An example DFA: reaching definitions

Transcription

1 Dataflow analysis: what is it? Dataflow analysis A common framework for expressg algorithms that compute formation ab a program Why is such a framework useful? Dataflow analysis: what is it? A common framework for expressg algorithms that compute formation ab a program Why is such a framework useful? Provides a common language, which makes it easier to: communicate your analysis to others compare analyses adapt techniques from one analysis to another reuse implementations (eg: dataflow analysis frameworks) Control Flow Graphs For now, we will use a Control Flow Graph representation of programs each statement becomes a node edges between nodes represent control flow Later we will see other program representations variations on the CFG (eg CFG with basic blocks) other graph based representations Example CFG if (...) { } else { * } if (...) * An example DFA: reachg defitions For each use of a variable, determe what assignments could have set the value beg read from the variable Information useful for: performg constant and copy prop detectg references to undefed variables presentg def/use chas to the programmer buildg other representations, like the DFG Let s try this on an example 1

2 Visual sugar 1: 2: 3: 1: 2: 3: if (...) 5: 6: 5: 6: * 8: 8: 5: 1: 2: 3: 8: 6: Safety When is computed fo safe? Recall tended use of this fo: performg constant and copy prop detectg references to undefed variables presentg def/use chas to the programmer buildg other representations, like the DFG Safety: can have more bdgs than the true answer, but can t miss any Reachg defitions generalized DFA framework geared to computg formation at each program pot (edge) the CFG So generalize problem by statg what should be computed at each program pot For each program pot the CFG, compute the set of defitions (statements) that may reach that pot Notion of safety remas the same Reachg defitions generalized Computed formation at a program pot is a set of var! stmt bdgs eg: { x! s 1, x! s 2, y! s 3 } How do we get the previous fo we wanted? if a var x is used a stmt whose comg fo is, then: 2

3 Reachg defitions generalized Computed formation at a program pot is a set of var! stmt bdgs eg: { x! s 1, x! s 2, y! s 3 } How do we get the previous fo we wanted? if a var x is used a stmt whose comg fo is, then: { s (x! s) 2 } This is a common pattern generalize the problem to defe what formation should be computed at each program pot use the computed formation at the program pots to get the origal fo we wanted 5: 1: 2: 3: 8: 6: 5: 1: 2: 3: 8: 6: Usg constrats to formalize DFA Now that we ve gone through some examples, let s try to precisely express the algorithms for computg dataflow formation We ll model DFA as solvg a system of constrats Each node the CFG will impose constrats relatg formation at predecessor and successor pots Solution to constrats is result of analysis Constrats for reachg defitions Constrats for reachg defitions s: s: = { x! s s 2 stmts } [ { x! s } s: * s: * Usg may-pot-to formation: = [ { x! s x 2 may-pot-to(p) } Usg must-pot-to aswell: = { x! s x 2 must-pot-to(p) Æ s 2 stmts } [ { x! s x 2 may-pot-to(p) } 3

4 Constrats for reachg defitions Constrats for reachg defitions s: if (...) [0] [1] s: if (...) [ 0 ] = Æ [ 1 ] = [0] [1] more generally: 8 i. [ i ] = [0] [1] merge [0] [1] merge = [ 0 ] [ [ 1 ] more generally: = i [ i ] Flow functions The constrat for a statement kd s often have the form: = F s () F s is called a flow function other names for it: dataflow function, transfer function Given formation before statement s, F s () returns formation after statement s Other formulations have the statement s as an explicit parameter to F: given a statement s and some formation, F(s,) returns the gog formation after statement s Flow functions, some issues put edges to a node? gog edges to a node? Flow functions, some issues put edges to a node? the flow functions takes as put a tuple of values, one value for each comg edge gog edges to a node? the flow function returns a tuple of values, one value for each gog edge can also have one flow function per gog edge Flow functions Flow functions are a central component of a dataflow analysis They state constrats on the formation flowg to and of a statement This version of the flow functions is local it applies to a particular statement kd we ll see global flow functions shortly... 4

5 Summary of flow functions Flow functions: Given formation before statement s, F s () returns formation after statement s Flow functions are a central component of a dataflow analysis They state constrats on the formation flowg to and of a statement 5