Lattice-Theoretic Framework for Data-Flow Analysis Last time - PDF document

Lattice-Theoretic Framework for Data-Flow Analysis � Last time – � Generalizing data-flow analysis � Today – � Introduce lattice-theoretic frameworks for data-flow analysis CS553 Lecture Lattice Theoretic Framework for DFA 1 Context for Lattice-Theoretic Framework � Goals – � Provide a single formal model that describes all data-flow analyses – � Formalize the notions of “correct,” “conservative,” and “optimistic” – � Correctness proof for IDFA (iterative data-flow analysis) – � Place bounds on time complexity of data-flow analysis � Approach – � Define domain of program properties (flow values) computed by data- flow analysis, and organize the domain of elements as a lattice – � Define flow functions and a merge function over this domain using lattice operations – � Exploit lattice theory in achieving goals CS553 Lecture Lattice Theoretic Framework for DFA 2

Lattices {} � Define lattice L = (V, ! ) – � V is a set of elements of the lattice {i} {j} {k} – � ! is a binary relation over the elements of V ( meet or greatest lower bound ) {i,j} {i,k} {j,k} {i,j,k} � Properties of ! – � x,y � V � x ! y � V (closure) – � x � V � x ! x = x (idempotence) – � x,y � V � x ! y = y ! x (commutativity) – � x,y,z � V � (x ! y) ! z = x ! (y ! z) (associativity) CS553 Lecture Lattice Theoretic Framework for DFA 3 Lattices (cont) ! = {} � Under ( " ) – � Imposes a partial order on V {i} {j} {k} – � x " y � x ! y = x {i,j} {i,k} {j,k} � Top ( ! ) – � A unique “greatest” element of V (if it exists) � = {i,j,k} – � � x � V – { ! }, x " ! � Bottom ( � ) – � A unique “least” element of V (if it exists) – � � x � V – { � }, � " x � Height of lattice L – � The longest path through the partial order from greatest to least element (top to bottom) CS553 Lecture Lattice Theoretic Framework for DFA 4

Data-Flow Analysis via Lattices � Relationship – � Elements of the lattice (V) represent flow values (in[] and out[] sets) – � e.g ., Sets of live variables for liveness – � ! represents “best-case” information (initial flow value) – � e.g ., Empty set {} – � � represents “worst-case” information – � e.g ., Universal set {i} {j} {k} – � ! (meet) merges flow values {i,j} {i,k} {j,k} – � e.g ., Set union – � If x " y, then x is a conservative approximation of y {i,j,k} – � e.g ., Superset CS553 Lecture Lattice Theoretic Framework for DFA 5 Data-Flow Analysis via Lattices (cont) � Remember what these flow values represent – � At each program point a lattice element represents {} an in[] set or an out[] set {x} {y} { } Initially x = y { } {x,y} { } { } print(x) print(y) { } { } { y } Finally x = y { x,y } { x } { y } print(x) print(y) { } { } CS553 Lecture Lattice Theoretic Framework for DFA 6

Data-Flow Analysis Frameworks � Data-flow analysis framework – � A set of flow values (V) – � A binary meet operator ( ! ) – � A set of flow functions (F) (also known as transfer functions ) � Flow Functions – � F = {f: V � V} f describes how each node in CFG affects the flow values – � Flow functions map program behavior onto lattices CS553 Lecture Lattice Theoretic Framework for DFA 7 Visualizing DFA Frameworks as Lattices Example : Liveness analysis with 3 variables U = {v1, v2, v3} � = ! – � V: 2 S = {{v1,v2,v3}, � {v1,v2},{v1,v3},{v2,v3}, { v1 } { v2 } {v1},{v2},{v3}, � } { v3 } – � Meet ( ! ): � � – � " : � ! { v2,v3 } { v1,v2 } { v1,v3 } � – � Top( T ): � � – � Bottom ( � ): U � { v1,v2,v3 } = � { f n (X) = Gen n � (X – Kill n ), � n } – � F: � Inferior solutions are lower on the lattice More conservative solutions are lower on the lattice CS553 Lecture Lattice Theoretic Framework for DFA 8

Lattice Example � What are the data-flow sets for liveness? � What is the meet operation for liveness? � What partial order does the meet operation induce? � What is the liveness lattice for this example? CS553 Lecture Lattice Theoretic Framework for DFA 9 Recall Liveness Analysis � Data-flow equations for liveness � in[n] = use [n] � (out[n] – def [n]) � out[n] = � in[s] s � succ[n] Liveness equations in terms of Gen and Kill � in[n] = gen [n] � (out[n] – kill [n]) A use of a variable generates liveness A def of a variable kills liveness � out[n] = � in[s] s � succ[n] � Gen: New information that’s added at a node � Kill: Old information that’s removed at a node � Can define many data-flow analysis in terms of Gen and Kill CS553 Lecture Lattice Theoretic Framework for DFA 10

Reaching Constants (aka Constant Propagation) � Goal – � Compute value of each variable at each program point (if possible) � Flow values – � Set of (variable,constant) pairs � Merge function – � Intersection � Data-flow equations – � Effect of node n x = c – � kill[n] = {(x,d)| � d} – � gen[n] = {(x,c)} – � Effect of node n x = y + z – � kill[n] = {(x,c)| � c} – � gen[n] = {(x,c) | c=val(y)+valz, (y, valy) � in[n], (z, valz) � in[n]} CS553 Lecture Generalizing Data-flow Analysis 11 Direction of Flow � Backward data-flow analysis – � Information at a node is based on what happens later in the flow graph i.e., in[] is defined in terms of out[] n in[n] = gen[n] � (out[n] – kill[n]) in � liveness out[n] = � in[s] out � s � succ[n] � Forward data-flow analysis – � Information at a node is based on what happens earlier in the flow graph i.e., out[] is defined in terms of in[] n in[n] = � out[p] in reaching � p � pred[n] out[n] = gen[n] � (in[n] – kill[n]) definitions out � � Some problems need both forward and backward analysis – � e.g., Partial redundancy elimination (uncommon) CS553 Lecture Lattice Theoretic Framework for DFA 12

More Examples � Reaching Constants � Reaching definitions – � V: 2 v � c , variables v and – � V: � 2 S (S = set of all defs) � constants c – � ! : � � – � ! : � � � – � " : � ! – � " : � � � – � Top( ! ): � � U – � Top( ! ): � – � Bottom ( � ): U � � – � Bottom ( � ): � – � F: . . . � – � F: . . . � CS553 Lecture Lattice Theoretic Framework for DFA 13 Merging Flow Values � Live variables and reaching definitions – � Merge flow values via set union Reaching Definitions Live Variables in[n] = � out[s] out[n] = � in[s] � � p � pred[n] s � succ[n] out[n] = gen[n] � (in[n] – kill[n]) in[n] = gen[n] � (out[n] – kill[n]) � � � Why? � When might this be inappropriate? CS553 Lecture Lattice Theoretic Framework for DFA 14

Reaching Defs Example � What is the lattice? � What is the initial guess? � What is the meet operation? CS553 Lecture Lattice Theoretic Framework for DFA 15 Available Expressions Iterative Algorithm Data-Flow Equations in[n] = � out[p] p � pred[n] out[n] = gen[n] � (in[n] – kill[n]) Plug it in to our general DFA algorithm for each node n in[n] = U; out[n] = U repeat for each n in � [n] = in[n] out � [n] = out[n] in[n] = � out[p] p � pred[n] out[n] = gen[n] � (in[n] – kill[n]) until in � [n]=in[n] and out � [n]=out[n] for all n CS553 Lecture Lattice Theoretic Framework for DFA 16

Available Expressions Example � What is the initial guess? � What is the meet operation? � What does the lattice look like? CS553 Lecture Lattice Theoretic Framework for DFA 17 Solving Data-Flow Analyses � Goal – � For a forward problem, consider all paths from the v entry entry to a given program point, compute the flow values at the end of each path, and then meet these entry values together – � Meet-over-all-paths (MOP) solution at each program point – � ! all paths n1, n2, ..., ni (f ni (...f n2 (f n1 (v entry )))) CS553 Lecture Lattice Theoretic Framework for DFA 18

Solving Data-Flow Analyses (cont) � Problems – � Loops result in an infinite number of paths – � Statements following merge must be analyzed for all preceding paths – � Exponential blow-up � Solution – � Compute meets early (at merge points) rather than at the end – � Maximum fixed-point (MFP) � Questions – � Is this correct? – � Is this efficient? – � Is this accurate? CS553 Lecture Lattice Theoretic Framework for DFA 19 Correctness � “Is v MFP correct?” � “Is v MFP " v MOP ?” p2 p1 v p1 v p2 � Look at Merges F r � v MOP = F r (v p1 ) ! F r (v p2 ) v MFP v MOP � v MFP = F r (v p1 ! v p2 ) � v MFP " v MOP � F r (v p1 ! v p2 ) " F r (v p1 ) ! F r (v p2 ) � Observation � x,y � V f(x ! y) " f(x) ! f(y) � x " y � f(x) " f(y) � � v MFP correct when F r (really, the flow functions) are monotonic CS553 Lecture Lattice Theoretic Framework for DFA 20