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Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Claus Brabrand ((( brabrand@itu.dk ))) Associate Professor, Ph.D. ((( Programming, Logic, and Semantics ))) IT University of Copenhagen A NALYSIS F LOW - D ATA [ HOW TO ANALYZE LANGUAGES AUTOMATICALLY ]

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[ 2 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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[ 3 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS

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[ 4 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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[ 5 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS All you need is…: int x = 1; int y = 3; x = x+2;x y; print(x,y); E. E[x 1 ] E. E[y 3 ] E. E[x E(x) 2 ] E. E[x E(y), y E(x)] [, ] ENV L [, ] x y... We (only) need 3 things: A control-flow graph A lattice Transfer functions int x = 1; int y = 3; if (...) { x = x+2; } else { x y; } print(x,y); Given program:

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[ 6 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Solve Equations :-) One big lattice: E.g., (L |VAR| ) |PP| 1 big abstract value vector: [,,..., ] (L |VAR| ) |PP| 1 big transfer function: T : (L |VAR| ) |PP| (L |VAR| ) |PP| Compute fixed-point (simply): Start with bottom value vector ( ) Iterate transfer function ‘T’ (until nothing changes) Done; print out (or use) solution…! :-) [, ] (L |VAR| ) |PP|

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[ 7 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS x = 0; x = x+1; output x; a = b = f x=0 (a) c = b d d = f x=x+1 (c) e = d The Entire Process :-) 3. Recursive equations: x = 0; do { x = x+1; } while (…); output x; Program: 1. Control-flow graph: T 4. one ”big” transfer function: T((a,b,c,d,e)) = (,f x=0 (a),b d,f x=x+1 (c),d) |VAR|*|PP| = 1*5 = 5 …over a ”big” power-lattice: T T T 0 ( ) T 1 ( ) T 2 ( ) T 3 ( ) T a = b = d = c = e = T 5 ( ) T = LEAST FIXED POINT ANOTHER FIXED POINT 5. Solve rec. equations…: 2. Transfer functions: solution T 4 ( ) T f x=0 ( l ) = f x=x+1 ( l ) = l L

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[ 8 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Exercise: Repeat this process for program (of two vars): i.e., determine…: 1) Control-flow graph 2) Transfer functions 3) Recursive equations 4) One ”big” transfer function 5) Solve recursive equations :-) x = 1; y = 0; while (v>w) { x y; } y = y+1; …using lattice:

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[ 9 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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[ 10 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS f x=0 f x=x+1 LEAST FIXED POINT = Naïve Fixed-Point Algorithm Slow! Naïve Fixed-Point Algorithm: …uses intermediate ”results” from previous iteration: a = b = f x=0 (a) c = b d d = f x=x+1 (c) e = d f x=0 ( l ) = f x=x+1 ( l ) = l L solution f x=0 f x=x+1 f x=0 f x=x+1 f x=0 f x=x+1 f x=0 f x=x+1 a b d c e

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[ 11 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Chaotic Iteration Algorithm Faster! Chaotic Iteration Algorithm: …exploits ”forward nature” of program control flow: a = b = f x=1 (a) c = b d d = f x=x+1 (c) e = d f x=1 ( l ) = f x=x+1 ( l ) = l L solution LEAST FIXED POINT (always uses ”latest” results) f x=0 f x=x+1 f x=0 f x=x+1 f x=0 f x=x+1 a b d c e

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[ 12 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Work-list Algorithm Fastest! Work-list Algorithm: …uses a ”queue” to control (optimize) computation: a = b = f x=1 (a) c = b d d = f x=x+1 (c) e = d f x=1 ( l ) = f x=x+1 ( l ) = l L solution a b d c e [a] Queue: [b] [d,c] Initialize queue with start point: Q := [a] Pop top element from queue and (re-)compute it; IF it changed THEN enqueue all points that depend on it’s value (if it isn’t already on the queue) Stop when queue is empty LEAST FIXED POINT [c,e] [] … (in general) [c]

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[ 13 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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[ 14 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS The Language ’C--’ Syntactic Categories: Expressions ( E EXP): Statements ( S STM): (…assume we only have integer variables ‘ x ’, ‘ y ’, ‘ z ’) E : n | v | E + E’ | – E | E * E’ | E == E’ | input S : skip ; | v := E ; | output E ; | if E then S else S’ | while E do S | { var v; S 1 … S n }

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[ 15 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Control-Flow Graph (for ’C--’) Inductively defined control-flow graph: if ( E ) S 1 else S 2 S1S1 E truefalse S2S2 confluence while ( E ) S S E true false confluence skip ; v := E 1 ; output E ; { var v; S 1 … S n } S1S1 SnSn var v; …

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[ 16 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Sign Analysis: Lattice Lattice: ENV Lattice : Confluence operator: = ’(,, )’ (pairwise) x y z L SIGN |VAR| L SIGN : VAR L SIGN x y z

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[ 17 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Sign Analysis: Transfer F’s Transfer Functions: Env. Env[x sign(Env, Exp )] x := Exp ; var x; Env. Env[x ] Env. Env output … ;

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[ 18 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Inductive definition of ’ sign ’ in the syntactic structure of Exp Syntax: sign(Env, n ) = sign(Env, input ) = sign(Env, v ) = Env( v ) sign(Env, E 1 +E 2 ) = sign(Env, E 1 ) L sign(Env, E 2 ) sign(Env, -E ) = - L sign(Env, E ) … E : n | input | v | E + E’ | E * E’ | E == E’ | - E

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[ 19 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Exercise: Come up with a program the analysis… A) can analyse precisely B) can’t analyse precisely

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[ 20 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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[ 21 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Const Propagation: Lattice Lattice: ENV Lattice : Confluence operator: = ’(,, )’ (pairwise) x y z L NUM |VAR| VAR L NUM L NUM :

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[ 22 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Const Propagation: Transfer F’s Transfer Functions: Env. Env[x eval(Env, Exp )] x := Exp ; var x; Env. Env[x ] Env. Env output … ;

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[ 23 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Inductive definition of ’ eval ’ in the syntactic structure of Exp Syntax: eval(Env, n ) = n eval(Env, input ) = eval(Env, v ) = Env( v ) eval(Env, E 1 +E 2 ) = eval(Env, E 1 ) L eval(Env, E 2 ) eval(Env, -E ) = - L eval(Env, E ) … E : n | input | v | E + E’ | E * E’ | E == E’ | - E, if n = m = n L m =, if n = m = r, o/w (where r = n + m) …i.e.:

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[ 24 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Exercise: Come up with a program the analysis… A) can analyse precisely B) can’t analyse precisely

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[ 25 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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[ 26 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Initialized Variables Analysis Lattice: ENV Lattice : Confluence operator: = ’(,, )’ (pairwise) x y z x y z definitely initialized possibly uninitialized Note: It’s always "safe" to answer "too high"

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[ 27 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Initialized Variables Analysis Transfer Functions: Env. Env x := … ; output … ; var x; Env. Env[x ] Env. Env[x init(Env, Exp )]

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[ 28 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Inductive definition of ’ init ’ in the syntactic structure of Exp Syntax: init(Env, n ) = init(Env, input ) = init(Env, v ) = Env( v ) init(Env, E 1 +E 2 ) = eval(Env, E 1 ) L eval(Env, E 2 ) init(Env, -E ) = - L eval(Env, E ) … E : n | input | v | E + E’ | E * E’ | E == E’ | - E LL

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[ 29 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Note: Isomorphism! With value lattice: ENV-lattice is isomorphic to: …for every program point: isomorphic ’x’’x’’y’’y’’z’’z’ Init VARS { x, y, z } Vars that are possibly uninitialized

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[ 30 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Initialized Variables Analysis (Revisited) ENV-lattice: Confluence operator: = ’ ’ (i.e., set union) Transfer Functions: S. S \ { x } S. S x := … ; output … ; var x; Vars that are possibly uninitialized S. S { x }

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[ 31 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Exercise: Come up with a program the analysis… A) can analyse precisely B) can’t analyse precisely

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[ 32 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Set-based analyses… {may,must} {forwards,backwards}

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[ 34 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Forwards vs. Backwards? What you have seen: Forwards: Some analyses…: Backwards: Analyze info that depends on future behavior Analyze info that depends on past behavior x = 0; x = x+1; output x; y = 2*x; x = x+1; output x; E.g.: - Live Variables - Very Busy Expressions ’ y ’ dead here

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[ 35 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS May vs. Must? What you have seen: Confluence: = ’ ’ (set union) Partial order: = ’ ’ (sub-set-eq) Some analyses…: Confluence: = ’ ’ (set intersection) Partial order: = ’ ’ (super-set-eq) Analyze info that must definitely be true paths Analyze info that may possibly be true paths E.g.: - Initialized Variables - Available Expressions - Very Busy Expressions E.g.: - Uninitialized Variables

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[ 36 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS DUALITY: Lattice Lattice M: Confluence: M Partial order: M ” ”: W Confluence: W = M Partial order: W = M (M, ) (W, ) Lattice Stand on head

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[ 37 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS DUALITY: Framework What we have done: Approximation: ”safe” to throw away info upwards With confluence: (least upper bound) Least fixed point: Computed as: ” ”: Approximation: ”safe” to throw away info downwards With confluence: ( greatest lower bound) Greatest fixed point: Computed as: Stand on head lfp(f) = f i ( ) i≥0 f( )... f(f( )) gfp(f) = f i ( ) i≥0 f( ) f(f( ))...

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[ 38 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Agenda Quick recap (of everything so far): "Putting it all together" Data-Flow Analysis Fixed-Point Iteration Strategies (3x) "Sign Analysis" "Constant Propagation Analysis" "Initialized Variables Analysis" Set-Based Analysis Framework WORKSHOP 3 Example Data-Flow Analyses

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[ 39 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS WORKSHOP Reaching Definitions: ' ' ' ' (forward), ' ' (may / smallest set) Live Variables: ' ' ' ' (backward), ' ' (may / smallest set) Available Expressions: ' ' ' ' (forward), ' ' (must / largest set) Very Busy Expressions: ' ' ' ' (backward), ' ' (must / largest set)

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[ 40 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Reaching Definitions Reaching Definitions: Example: int x = input; int y; while (x>1) { y = x / 2; if (y>2) x = x - y; } output y; int x = input; y = x / 2; x = x – y; x>1 y>2 output y; int y; DEF-USE graph: The reaching definitions (for a given program point) are those assignments that may have defined the current vals of vars

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[ 41 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS WHY do we do this? ”Learning takes place through the active behavior of the student: it is what (s)he does that (s)he learns, not what the teacher does.” -- Ralph W. Tyler (1949)

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[ 42 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS WORKSHOP 1) Define the problem 2) Show that the problem is undecidable 3) Define a Lattice Check that it is a lattice (and explain how) 4) Define monotone transfer functions Check that they are monotone (and explain how) 5) Pick a program the analysis can analyze Make a "The Entire Process” diagram (cf. slide #5) 6) Repeat 5) for program the analysis can’t… 7) Explain possible uses of the analysis

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[ 43 ] Claus Brabrand, UFPE, Brazil Aug 11, 2010DATA-FLOW ANALYSIS Now, please: 3’ recap Please spend 3' on thinking about and writing down the main ideas and points from the lecture – now!: After 1 day After 1 week After 3 weeks After 2 weeks Immediately

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