Static Analysis of Programs: A Heap-Centric View
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1 Static Analysis of Programs: A Heap-Centric View ( uday) Department of Computer Science and Engineering, Indian Institute of Technology, Bombay 5 April 2008
2 Part 1 Introduction
3 ETAPS 08 Static Analysis: Introduction 1/140 Copyright A large part of these slides are based on the following material: 1., Amitabha Sanyal, and Amey Karkare. Heap Reference Analysis Using Access Graphs. ACM Transactions on Programming Languages and Systems, 30(1), Nov , Amitabha Sanyal, and Bageshri Karkare. Data Flow Analysis: Theory and Practice. CRC Press USA. (Under Publication) 3.. Data Flow Analysis. In The Compiler Design Handbook. Editors: Priti Shankar and Y. N. Srikant. CRC Press USA (2) is not yet published and the copyrights are reserved by the authors. The copyright for (1) is held by ACM and that for (3) is held by CRC Press USA. These slides may be used only for academic purposes.
4 ETAPS 08 Static Analysis: Introduction 2/140 Outline Motivation: The need of heap data flow analysis Formulating data flow analysis Mathematical foundations of data flow analysis The set of data flow values The set of flow functions Solutions of data flow analyses Algorithms for performing data flow analysis Complexity of data flow analysis Data flow analysis for heap references Conclusions
5 Part 2 Motivation
6 ETAPS 08 Static Analysis: Motivation 3/140 Motivation Heap memory allocation Limitations of garbage collection Optimization to improve garbage collection
7 ETAPS 08 Static Analysis: Motivation 4/140 Standard Memory Architecture of Programs Static Data Heap allocation provides the flexibility of Stack Variable Sizes. Data structures can grow or shrink as desired at runtime. (Not bound to the declarations in program.) Heap Code Variable Lifetimes. Data structures can be created and destroyed as desired at runtime. (Not bound to the activations of procedures.)
8 ETAPS 08 Static Analysis: Motivation 5/140 Managing Heap Memory Decision 1: When to Allocate? Explicit. Specified in the programs. (eg. Imperative/OO languages) Implicit. Decided by the language processors. (eg. Declarative Languages)
9 ETAPS 08 Static Analysis: Motivation 5/140 Managing Heap Memory Decision 1: When to Allocate? Explicit. Specified in the programs. (eg. Imperative/OO languages) Implicit. Decided by the language processors. (eg. Declarative Languages) Decision 2: When to Deallocate? Explicit. Manual Memory Management (eg. C/C++) Implicit. Automatic Memory Management aka Garbage Collection (eg. Java/Declarative languages)
10 ETAPS 08 Static Analysis: Motivation 6/140 State of Art in Manual Deallocation Memory leaks 10% to 20% of last development effort goes in plugging leaks Tool assisted manual plugging Purify, Electric Fence, RootCause, GlowCode, yaktest, Leak Tracer, BDW Garbage Collector, mtrace, memwatch, dmalloc etc. All leak detectors are dynamic (and hence specific to execution instances) generate massive reports to be perused by programmers usually do not locate last use but only allocation escaping a call At which program point should a leak be plugged?
11 ETAPS 08 Static Analysis: Motivation 7/140 Garbage Collection Automatic Deallocation Retain active data structure. Deallocate inactive data structure. What is an Active Data Structure?
12 ETAPS 08 Static Analysis: Motivation 7/140 Garbage Collection Automatic Deallocation Retain active data structure. Deallocate inactive data structure. What is an Active Data Structure? If an object does not have an access path, (i.e. it is unreachable) then its memory can be reclaimed.
13 ETAPS 08 Static Analysis: Motivation 7/140 Garbage Collection Automatic Deallocation Retain active data structure. Deallocate inactive data structure. What is an Active Data Structure? If an object does not have an access path, (i.e. it is unreachable) then its memory can be reclaimed. What if an object has an access path, but is not accessed after the given program point?
14 ETAPS 08 Static Analysis: Motivation 8/140 What is Garbage? Garbage 1 w = x // x points to m a 2 if (x.data < max) 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data Garbage z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
15 ETAPS 08 Static Analysis: Motivation 8/140 What is Garbage? False Garbage 1 w = x // x points to m a 2 if (x.data < max) 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data Garbage z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
16 ETAPS 08 Static Analysis: Motivation 8/140 What is Garbage? True Garbage 1 w = x // x points to m a 2 if (x.data < max) 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data Garbage z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
17 ETAPS 08 Static Analysis: Motivation 8/140 What is Garbage? Garbage 1 w = x // x points to m a 2 if (x.data < max) 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data Garbage z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o All white nodes are unused and should be considered garbage
18 ETAPS 08 Static Analysis: Motivation 9/140 Is Reachable Same as Live? From live (also known as alive, active) : Memory(2) or an object is live if the program will read from it in future. The term is often used more broadly to mean reachable. It is not possible, in general, for garbage collectors to determine exactly which objects are still live. Instead, they use some approximation to detect objects that are provably dead, such as those that are not reachable. Similar terms: reachable. Opposites: dead. See also: undead.
19 ETAPS 08 Static Analysis: Motivation 10/140 Is Reachable Same as Live? Not really. Most of us know that. Even with the state of art of garbage collection, 24% to 76% unused memory remains unclaimed Yet we have no way of distinguishing. Over a dozen reported approaches (since 1996), no real success.
20 ETAPS 08 Static Analysis: Motivation 11/140 Cedar Mesa Folk Wisdom Make the unused memory unreachable by setting references to NULL. (GC FAQ: z x w y p a q rptr lptr b i X lptr X lptr c f j m rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o Stack Heap
21 ETAPS 08 Static Analysis: Motivation 11/140 Cedar Mesa Folk Wisdom Make the unused memory unreachable by setting references to NULL. (GC FAQ: z x w y p a q rptr lptr b i lptr lptr c f j m rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o Stack Heap
22 ETAPS 08 Static Analysis: Motivation 12/140 Cedar Mesa Folk Wisdom Most promising, simplest to understand, yet the hardest to implement. Which references should be set to NULL? Most approaches rely on feedback from profiling. No systematic and clean solution.
23 ETAPS 08 Static Analysis: Motivation 13/140 Distinguishing Between Reachable and Live The state of art Eliminating objects reachable from root variables which are not live. Implemented in current Sun JVMs. Uses liveness data flow analysis of root variables (stack data). What about liveness of heap data?
24 ETAPS 08 Static Analysis: Motivation 14/140 Liveness of Stack Data 1 w = x // x points to m a 2 while (x.data < max) Heap 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr w x y z Stack 7 z.sum = x.data + y.data if changed to while
25 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) x = x.rptr y = x.lptr z = New class of z y = y.lptr z.sum = x.data + y.data No variable is used beyond this program point w x y z
26 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) x = x.rptr y = x.lptr z = New class of z Current values of x, y, and z are used beyond this program point y = y.lptr z.sum = x.data + y.data w x y z Live Dead
27 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) y = x.lptr x = x.rptr Current values of x, y, and z are used beyond this program point The value of y is different before and after the assignment to y z = New class of z w x y z y = y.lptr z.sum = x.data + y.data
28 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) y = x.lptr x = x.rptr The current values of x and y are used beyond this program point The current value of z is not used beyond this program point w x y z z = New class of z y = y.lptr z.sum = x.data + y.data
29 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) x = x.rptr y = x.lptr z = New class of z y = y.lptr w x y z The current values of x is used beyond this program point Current values of y and z are not used beyond this program point z.sum = x.data + y.data
30 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) x = x.rptr w x y z y = x.lptr z = New class of z y = y.lptr Nothing is known as of now Some information will be available in the next iteration point z.sum = x.data + y.data
31 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) y = x.lptr x = x.rptr z = New class of z w x y z Current value of x is used beyond this program point However its value is different before and after the assignment y = y.lptr z.sum = x.data + y.data
32 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) x = x.rptr y = x.lptr z = New class of z w x y z Current value of x is used beyond this program point There are two control flow paths beyond this program point y = y.lptr z.sum = x.data + y.data
33 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x while (x.data < max) w x y z x = x.rptr Current value of x is used beyond this program point y = x.lptr z = New class of z y = y.lptr z.sum = x.data + y.data
34 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x w x y z while (x.data < max) Current value of x is used beyond this program point x = x.rptr y = x.lptr z = New class of z y = y.lptr z.sum = x.data + y.data
35 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x w x y z w x y z while (x.data < max) x = x.rptr y = x.lptr z = New class of z y = y.lptr z.sum = x.data + y.data w x y z w x y z w x y z w x y z w x y z w x y z w x y z w x y z End of iteration #1 Live Dead
36 ETAPS 08 Static Analysis: Motivation 15/140 Liveness of Stack Data w = x w x y z w x y z while (x.data < max) x = x.rptr y = x.lptr z = New class of z y = y.lptr z.sum = x.data + y.data w x y z w x y z w x y z w x y z w x y z w x y z w x y z w x y z End of iteration #2 Live Dead
37 ETAPS 08 Static Analysis: Motivation 16/140 Applying Cedar Mesa Folk Wisdom to Heap Data Liveness Analysis of Heap Data If the while loop is not executed even once. 1 w = x // x points to m a 2 while (x.data < max) 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data z x w y p a q rptr lptr b i rptr lptr rptr lptr c f j m rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o Stack Heap
38 ETAPS 08 Static Analysis: Motivation 16/140 Applying Cedar Mesa Folk Wisdom to Heap Data Liveness Analysis of Heap Data If the while loop is executed once. 1 w = x // x points to m a 2 while (x.data < max) 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data z x w y p a q rptr lptr b i rptr lptr rptr lptr c f j m rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o Stack Heap
39 ETAPS 08 Static Analysis: Motivation 16/140 Applying Cedar Mesa Folk Wisdom to Heap Data Liveness Analysis of Heap Data If the while loop is executed twice. 1 w = x // x points to m a 2 while (x.data < max) 3 x = x.rptr 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data z x w y p a q rptr lptr b i rptr lptr rptr lptr c f j m rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o Stack Heap
40 ETAPS 08 Static Analysis: Motivation 17/140 The Moral of the Story Mappings between access expressions and l-values keep changing This is a rule for heap data For stack and static data, it is an exception! Static analysis of programs has made significant progress for stack and static data. What about heap data? Given two access expressions at a program point, do they have the same l-value? Given the same access expression at two program points, does it have the same l-value?
41 ETAPS 08 Static Analysis: Motivation 18/140 Our Solution 1 w = x 2 while (x.data < max) y = z = null w = null { x.lptr = null 3 x = x.rptr } 4 y = x.lptr 5 z = New class of z 6 y = y.lptr 7 z.sum = x.data + y.data x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null z.lptr = z.rptr = null y.lptr = y.rptr = null x = y = z = null
42 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null While loop is not executed even once z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
43 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null While loop is not executed even once z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
44 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null While loop is not executed even once z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
45 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null While loop is not executed even once z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
46 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null While loop is not executed even once z x w y Stack p a q rptr lptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
47 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null While loop is not executed even once z x w y Stack p a q rptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
48 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null While loop is not executed even once z x w y Stack p a q b i rptr lptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr d e g h k l n o
49 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null z x w y Stack While loop is executed once p a q rptr b i rptr lptr c f j m Heap rptr lptr lptr rptr lptr rptr lptr d e g h k l n o
50 ETAPS 08 Static Analysis: Motivation 19/140 Our Solution y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null z x w y Stack While loop is executed twice p a q rptr b i rptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr d e g h k l n o
51 ETAPS 08 Static Analysis: Motivation 20/140 Some Observations y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null Node i is live but link a i is nullified z x w y Stack p a q rptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
52 ETAPS 08 Static Analysis: Motivation 20/140 Some Observations y = z = null 1 w = x w = null 2 while (x.data < max) { x.lptr = null 3 x = x.rptr } x.rptr = x.lptr.rptr = null x.lptr.lptr.lptr = null x.lptr.lptr.rptr = null 4 y = x.lptr x.lptr = y.rptr = null y.lptr.lptr = y.lptr.rptr = null 5 z = New class of z z.lptr = z.rptr = null 6 y = y.lptr y.lptr = y.rptr = null 7 z.sum = x.data + y.data x = y = z = null New access expressions are created. Can they cause exceptions? z x w y Stack p a q rptr b i rptr lptr rptr lptr c f j m Heap rptr lptr rptr lptr rptr lptr rptr lptr d e g h k l n o
53 ETAPS 08 Static Analysis: Motivation 21/140 Goals of This Tutorial Main Goal: Explain how to get the modified program Side goals:
54 ETAPS 08 Static Analysis: Motivation 21/140 Goals of This Tutorial Main Goal: Explain how to get the modified program Side goals: Data Flow Values Acceptable Operations Desired Solutions Practical Algorithms Explain the intuitions Relate them to mathematical rigour Highlight the frontiers
55 Part 3 Formulating Data Flow Analysis
56 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 22/140 Formulating Data Flow Analysis Live variables analysis and available expressions analysis Local and global data flow properties Common form of data flow equations
57 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 23/140 Live Variables Analysis A variable v is live at a program point p, if some path from p to program exit contains an r-value occurrence of v which is not preceded by an l-value occurrence of v. Entry Entry Entry v =a b p v =a b p p Exit a=v +2 Exit v =a+2 Exit v =v = v +2+ 2
58 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 23/140 Live Variables Analysis A variable v is live at a program point p, if some path from p to program exit contains an r-value occurrence of v which is not preceded by an l-value occurrence of v. v is live at p Entry Entry Entry v =a b p v =a b p p Exit a=v +2 Exit v =a+2 Exit v =v = v +2+ 2
59 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 23/140 Live Variables Analysis A variable v is live at a program point p, if some path from p to program exit contains an r-value occurrence of v which is not preceded by an l-value occurrence of v. v is live at p Entry v is not live at p Entry Entry v =a b p v =a b p p Exit a=v +2 Exit v =a+2 Exit v =v = v +2+ 2
60 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 23/140 Live Variables Analysis A variable v is live at a program point p, if some path from p to program exit contains an r-value occurrence of v which is not preceded by an l-value occurrence of v. v is live at p v is not live at p v is live at p Entry Entry Entry v =a b p v =a b p p Exit a=v +2 Exit v =a+2 Exit v =v = v +2+ 2
61 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 23/140 Live Variables Analysis A variable v is live at a program point p, if some path from p to program exit contains an r-value occurrence of v which is not preceded by an l-value occurrence of v. Path based specification v is live at p v is not live at p v is live at p Entry Entry Entry v =a b p v =a b p p Exit a=v +2 Exit v =a+2 Exit v =v = v +2+ 2
62 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 24/140 Defining Data Flow Analysis for Live Variables Analysis LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j
63 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 24/140 Defining Data Flow Analysis for Live Variables Analysis Basic Blocks Single statements or Maximal groups of sequentially executed statements LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j
64 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 24/140 Defining Data Flow Analysis for Live Variables Analysis Basic Blocks Single statements or Maximal groups of sequentially executed statements LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j Control Transfer
65 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 24/140 Defining Data Flow Analysis for Live Variables Analysis LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j
66 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 24/140 Defining Data Flow Analysis for Live Variables Analysis LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j Local Data Flow Properties
67 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 25/140 Local Data Flow Properties for Live Variables Analysis LGen b = { v variable v is used in basic block b and is not preceded by a definition of v } LKill b = { v basic block b contains a definition of v }
68 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 25/140 Local Data Flow Properties for Live Variables Analysis r-value occurrence Value is only read, e.g. x,y,z in x.sum = y.data + z.data LGen b = { v variable v is used in basic block b and is not preceded by a definition of v } LKill b = { v basic block b contains a definition of v }
69 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 25/140 Local Data Flow Properties for Live Variables Analysis l-value occurrence r-value occurrence Value is modified e.g. y in Value is only read, e.g. x,y,z in y = x.lptr x.sum = y.data + z.data LGen b = { v variable v is used in basic block b and is not preceded by a definition of v } LKill b = { v basic block b contains a definition of v }
70 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 25/140 Local Data Flow Properties for Live Variables Analysis l-value occurrence r-value occurrence Value is modified e.g. y in Value is only read, e.g. x,y,z in y = x.lptr x.sum = y.data + z.data LGen b = { v variable v is used in basic block b and is not preceded by a definition of v } LKill b = { v basic block b contains a definition of v } within b
71 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 25/140 Local Data Flow Properties for Live Variables Analysis l-value occurrence r-value occurrence Value is modified e.g. y in Value is only read, e.g. x,y,z in y = x.lptr x.sum = y.data + z.data LGen b = { v variable v is used in basic block b and is not preceded by a definition of v } LKill b = { v basic block b contains a definition of v } within b anywhere in b
72 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 26/140 Defining Data Flow Analysis for Live Variables Analysis LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j
73 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 26/140 Defining Data Flow Analysis for Live Variables Analysis Global Data Flow Properties LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j
74 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 26/140 Defining Data Flow Analysis for Live Variables Analysis Global Data Flow Properties Edge based specifications LIn k = LGen k (LOut k LKill k ) LGen k, LKill k LOut k = LIn i LIn j LIn i LGen i, LKill i LOut i LIn j LGen j, LKill j LOut j
75 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 27/140 Data Flow Equations For Live Variables Analysis LIn b = LGen b (LOut b LKill b ) LOut b = s succ(b) LIn s b is the exit node otherwise
76 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 27/140 Data Flow Equations For Live Variables Analysis LIn b = LGen b (LOut b LKill b ) LOut b = s succ(b) LIn s b is the exit node otherwise Alternatively, LIn b = f b (LOut b ), where f b (X) = LGen b (X LKill b )
77 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 27/140 Data Flow Equations For Live Variables Analysis LIn b = LGen b (LOut b LKill b ) LOut b = s succ(b) LIn s b is the exit node otherwise Alternatively, LIn b = f b (LOut b ), where f b (X) = LGen b (X LKill b ) LIn b and LOut b are sets of variables.
78 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 28/140 Performing Live Variables Analysis LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) LGen={x}, LKill = {y} y = x.lptr LGen={x}, LKill = {x} x = x.rptr LGen=, LKill = {z} z = New class of z LGen ={y}, LKill = {y} y = y.lptr LGen ={x,y, z},lkill = z.sum = x.data + y.data
79 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 28/140 Performing Live Variables Analysis LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) LGen={x}, LKill = {y} y = x.lptr LGen=, LKill = {z} z = New class of z LGen ={y}, LKill = {y} y = y.lptr LGen={x}, LKill = {x} x = x.rptr LGen and LKill need not be mutually exclusive LGen ={x,y, z},lkill = z.sum = x.data + y.data
80 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 28/140 Performing Live Variables Analysis LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) LGen={x}, LKill = {y} y = x.lptr LGen={x}, LKill = {x} x = x.rptr LGen=, LKill = {z} z = New class of z LGen ={y}, LKill = {y} y = y.lptr LGen ={x,y, z},lkill = z.sum = x.data + y.data z is an r-value occurrence and not an l-value occurrence
81 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 28/140 Performing Live Variables Analysis LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) LGen={x}, LKill = {y} y = x.lptr LGen=, LKill = {z} z = New class of z LGen ={y}, LKill = {y} y = y.lptr LGen ={x,y, z},lkill = z.sum = x.data + y.data LGen={x}, LKill = {x} x = x.rptr x,y, z are considered to be used based purely on local use even if the value of z is not use later. A different analysis called faint variables analysis improves on this.
82 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 28/140 Performing Live Variables Analysis LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) LGen={x}, LKill = {y} y = x.lptr LGen={x}, LKill = {x} x = x.rptr LGen=, LKill = {z} z = New class of z LGen ={y}, LKill = {y} y = y.lptr LGen ={x,y, z},lkill = z.sum = x.data + y.data Initialization
83 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 28/140 Performing Live Variables Analysis {x} {x,y} {x,y} {x,y,z} {x,y,z} {x,y,z} {x,y,z} {x} {x} {x} {x} LGen={x}, LKill = {y} y = x.lptr LGen=, LKill = {z} z = New class of z LGen ={y}, LKill = {y} y = y.lptr LGen ={x,y, z},lkill = z.sum = x.data + y.data LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) {x} LGen={x}, LKill = {x} x = x.rptr Iteration #1 Traversal
84 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 28/140 Performing Live Variables Analysis {x} {x,y} {x,y} {x,y,z} {x,y,z} {x,y,z} {x,y,z} {x} {x} {x} {x} LGen={x}, LKill = {y} y = x.lptr LGen=, LKill = {z} z = New class of z LGen ={y}, LKill = {y} y = y.lptr LGen ={x,y, z},lkill = z.sum = x.data + y.data LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) {x} {x,y} {x,y} {x,y,z} {x,y,z} {x,y,z} {x,y,z} {x} {x} {x} {x} {x} LGen={x}, LKill = {x} x = x.rptr Iteration #1 #2 {x} {x} Traversal
85 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 29/140 Performing Live Variables Analysis LGen ={x}, LKill = {w} w = x LGen ={x}, LKill = while (x.data < max) LGen={x}, LKill = {y,z} y = x.lptr z = New class of z y = y.lptr z.sum = x.data + y.data LGen={x}, LKill = {x} x = x.rptr
86 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 30/140 Available Expressions Analysis An expression e is available at a program point p, if every path from program entry to p contains an evaluation of e which is not followed by a definition of any operand of e. Entry Entry Entry a b a b a b a b a b a b a = a b a b p p p Exit Exit Exit
87 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 30/140 Available Expressions Analysis An expression e is available at a program point p, if every path from program entry to p contains an evaluation of e which is not followed by a definition of any operand of e. a b is available at p Entry Entry Entry a b a b a b a b a b a b a = a b a b p p p Exit Exit Exit
88 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 30/140 Available Expressions Analysis An expression e is available at a program point p, if every path from program entry to p contains an evaluation of e which is not followed by a definition of any operand of e. a b is available at p Entry a b is not available at p Entry Entry a b a b a b a b a b a b a = a b a b p p p Exit Exit Exit
89 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 30/140 Available Expressions Analysis An expression e is available at a program point p, if every path from program entry to p contains an evaluation of e which is not followed by a definition of any operand of e. a b is available at p a b Entry a b a b is not available at p a b Entry a b a b is not available at p a b Entry a b a = a b a b p p p Exit Exit Exit
90 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 31/140 Local Data Flow Properties for Available Expressions Analysis AGen b = { e expression e is evaluated in basic block b and this evaluation is not followed by a definition of any operand of e} AKill b = { e basic block b contains a definition of an operand of e}
91 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 32/140 Data Flow Equations For Available Expressions Analysis AIn b = AOut p p pred(b) AOut b = AGen b (AIn b AKill b ) b is the entry node otherwise
92 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 32/140 Data Flow Equations For Available Expressions Analysis AIn b = AOut p p pred(b) AOut b = AGen b (AIn b AKill b ) b is the entry node otherwise Alternatively, AOut b = f b (AIn b ), where f b (X) = AGen b (X AKill b )
93 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 32/140 Data Flow Equations For Available Expressions Analysis AIn b = AOut p p pred(b) AOut b = AGen b (AIn b AKill b ) b is the entry node otherwise Alternatively, AOut b = f b (AIn b ), where f b (X) = AGen b (X AKill b ) AIn b and AOut b are sets of expressions.
94 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 33/140 An Example of Available Expressions Analysis 1 a b b c 1 Let e 1 a b, e 2 b c, e 3 c d, e 4 d e 2 c d 2 3 c = d = d e a b 5 Node Computed Killed Available Redund. 1 {e 1,e 2 } {e 3 } {e 1 } {e 2,e 3 } 0110 {e 1,e 3 } {e 3,e 4 } 0011 {e 1,e 3 } {e 1,e 4 } {e 1 } 1000 {e 1 } {e 4 } {e 1,e 4 } 1001 {e 4 } d e 6
95 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 33/140 An Example of Available Expressions Analysis Initialisation a b b c c d c = d = d e 5 a b d e Node Let e 1 a b, e 2 b c, e 3 c d, e 4 d e Computed Killed Available Redund. 1 {e 1,e 2 } {e 3 } {e 1 } {e 2,e 3 } 0110 {e 1,e 3 } {e 3,e 4 } 0011 {e 1,e 3 } {e 1,e 4 } {e 1 } 1000 {e 1 } {e 4 } {e 1,e 4 } 1001 {e 4 } 0001
96 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 33/140 An Example of Available Expressions Analysis Iteration # a b b c c d c = d = d e 5 a b d e Node Let e 1 a b, e 2 b c, e 3 c d, e 4 d e Computed Killed Available Redund. 1 {e 1,e 2 } {e 3 } {e 1 } {e 2,e 3 } 0110 {e 1,e 3 } {e 3,e 4 } 0011 {e 1,e 3 } {e 1,e 4 } {e 1 } 1000 {e 1 } {e 4 } {e 1,e 4 } 1001 {e 4 } 0001
97 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 33/140 An Example of Available Expressions Analysis Iteration # a b b c c d c = d = d e 5 a b d e Node Let e 1 a b, e 2 b c, e 3 c d, e 4 d e Computed Killed Available Redund. 1 {e 1,e 2 } {e 3 } {e 1 } {e 2,e 3 } 0110 {e 1,e 3 } {e 3,e 4 } 0011 {e 1,e 3 } {e 1,e 4 } {e 1 } 1000 {e 1 } {e 4 } {e 1,e 4 } 1001 {e 4 } 0001
98 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 33/140 An Example of Available Expressions Analysis Final Result a b b c c d c = d = d e 5 a b d e Node Let e 1 a b, e 2 b c, e 3 c d, e 4 d e Computed Killed Available Redund. 1 {e 1,e 2 } {e 3 } {e 1 } {e 2,e 3 } 0110 {e 1,e 3 } {e 3,e 4 } 0011 {e 1,e 3 } {e 1,e 4 } {e 1 } 1000 {e 1 } {e 4 } {e 1,e 4 } 1001 {e 4 } 0001
99 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 34/140 Using Data Flow Information Available Expressions Analysis. Used for common subsexpression elimination.
100 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 34/140 Using Data Flow Information Available Expressions Analysis. Used for common subsexpression elimination. If an expression is available at the entry of a block b and
101 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 34/140 Using Data Flow Information Available Expressions Analysis. Used for common subsexpression elimination. If an expression is available at the entry of a block b and a computation of the expression exists in b such that
102 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 34/140 Using Data Flow Information Available Expressions Analysis. Used for common subsexpression elimination. If an expression is available at the entry of a block b and a computation of the expression exists in b such that it is not preceded by a definition of any of its operands
103 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 34/140 Using Data Flow Information Available Expressions Analysis. Used for common subsexpression elimination. If an expression is available at the entry of a block b and a computation of the expression exists in b such that it is not preceded by a definition of any of its operands Then the expression is redundant.
104 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 34/140 Using Data Flow Information Available Expressions Analysis. Used for common subsexpression elimination. If an expression is available at the entry of a block b and a computation of the expression exists in b such that it is not preceded by a definition of any of its operands Then the expression is redundant. Expression must be upwards exposed or locally anticipable.
105 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 34/140 Using Data Flow Information Available Expressions Analysis. Used for common subsexpression elimination. If an expression is available at the entry of a block b and a computation of the expression exists in b such that it is not preceded by a definition of any of its operands Then the expression is redundant. Expression must be upwards exposed or locally anticipable. Expressions in Gen b are downwards exposed.
106 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 35/140 Using Data Flow Information Live Variables Analysis. Used for register allocation. If variable x is live in a basic block b, it is a potential candidate for register allocation.
107 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 35/140 Using Data Flow Information Live Variables Analysis. Used for register allocation. If variable x is live in a basic block b, it is a potential candidate for register allocation. Used for dead code elimination. If variable x is not live after an assignment x =..., then the assginment is redundant and can be deleted as dead code.
108 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 36/140 Reaching Definitions Analysis A definition d x : x = y reaches a program point u if it appears (without a refefinition of x) on some path from program entry to u Application : Copy Propagation A use of a variable x at a program point u can be replaced by y if d x : x = y is the only definition which reaches p and y is not modified between the point of d x and p.
109 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 37/140 Defining Data Flow Analysis for Reaching Definitions Analysis Let d v be a definition of variable v Gen b = { d v variable v is defined in basic block b and this definition is not followed (within b) by a definition of v} Kill b = { d v basic block b contains a definition of v}
110 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 38/140 Data Flow Equations for Reaching Definitions Analysis IN b = {} b is the entry node OUT p otherwise p pred(b) OUT b = Gen b (IN b Kill b ) IN b and OUT b are sets of definitions
111 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 39/140 Very Busy Expressions Analysis An expression e is very busy at a program point u, if every path from u to the program exit contains an evaluation of e which is not preceded by a redefinition of any operand of e. Application : Safety of Code Motion
112 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 40/140 Safety of Code Motion 1 if (b == 0) 1 False True 2 c = a/b 2 3 c = a/b 3 Expression a/b is not very busy at the exit of 1. Moving a/b to the exit of 1 is unsafe.
113 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 41/140 Defining Data Flow Analysis for Very Busy Expressions Analysis Gen b = { e expression e is evaluated in basic block b and this evaluation is not preceded (within b) by a definition of any operand of e} Kill b = { e basic block b contains a definition of an operand of e}
114 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 42/140 Data Flow Equations for Very Busy Expressions Analysis IN b = Gen b (OUT b Kill b ) {} b is the exit node OUT b = IN s otherwise a succ(b) IN b and OUT b are sets of expressions
115 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 43/140 Common Form of Data Flow Equations X i = f (Y i ) Y i = X j
116 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 43/140 Common Form of Data Flow Equations Data Flow Information So far we have seen sets (or bit vectors). Could be entities other than sets. X i = f (Y i ) Y i = X j
117 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 43/140 Common Form of Data Flow Equations Data Flow Information So far we have seen sets (or bit vectors). Could be entities other than sets. X i = f (Y i ) Flow Function Y i = X j
118 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 43/140 Common Form of Data Flow Equations Data Flow Information So far we have seen sets (or bit vectors). Could be entities other than sets. X i = f (Y i ) Flow Function Y i = X j Confluence So far we have seen and. Could be other operations.
119 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 44/140 A Taxonomy of Bit Vector Data Flow Frameworks Confluence Union Intersection Forward Reaching Definitions Available Expressions Backward Live Variables Anticipable Exressions Bidirectional Partial Redundancy Elimination (limited) (Original M-R Formulation)
120 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 44/140 A Taxonomy of Bit Vector Data Flow Frameworks Any Path Confluence Union Intersection Forward Reaching Definitions Available Expressions Backward Live Variables Anticipable Exressions Bidirectional Partial Redundancy Elimination (limited) (Original M-R Formulation)
121 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 44/140 A Taxonomy of Bit Vector Data Flow Frameworks Any Path All Paths Confluence Union Intersection Forward Reaching Definitions Available Expressions Backward Live Variables Anticipable Exressions Bidirectional Partial Redundancy Elimination (limited) (Original M-R Formulation)
122 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 44/140 A Taxonomy of Bit Vector Data Flow Frameworks Any Path All Paths Confluence Union Intersection Forward Reaching Definitions Available Expressions Backward Live Variables Anticipable Exressions Bidirectional Partial Redundancy Elimination (limited) (Original M-R Formulation)
123 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 44/140 A Taxonomy of Bit Vector Data Flow Frameworks Any Path All Paths Confluence Union Intersection Forward Reaching Definitions Available Expressions Backward Live Variables Anticipable Exressions Bidirectional Partial Redundancy Elimination (limited) (Original M-R Formulation)
124 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 44/140 A Taxonomy of Bit Vector Data Flow Frameworks Any Path All Paths Confluence Union Intersection Forward Reaching Definitions Available Expressions Backward Live Variables Anticipable Exressions Bidirectional Partial Redundancy Elimination (limited) (Original M-R Formulation)
125 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 n 2 c = a + b d = a b n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3
126 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN n 1 n 2 c = a + b d = a b n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3
127 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 n 2 c = a + b d = a b n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3
128 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 n 2 c = a + b d = a b n 2 1, 2, 3, 2 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3
129 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 1, 2, 3, 2 n 2 n 2 c = a + b d = a b n 2 2, 1, 3, 2 n 3 d = c 1 a = 2 n 3 b = 1 c = a + b n 3
130 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 1, 2, 3, 2 n 2 n 2 c = a + b d = a b n 2 2, 1, 3, 2 n 3 2, 1, 3, 2 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3
131 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 1, 2, 3, 2 n 2 n 2 c = a + b d = a b n 2 2, 1, 3, 2 n 3 2, 1, 3, 2 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3 2, 1, 3, 2 n 3
132 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 1, 2, 3, 2 n 2 n 2 c = a + b d = a b n 2 2, 1, 3, 2 n 3 2, 1, 3, 2 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2
133 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 1, 2, 3, 2 n 2 n 2 c = a + b d = a b n 2 2, 1, 3, 2 n 3 2, 1, 3, 2 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2 n 3...
134 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b Summary Values?,?,?,? n 1 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT n 1 1, 2, 3, 2 n 2 n 2 c = a + b d = a b n 2 2, 1, 3, 2 n 3 2, 1, 3, 2 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2 n 3...
135 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 2 c = a + b d = a b Summary Values?,?,?,? n 1 1, 2, 3,? n 2 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT 1, 2, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 1 n 2 n 3 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2 n 3...
136 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 2 c = a + b d = a b Summary Values?,?,?,? n 1 1, 2, 3,?,, 3, 2 n 2 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT 1, 2, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 1 n 2 n 3 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2 n 3...
137 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 2 c = a + b d = a b Summary Values?,?,?,? n 1 1, 2, 3,?,, 3, 2 n 2,, 3, 2 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT 1, 2, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 1 n 2 n 3 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b n 3 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2 n 3...
138 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 2 c = a + b d = a b Summary Values?,?,?,? n 1 1, 2, 3,?,, 3, 2 n 2,, 3, 2 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT 1, 2, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 1 n 2 n 3 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b,, 3, 2 n 3 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2 n 3...
139 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 45/140 An Introduction to Constant Propagation n 1 a = 1 b = 2 c = a + b n 2 c = a + b d = a b Summary Values?,?,?,? n 1 1, 2, 3,?,, 3, 2 n 2,, 3, 2 a, b, c, d?,?,?,? Execution Sequence IN 1, 2, 3,? OUT 1, 2, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 1 n 2 n 3 n 2 d = c 1 a = 2 n 3 b = 1 c = a + b,, 3, 2 n 3 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 2, 1, 3, 2 n 3 n 2 n 3...
140 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 46/140 Issues Data Flow Values Acceptable Operations Desired Solutions Practical Algorithms
141 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 46/140 Issues Representation Lattice Partial Order, Top, Bottom Data Flow Values Acceptable Operations Desired Solutions Practical Algorithms
142 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 46/140 Issues Representation Lattice Partial Order, Top, Bottom Merge Commutativity, Associativity, Idempotence Data Flow Values Acceptable Operations Desired Solutions Practical Algorithms
143 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 46/140 Issues Representation Lattice Partial Order, Top, Bottom Merge Commutativity, Associativity, Idempotence Data Flow Values Acceptable Operations Desired Solutions Practical Algorithms Flow Functions Monotonicity, Distributivity, k-boundedness, Separability
144 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 46/140 Issues Representation Existence Lattice Partial Order, Top, Bottom Merge Commutativity, Associativity, Idempotence Data Flow Values Acceptable Operations Desired Solutions Practical Algorithms Safety Precision Flow Functions Monotonicity, Distributivity, k-boundedness, Separability
145 ETAPS 08 Static Analysis: Formulating Data Flow Analysis 46/140 Issues Representation Existence Lattice Partial Order, Top, Bottom Merge Commutativity, Associativity, Idempotence Data Flow Functions Flow Values Acceptable Operations Monotonicity, Distributivity, k-boundedness, Separability Desired Solutions Practical Algorithms Safety Initialisation Precision Complexity Convergence
146 Part 4 Data Flow Values
147 ETAPS 08 Static Analysis: Data Flow Values 47/140 The Set of Data Flow Values Properties of the data flow values The notion of approximations Combining data flow values
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