ) ′ 𝑦 𝑜 0 ≔ 𝑦 𝑜 𝛿∈Γ 0  MOP solution using only paths in 𝐽𝑊𝑄 ′ 𝑟 0 | 𝑟 ∈ 𝐽𝑊𝑄 ′ 𝑠 ( MOP CS0 ) ′ 𝑧 𝑜 0 ≔ 𝑔 𝑛𝑏𝑗𝑜 , 𝑜 12.06.2010 Nikolai Knopp 74

MOP CS0 = MFP CS0 Theorem: ∀𝑜 ∈ 𝑂 ∗ : 𝑦 𝑜 ′ ′ 0 = 𝑧 𝑜 0 Proof:  Completely analogous to MOP CS =MFP CS proof by replacing Γ, 𝐽𝑊𝑄,∘ by Γ 0 , 𝐽𝑊𝑄 ′ ,∘ 0  No reasoning about infinite meets or continuity of 𝐺 0 ∗ required ∎ 12.06.2010 Nikolai Knopp 75

Motivat ation: CS of limi mited length main p 〈𝑑 1 〉, 1 , 𝑞 𝑑 1 𝑑 2 , ⊤ if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉, 1 , ∗ 𝑑 1 𝑑 2 , ⊤ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉, 1 , 𝑑 2 𝑑 1 𝑑 2 , ⊤ call p 12.06.2010 Nikolai Knopp 76

Motivat ation: CS of limi mited length main p 〈𝑑 1 〉,1 , 𝑞 𝑑 1 𝑑 2 ,⊤ , if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑑 1 𝑑 2 𝑑 2 , ⊤ 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉, 1 , ∗ 𝑑 1 𝑑 2 , ⊤ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉, 1 , 𝑑 2 𝑑 1 𝑑 2 , ⊤ call p 12.06.2010 Nikolai Knopp 77

Motivat ation: CS of limi mited length main p 〈𝑑 1 〉,1 , 𝑞 𝑑 1 𝑑 2 ,⊤ , if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑑 1 𝑑 2 𝑑 2 , ⊤ 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉,1 , ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 𝑑 1 𝑑 2 ,⊤ , 𝑑 1 𝑑 2 𝑑 2 , ⊤ ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉, 1 , 𝑑 2 𝑑 1 𝑑 2 , ⊤ call p 12.06.2010 Nikolai Knopp 78

Motivat ation: CS of limi mited length main p 〈𝑑 1 〉,1 , 𝑞 𝑑 1 𝑑 2 ,⊤ , if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑑 1 𝑑 2 𝑑 2 , ⊤ 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉,1 , ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 𝑑 1 𝑑 2 ,⊤ , 𝑑 1 𝑑 2 𝑑 2 , ⊤ ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉,1 , 𝑑 2 𝑑 1 𝑑 2 ,⊤ , call p 𝑑 1 𝑑 2 𝑑 2 , ⊤ 12.06.2010 Nikolai Knopp 79

Motivat ation: CS of limi mited length 〈𝑑 1 〉, 1 , main p 𝑑 1 𝑑 2 ,⊤ , 𝑞 if a=0 T 𝑠 read a, b 𝜇, ⊤ 𝑑 1 𝑑 2 𝑑 2 , ⊤ , F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑑 1 𝑑 2 𝑑 2 𝑑 2 ,⊤ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉,1 , ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 𝑑 1 𝑑 2 ,⊤ , 𝑑 1 𝑑 2 𝑑 2 , ⊤ ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉,1 , 𝑑 2 𝑑 1 𝑑 2 ,⊤ , call p 𝑑 1 𝑑 2 𝑑 2 , ⊤ 12.06.2010 Nikolai Knopp 80

Motivat ation: CS of limi mited length 〈𝑑 1 〉, 1 , main p 𝑑 1 𝑑 2 ,⊤ , 𝑞 if a=0 T 𝑠 read a, b 𝜇, ⊤ 𝑑 1 𝑑 2 𝑑 2 , ⊤ , F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑑 1 𝑑 2 𝑑 2 𝑑 2 ,⊤ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉,1 , ∗ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 𝑑 1 𝑑 2 ,⊤ , 𝑑 1 𝑑 2 𝑑 2 , ⊤ ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉,1 , 𝑑 2 𝑑 1 𝑑 2 ,⊤ , call p 𝑑 1 𝑑 2 𝑑 2 , ⊤  No information gain on further iteration  Stop if data tagged by longer CS „irrelevant“ 12.06.2010 Nikolai Knopp 81

Motivat ation: CS of limi mited length main p 〈𝑑 1 〉, 1 , 𝑞 𝑑 1 𝑑 2 , ⊤ if a=0 T 𝑠 read a, b 𝜇, ⊤ F t := a * b 𝑛𝑏𝑗𝑜 𝑠 ∗ 𝑔 𝑠 𝑞 ,𝑜 2 〈𝑑 1 〉, 1 , ∗ 𝑑 1 𝑑 2 , ⊤ 𝑜 2 𝑔 𝑠 𝑛𝑏𝑗𝑜 ,𝑑 1 a := a - 1 ∗ ∗ ∗ 𝑔 𝑑 1 ,𝑠 𝑞 𝑔 𝑑 2 ,𝑠 𝑞 𝑔 𝑜 2 ,𝑑 2 𝑑 1 𝜇, 1 call p 〈𝑑 1 〉, 1 , 𝑑 2 𝑑 1 𝑑 2 , ⊤ call p  No information gain on further iteration  Stop if data tagged by longer CS „irrelevant“ 12.06.2010 Nikolai Knopp 82

Redundan ant inform ormat ation on in anal alysis  Why paths of arbitrary length?  𝐿 ≔ number of call sites in program ∈ ℕ ⇒ only possible by recurring on call site(s)  At each call, analysis arrives with value 𝑦 ∈ 𝑀 and returns with value 𝑧 ∈ 𝑀 . 𝑀 is finite ⇒ 𝑀 2 many distinct begin/end value combinations per call  Idea: Recurring on any of the 𝐿 calls more than 𝑀 2 times is redundant. 12.06.2010 Nikolai Knopp 83

Lemm mma: a: Path short ortening Let 𝑛𝑏𝑗𝑜 𝑞 1  (𝑀, 𝐺) with 𝑀 finite  𝑁 ≔ 𝐿 ∗ 𝑀 2 𝑞 2 𝑑 1 call 𝑞 1 𝑠  𝐿 ≔ # call blocks in 𝐻 ∗  Γ 0 = Γ 𝑁 . 𝑑 2 call 𝑞 1 Then ∀ 𝑜 ∈ 𝑂 ∗ : 𝑜 2 If 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 then exists 𝑟 ′ ∈ 𝐽𝑊𝑄 ′ (𝑠 𝑛𝑏𝑗𝑜 , 𝑜) with 𝑜 1 𝑓 𝑞 2 𝑟 0 = 𝑔 𝑟 ′ 0 . 𝑔 return 12.06.2010 Nikolai Knopp 84

Lemm mma: a: Path short ortening Let 𝑛𝑏𝑗𝑜 𝑞 1  (𝑀, 𝐺) with 𝑀 finite  𝑁 ≔ 𝐿 ∗ 𝑀 2 𝑞 2 𝑑 1 call 𝑞 1 𝑠  𝐿 ≔ # call blocks in 𝐻 ∗  Γ 0 = Γ 𝑁 . 𝑑 2 call 𝑞 1 𝑟 Then ∀ 𝑜 ∈ 𝑂 ∗ : 𝑜 2 If 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 then exists 𝑟 ′ ∈ 𝐽𝑊𝑄 ′ (𝑠 𝑛𝑏𝑗𝑜 , 𝑜) with 𝑜 1 𝑓 𝑞 2 𝑟 0 = 𝑔 𝑟 ′ 0 . 𝑔 return 12.06.2010 Nikolai Knopp 85

Lemm mma: a: Path short ortening Let 𝑛𝑏𝑗𝑜 𝑞 1  (𝑀, 𝐺) with 𝑀 finite  𝑁 ≔ 𝐿 ∗ 𝑀 2 𝑞 2 𝑑 1 call 𝑞 1 𝑠  𝐿 ≔ # call blocks in 𝐻 ∗  Γ 0 = Γ 𝑁 . 𝑑 2 call 𝑞 1 𝑟 ′ Then ∀ 𝑜 ∈ 𝑂 ∗ : 𝑜 2 If 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 then exists 𝑟 ′ ∈ 𝐽𝑊𝑄 ′ (𝑠 𝑛𝑏𝑗𝑜 , 𝑜) with 𝑜 1 𝑓 𝑞 2 𝑟 0 = 𝑔 𝑟 ′ 0 . 𝑔 return 12.06.2010 Nikolai Knopp 86

Lemm mma: a: Path short ortening (proo oof)  Proof: By induction on 𝑚(𝑟) ≔ length of 𝑟 .  𝒎 𝒓 = 𝟏 : 𝜇 ∈ Γ and 𝜇 ∈ Γ 0 .  IH: Lemma holds for all paths with 𝑚 𝑟 < 𝑙, 𝑙 ∈ ℕ + .  IS: Let 𝑜 ∈ 𝑂 ∗ , 𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 , 𝑚 𝑟 = 𝑙. Assume 𝑟 ∉ 𝐽𝑊𝑄 ′ 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 . Let 𝑟 ′ the shortest prefix with 𝑟 ′ ∉ 𝐽𝑊𝑄 ′ 𝑠 𝑛𝑏𝑗𝑜 , 𝑜 . Then 𝑟′ contains 𝑁 + 1 unreturned calls. By decomposition Lemma: 𝑟 ′ = 𝑟 0 || 𝑑 1 , 𝑠 𝑞 𝑁+1 ||𝑟 𝑁+1 𝑞1 ||𝑟 1 || 𝑑 2 , 𝑠 𝑞2 || … || 𝑑 𝑁+1 , 𝑠 12.06.2010 Nikolai Knopp 87

Lemm mma: a: Path short ortening (proo oof) 𝑟 ′ = 𝑟 0 || 𝑑 1 , 𝑠 𝑞 𝑁+1 ||𝑟 𝑁+1 𝑞1 ||𝑟 1 || 𝑑 2 , 𝑠 𝑞2 || … || 𝑑 𝑁+1 , 𝑠  𝑁 + 1 calls, but max. 𝑁 distinct elements. Let 𝑑 , 𝑠 𝑞 a duplicate call. If 𝑑 returns, let 𝑓 𝑞 , 𝑜 the return edge.  Rewrite 𝑟 as ′ || 𝑓 𝑞 ′ || 𝑑 , 𝑠 𝑞 ′ || 𝑑 , 𝑠 𝑞 ′ || 𝑓 𝑞 ′ 𝑟 0 || 𝑟 1 || 𝑟 2 , 𝑜 || 𝑟 3 ,𝑜 || 𝑟 4  Shorter 𝑟 𝑛𝑏𝑗𝑜 , 𝑜 with 𝑔 𝑟 by dropping ∈ 𝐽𝑊𝑄 𝑠 = 𝑔 𝑟 redundant parts IH, ∃𝑟 ′ ∈ 𝐽𝑊𝑄 ′ 𝑠  By IH 𝑛𝑏𝑗𝑜 ,𝑜 for 𝑟 with 𝑔 𝑟 ′ = 𝑔 and thus 𝑟 𝑟 . 𝑔 𝑟 ′ = 𝑔 ∎ 12.06.2010 Nikolai Knopp 88

Lemm mma: a: Path short ortening (proo oof) 𝑟 ′ = 𝑟 0 || 𝑑 1 , 𝑠 𝑞 𝑁+1 ||𝑟 𝑁+1 𝑞1 ||𝑟 1 || 𝑑 2 , 𝑠 𝑞2 || … || 𝑑 𝑁+1 , 𝑠  𝑁 + 1 calls, but max. 𝑁 distinct elements. Let 𝑑 , 𝑠 𝑞 a duplicate call. If 𝑑 returns, let 𝑓 𝑞 , 𝑜 the return edge.  Rewrite 𝑟 as ′ || 𝑓 𝑞 ′ || 𝑑 , 𝑠 𝑞 ′ || 𝑑 , 𝑠 𝑞 ′ || 𝑓 𝑞 ′ 𝑟 0 || 𝑟 1 || 𝑟 2 , 𝑜 || 𝑟 3 ,𝑜 || 𝑟 4  Shorter 𝑟 𝑛𝑏𝑗𝑜 , 𝑜 with 𝑔 𝑟 by dropping ∈ 𝐽𝑊𝑄 𝑠 = 𝑔 𝑟 redundant parts IH, ∃𝑟 ′ ∈ 𝐽𝑊𝑄 ′ 𝑠  By IH 𝑛𝑏𝑗𝑜 ,𝑜 for 𝑟 with 𝑔 𝑟 ′ = 𝑔 and thus 𝑟 𝑟 . 𝑔 𝑟 ′ = 𝑔 ∎ 12.06.2010 Nikolai Knopp 89

MOP CS0 =MOP  For 𝑁 ∈ ℕ ≥ 0 , define Γ 𝑁 as set of all 𝛿 ∈ Γ with length ≤ 𝑁 .  Theorem: Let 𝑀, 𝐺 a distributive DF framework with 𝑀 finite, and Γ 0 = Γ 𝑁 with 𝑁 = 𝐿 ∗ 𝑀 2 . Then 0 = 𝑧 𝑜 ∀𝑜 ∈ 𝑂 ∗ : 𝑦 𝑜 ′ ⇒ “If 𝛿 is too long and gets discarded, a shorter one in the set represents the same data.” 12.06.2010 Nikolai Knopp 90

Better bounds s for 𝑁  𝑁 impractically large for most problems  Lower bounds exists for certain problem classes. Example: Decomposable frameworks 12.06.2010 Nikolai Knopp 91

Decom omposab sable fram amework orks main p read a,b 𝑞 if … T 𝑠 t := a * b F 𝑛𝑏𝑗𝑜 𝑠 u := 2 * b a := a - 1 𝑑 1 call p 𝑑 2 call p t := a * b a*b,2*b 𝑜 2 avail.? b := b + 1 u := 2 * b t := a * b print t,u 𝑓 𝑛𝑏𝑗𝑜 𝑓 𝑞 stop return , 𝑢 , 𝑣 , 𝑢, 𝑣 , ⊥ 𝑁 = 50 𝑀 = 12.06.2010 Nikolai Knopp 92

Decom omposab sable fram amework orks main p read a,b 𝑞 if … T 𝑠 t := a * b F 𝑛𝑏𝑗𝑜 𝑗𝑒 𝑠 u := 2 * b +𝑢 + 𝑣 a := a - 1 𝑑 1 −𝑢 call p 𝑑 2 𝑗𝑒 call p t := a * b a*b,2*b 𝑜 2 avail.? b := b + 1 u := 2 * b t := a * b print t,u +𝑢 − 𝑣 𝑓 𝑛𝑏𝑗𝑜 𝑓 𝑞 stop return , 𝑢 , 𝑣 , 𝑢, 𝑣 , ⊥ 𝑁 = 50 𝑀 = 12.06.2010 Nikolai Knopp 93

Decom omposab sable fram amework orks main p read a,b 𝑞 𝑣 if … T 𝑠 t := a * b F 𝑛𝑏𝑗𝑜 𝑗𝑒 𝑠 u := 2 * b +𝑢 + 𝑣 𝑣 a := a - 1 𝑑 1 𝑢, 𝑣 −𝑢 call p 𝑑 2 𝑣 𝑗𝑒 call p t := a * b a*b,2*b 𝑜 2 𝑢 avail.? b := b + 1 u := 2 * b t := a * b print t,u +𝑢 − 𝑣 𝑓 𝑛𝑏𝑗𝑜 𝑓 𝑞 stop return , 𝑢 , 𝑣 , 𝑢, 𝑣 , ⊥ 𝑁 = 50 𝑀 = 12.06.2010 Nikolai Knopp 94

Decom omposab sable fram amework orks main p read a,b 𝑞 𝑣 if … T 𝑠 t := a * b F 𝑛𝑏𝑗𝑜 id 𝑗𝑒 𝑠 u := 2 * b 𝑣 +𝑢 +𝑣 a := a - 1 𝑑 1 𝑢 𝑣 −𝑢 𝑗𝑒 call p 𝑑 2 𝑣 𝑗𝑒 𝑗𝑒 call p t := a * b a*b,2*b 𝑜 2 𝑢 avail.? b := b + 1 u := 2 * b t := a * b print t,u +𝑢 −𝑣 𝑓 𝑛𝑏𝑗𝑜 𝑓 𝑞 stop return , 𝑢 ,⊥ , 𝑀 𝑣 = , 𝑣 , ⊥ , ⇒ 𝑀 = 𝑀 𝑢 × 𝑀 𝑣 𝑁 = 18 𝑀 𝑢 = 12.06.2010 Nikolai Knopp 95

Bound for decom compos osab able framew mework orks  Theorem: Let 𝑀, 𝐺 decomposable into 𝑙 frameworks 𝑀 𝑗 ,𝐺 𝑗 𝑗=1 . 𝑙 Setting the maximum callstring length to 2 𝑁 = 𝐿 ∗ max 𝑀 𝑗 𝑗∈ 1..𝑙 yields 𝑧 𝑜 0 = 𝑧 𝑜 ∀𝑜 ∈ 𝑂 ∗ . ′ 12.06.2010 Nikolai Knopp 96

1-relat ated fram amework orks  Defini nition: n: A decomposable DF framework (L,F) is 1-related if each 𝐺 𝑗 only consists of constant and identity functions.  Theorem: In this case, using Γ 0 = Γ 3𝐿 yields 0 = 𝑧 𝑜 ∀𝑜 ∈ 𝑂 ∗ . ′ 𝑧 𝑜  Example: Available expressions is 1-related. Decomp. into subproblems 𝑀 𝑓 ,𝐺 𝑓 for each expression 𝑓 with 𝑀 𝑓 = ⊤, 1,⊥ , 𝐺 ⊥ 𝑓 = 𝑗𝑒, 𝑔 ⊤ , 𝑔 1 , 𝑔 12.06.2010 Nikolai Knopp 97

Chap apter summ mmar ary  If 𝑀 finite, MFP CS iteratively computable on Γ 0  Precision/safeness depending on choice for Γ 0  Enforce termination by limiting max. CS length  Precision preserving bounds exist, but only of theoretical value  Better bounds exist for special problem classes 12.06.2010 Nikolai Knopp 98

Struct cture 1. Definition of a new DF problem (𝑀 ∗ , 𝐺 ∗ ) 2. Proof: Solution to 𝑀 ∗ ,𝐺 ∗ ≡ MOP solution 3. Feasibility and precise variants 4. Approximative solut utions ns 4. 12.06.2010 Nikolai Knopp 99

Motivat ation  Prefer a safe approximative solution 𝑦 𝑜 ≤ 𝑧 𝑜 over MFP CS if  MFP CS not (iteratively) computable  computation not feasible MFP CS by time/space constraints and computation of 𝑦 𝑜 ≤ 𝑧 𝑜 feasible. ⇒ Accept precision loss for less complexity. 12.06.2010 Nikolai Knopp 100

Reduci cing comp mplexity of CS approa oach ch 𝜏 Γ Γ 12.06.2010 Nikolai Knopp 101