New Algorithms for Static Analysis via Dyck Reachability

Transcription

1 New Algorithms for Static Analysis via Dyck Reachability Andreas Pavlogiannis 4th Inria/EPFL Workshop February 15, 2018

2 A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 2

3 A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 2

4 Static Analysis Program analysis is hard All non-trivial problems undecidable Relax... A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 3

5 Static Analysis Program analysis is hard All non-trivial problems undecidable Relax your ambitions... your model A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 3

6 Static Analysis Program analysis is hard All non-trivial problems undecidable Relax your ambitions... your model Static analysis Lightweight, clever scans of the program Detect obvious bugs 1 Fast 2 On demand 3 Dynamic A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 3

7 Static Analysis via Dyck Reachability A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 4

8 Dyck Reachability Σ = {( 1, ) 1,... ( k, ) k } {ɛ} G = (V, E, λ : E Σ) S S S ( 1 S ) 1... ( k S ) k ɛ y ( 2 ) 2 ) 2 ) 1 z w ( 1 x A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 5

9 Dyck Reachability Σ = {( 1, ) 1,... ( k, ) k } {ɛ} G = (V, E, λ : E Σ) S S S ( 1 S ) 1... ( k S ) k ɛ y ) 2 ) 1 z P : x z with λ(p) = ( 1 ( 2 ) 2 ) 1 ( 2 ) 2 w ( 1 x A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 5

10 Dyck Reachability Σ = {( 1, ) 1,... ( k, ) k } {ɛ} G = (V, E, λ : E Σ) S S S ( 1 S ) 1... ( k S ) k ɛ y ) 2 ) 1 z P : x z with λ(p) = ( 1 ( 2 ) 2 ) 1 ( 2 ) 2 w ( 1 x Alias analysis Data-dependence analysis Data-flow analysis Shape analysis Impact analysis Bloat analysis Type-based flow analysis Program slicing A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 5

11 Algorithms Q: Is v Dyck-reachable from u? Dyck languages are a class of CFL Solution similar to CYK for CFL parsing O(n 3 ) [Yannakakis 90] O(n 3 / log n) [Chaudhuri 08] Often prohibitive for lightweight static analysis A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 6

12 Algorithms Q: Is v Dyck-reachable from u? Dyck languages are a class of CFL Solution similar to CYK for CFL parsing O(n 3 ) [Yannakakis 90] O(n 3 / log n) [Chaudhuri 08] Often prohibitive for lightweight static analysis 1 Low treewidth 2 Bidirected Can do better if the graph is simple! A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 6

13 Static Analysis Analyze source code without executing it Typical paradigm in static analysis: reduce the problem to a graph problem P: Input: A program of methods M i 1 Extract control flow graphs G i 2 Annotate G i 3 Run best graph algorithm for P on G A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 7

14 Static Analysis Analyze source code without executing it Typical paradigm in static analysis: reduce the problem to a graph problem P: Input: A program of methods M i 1 Extract control flow graphs G i 2 Annotate G i 3 Run best graph algorithm for P on G Data-dependence analysis Which variable depends on which others Identify Def-Use chains in a program LHS depends on RHS 1 x 2 2 y x A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 7

15 Static Analysis Analyze source code without executing it Typical paradigm in static analysis: reduce the problem to a graph problem P: Input: A program of methods M i 1 Extract control flow graphs G i 2 Annotate G i 3 Run best graph algorithm for P on G Data-dependence analysis Which variable depends on which others Identify Def-Use chains in a program LHS depends on RHS y depends on x if x y 1 x 2 2 y x A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 7

16 Intra-procedural Finite Graphs Method 1: dot vector 1 result 0 2 for i 1 to n do 3 z x[i] y[i] 4 result result + z 5 end 6 return result Method 2: dot matrix 1 C zero matrix of size n m 2 for i 1 to n do 3 for j 1 to m do 4 Call dot vector(a[i, :], B[:, j]) 5 C[i, j] the value returned by the call of line 4 6 end 7 end 8 return C A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 8

17 Inter-procedural Recursive Graphs Method 1: f 1 () a g() 3... Method 3: g() 1... Method 2: f 2 () b g() 3... f 1 g f 2 x y u v Filter out interprocedurally invalid paths A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 9

18 Inter-procedural Recursive Graphs Method 1: f 1 () a g() 3... Method 3: g() 1... Method 2: f 2 () b g() 3... f 1 g f 2 ( 1 ( 2 x u y ) 1 ) 2 v Filter out interprocedurally invalid paths A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 9

19 Inter-procedural Recursive Graphs Method 4: dot vector 1 result 0 2 for i 1 to n do 3 z x[i] y[i] 4 result result + z 5 end 6 return result ( 1 Method 5: dot matrix 1 C zero matrix of size n m 2 for i 1 to n do 3 for j 1 to m do 4 Call dot vector(a[i, :], B[:, j]) 5 C[i, j] the value returned by the call of line 4 6 end 7 end 8 return C ) 1 A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 10

20 Control-flow graphs are simple Similar to trees Precise notion: treewidth A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 11

21 Treewidth A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 12

22 Cops and Robber A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 13

23 Cops and Robber You enter the metro and forget to buy a ticket t inspectors are looking for you in the metro stations Everyone is aware of everyone else s position A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 13

24 Cops and Robber You enter the metro and forget to buy a ticket t inspectors are looking for you in the metro stations Everyone is aware of everyone else s position In every round Some inspectors move between stations to reach you You use the metro system to move between stations You cannot cross an occupied station A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 13

25 Cops and Robber You enter the metro and forget to buy a ticket t inspectors are looking for you in the metro stations Everyone is aware of everyone else s position In every round Some inspectors move between stations to reach you You use the metro system to move between stations You cannot cross an occupied station The treewidth of the metro graph is the largest t from which you can always escape A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 13

26 Cops and Robber A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 13

27 Tree Decompositions Definition (Tree decomposition) Given a graph G = (V, E), a tree-decomposition Tree(G) = (V T, E T ) is a tree of bags B i V G Tree(G) 2, 8, , 9, 10 2, 3, , 8, 9 1, 8, , 7, 9 4 4, 6, 9 5, 6, 7 A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 14

28 Treewidth Definition (Treewidth) The width of Tree(G) is max i B i 1. The treewidth of G is the minimum width of a tree decomposition of G. G Tree(G) 2, 8, , 9, 10 2, 3, , 8, 9 1, 8, , 7, 9 4 4, 6, 9 5, 6, 7 A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 15

29 Treewidth Definition (Treewidth) The width of Tree(G) is max i B i 1. The treewidth of G is the minimum width of a tree decomposition of G. CFGs of typical imperative programs have small treewidth For goto-free programs [Thorup 98] Pascal 3 C 6 In practice small in imperative programs (e.g. Java 7) A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 15

30 Treewidth Definition (Treewidth) The width of Tree(G) is max i B i 1. The treewidth of G is the minimum width of a tree decomposition of G. CFGs of typical imperative programs have small treewidth For goto-free programs [Thorup 98] Pascal 3 C 6 In practice small in imperative programs (e.g. Java 7) Theorem (Tree decomposition) For constant treewidth graphs,tree(g) can be constructed in O(n) time A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 15

31 Algorithmic Principle G h 7 8 ( 2 ) 2 G g ( 1 ) 1 G f Reachability ignoring parenthesis edges 2 If Entry Exit Insert summary edge A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 16

32 Algorithmic Principle G h 7 8 ( 2 ) 2 G g ( 1 ) 1 G f Reachability ignoring parenthesis edges 2 If Entry Exit Insert summary edge A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 16

33 Algorithmic Principle G h 7 8 ( 2 ) 2 G g ( 1 ) 1 G f Reachability ignoring parenthesis edges 2 If Entry Exit Insert summary edge A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 16

34 Algorithmic Principle G h 7 8 ( 2 ) 2 G g ( 1 ) 1 G f Reachability ignoring parenthesis edges 2 If Entry Exit Insert summary edge A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 16

35 Algorithmic Principle G h 7 8 ( 2 ) 2 G g ( 1 ) 1 G f Reachability ignoring parenthesis edges 2 If Entry Exit Insert summary edge A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 16

36 Algorithmic Principle G h 7 8 ( 2 ) 2 G g ( 1 ) 1 G f Reachability ignoring parenthesis edges 2 If Entry Exit Insert summary edge A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 16

37 Algorithmic Principle G h 7 8 ( 2 ) 2 G g ( 1 ) 1 G f Reachability ignoring parenthesis edges 2 If Entry Exit Insert summary edge Most time spent in (1) Incremental reachability Low treewidth A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 16

38 Reachability on low-treewidth graphs A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 17

39 Reachability [ESA 16] Given a graph G = (V, E) of n nodes and bounded treewidth Preprocessing time Space Pair query time Single-source query time O(n) O(n) O(n) ( O n 2 log n ) ( O n 2 log n ) O(1) ( O n log n O(n log n) O(n log n) O(log n) O(n) ( ) O(n) O(n) O (1) O n log n ) Optimal Faster than DFS/BFS after a constant number of queries A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 18

40 Reachability [ESA 16] Given a graph G = (V, E) of n nodes and bounded treewidth Preprocessing time Space Pair query time Single-source query time O(n) O(n) O(n) ( O n 2 log n ) ( O n 2 log n ) O(1) ( O n log n O(n log n) O(n log n) O(log n) O(n) ( ) O(n) O(n) O (1) O n log n ) Optimal Faster than DFS/BFS after a constant number of queries A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 18

41 Reachability [ESA 16] Given a graph G = (V, E) of n nodes and bounded treewidth Preprocessing time Space Pair query time Single-source query time O(n) O(n) O(n) ( O n 2 log n ) ( O n 2 log n ) O(1) ( O n log n O(n log n) O(n log n) O(log n) O(n) ( ) O(n) O(n) O (1) O n log n ) Optimal Faster than DFS/BFS after a constant number of queries A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 18

42 Reachability [ESA 16] Given a graph G = (V, E) of n nodes and bounded treewidth Preprocessing time Space Pair query time Single-source query time O(n) O(n) O(n) ( O n 2 log n ) ( O n 2 log n ) O(1) ( O n log n O(n log n) O(n log n) O(log n) O(n) ( ) O(n) O(n) O (1) O n log n ) Optimal Faster than DFS/BFS after a constant number of queries A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 18

43 Reachability [ESA 16] Given a graph G = (V, E) of n nodes and bounded treewidth Preprocessing time Space Pair query time Single-source query time O(n) O(n) O(n) ( O n 2 log n ) ( O n 2 log n ) O(1) ( O n log n O(n log n) O(n log n) O(log n) O(n) ( ) O(n) O(n) O (1) O n log n ) Optimal Faster than DFS/BFS after a constant number of queries A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 18

44 Interprocedural Dataflow Analysis on Recursive Graphs of Low Treewidth A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 19

45 Interprocedural Dataflow Analysis Input: Recursive graph G with n nodes Finite graphs G i of n i nodes and bounded treewidth Every G i has a unique entry and exit A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 20

46 Interprocedural Dataflow Analysis Input: Recursive graph G with n nodes Finite graphs G i of n i nodes and bounded treewidth Every G i has a unique entry and exit D data fact set e.g. variable x is uninitialized, variable y is constant Weighted graph Semiring distance on lattice over D A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 20

47 Interprocedural Dataflow Analysis Input: Recursive graph G with n nodes Finite graphs G i of n i nodes and bounded treewidth Every G i has a unique entry and exit D data fact set e.g. variable x is uninitialized, variable y is constant Weighted graph Semiring distance on lattice over D Theorem (POPL 15) For low-treewidth graphs we construct dynamic reachability oracles: Preprocess G i in O(n i D 3 ) time Query for the distance d(u, v) in O( D 3 log n i ) time Update the weight of an edge (u, v) in O( D 3 log n i ) time A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 20

48 Interprocedural Dataflow Analysis 0 G 1 G 2... G k A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

49 Interprocedural Dataflow Analysis 0 G 1 G 2... G k Preprocess Preprocess Preprocess A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

50 Interprocedural Dataflow Analysis 0 G 1 G 2... G k 1 G 1 G 2... G k A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

51 Interprocedural Dataflow Analysis 0 G 1 G 2... G k Query d(en 1, Ex 1 ) Query d(en i, Ex i ) 1 G 1 G 2... G k A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

52 Interprocedural Dataflow Analysis 0 G 1 G 2... G k Query d(en 1, Ex 1 ) 1 G 1 G 2... G k Update wt 2 (C, R) A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

53 Interprocedural Dataflow Analysis 0 G 1 G 2... G k 1 G 1 G 2... G k h G 1 G 2... G k A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

54 Interprocedural Dataflow Analysis 0 G 1 G 2... G k 1 G 1 G 2... G k Query d(u, v) in O(log n 2 ) h G 1 G 2... G k A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

55 Interprocedural Dataflow Analysis 0 G 1 G 2... G k 1 G 1 G 2... G k Query d(u, v) in O(1) h G 1 G 2... G k A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 21

56 Dataflow Analysis [POPL 15] Input graph with n nodes, D data fact set, D = Ω(log n) Preprocessing time Space Query Single-source Pair No - O(n D 2 ) O(n D 3 ) O(n D 3 ) preprocessing Complete O(n 2 D 3 ) O(n 2 D 2 ) O(n D 2 ) O( D 2 ) preprocessing Ours O(n D 3 ) O(n D 2 ) O(n D 2 ) O( D 2 ) A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 22

57 Dataflow Analysis [POPL 15] Input graph with n nodes, D data fact set, D = Ω(log n) Preprocessing time Space Query Single-source Pair No - O(n D 2 ) O(n D 3 ) O(n D 3 ) preprocessing Complete O(n 2 D 3 ) O(n 2 D 2 ) O(n D 2 ) O( D 2 ) preprocessing Ours O(n D 3 ) O(n D 2 ) O(n D 2 ) O( D 2 ) A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 22

58 Dyck Reachability on Bidirected Graphs A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 23

59 Bidirected graphs ( i x y u, v V : λ(u, v) = ( i λ(v, u) =) i ) i A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 24

60 Bidirected graphs ( i x y u, v V : λ(u, v) = ( i λ(v, u) =) i ā a 1 a 1 ā 1 ā 2 a 2 ) i 5 1 a 1 ā ā 1 a 2 (b) The bidirected graph case. illustrating a directed graph and its ph. Demand-driven field-sensitive alias analysis [YXR, ISSTA 11] [ZLYS, PLDI 13] lity is defined similarly by restricting k language, which generates strings of es. Consider an alphabet over the set {a 1,a 2,...,a k } and the set of their A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 24

61 Reachability is Equivalence Theorem (ZLYS 13) Dyck reachability on bidirected graphs is an equivalence relation. A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 25

62 Reachability is Equivalence Theorem (ZLYS 13) Dyck reachability on bidirected graphs is an equivalence relation. x ( ) y A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 25

63 Reachability is Equivalence Theorem (ZLYS 13) Dyck reachability on bidirected graphs is an equivalence relation. x ( ) ) ( y A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 25

64 Dyck Reachability on Bidirected Graphs [POPL 18] Graphs of n nodes, α(n) the inverse Ackermann Source Worst-case Time Average-case Time [ZLYS 13] O ( n 2) O (n log n) Our Result O(n α(n)) O(n) A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 26

65 Dyck Reachability on Bidirected Graphs [POPL 18] Graphs of n nodes, α(n) the inverse Ackermann Source Worst-case Time Average-case Time [ZLYS 13] O ( n 2) O (n log n) Our Result O(n α(n)) O(n) optimal! optimal! A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 26

66 Hardness? A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 27

67 Complexity of Dyck Reachability [POPL 18] O(n 3 ) 2NPDA-hard Conditional cubic lower bound A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 28

68 Complexity of Dyck Reachability [POPL 18] O(n 3 ) 2NPDA-hard Conditional cubic lower bound Theorem Dyck reachability is Boolean Matrix Multiplication - hard. A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 28

69 Complexity of Dyck Reachability [POPL 18] O(n 3 ) 2NPDA-hard Conditional cubic lower bound Theorem Dyck reachability is Boolean Matrix Multiplication - hard. A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 28

70 Complexity of Dyck Reachability [POPL 18] O(n 3 ) 2NPDA-hard Conditional cubic lower bound Theorem Dyck reachability on low-treewidth graphs is Boolean Matrix Multiplication - hard. Can we do O(n 3 / log 2 n)? A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 28

71 Complexity on Bidirected Graphs [POPL 18] Theorem Dyck reachability on constant-treewidth, bidirected graphs requires Ω(n α(n)) time. A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 29

72 Complexity on Bidirected Graphs [POPL 18] Theorem Dyck reachability on constant-treewidth, bidirected graphs requires Ω(n α(n)) time. Compare with Theorem (ZLYZ 13) Dyck reachability on bidirected trees solvable in O(n) time. Typically complexity on low-treewidth graphs = complexity on trees A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 29

73 Implementation & Experiments A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 30

74 Implementation & Experiments Alias Analysis Implementation in C++ Compared with [ZLYS, PLDI 13] DaCapo-2006 benchmarks SPGs from [YXR, ISSTA 11] A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 30

75 Implementation & Experiments Alias Analysis Implementation in C++ Compared with [ZLYS, PLDI 13] DaCapo-2006 benchmarks SPGs from [YXR, ISSTA 11] Data-dependence Analysis Implementation in Java Compared with TAL reachability [TWZXZM, POPL 15] Benchmarks: SPECjvm randomly chosen GitHub projects A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 30

76 Alias Analysis Time-usage comparison (ms) Time % antlr bloat chart eclipse fop hsqldb jython luindex lusearch pmd xalan Our Algorithm Existing A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 31

77 Data-dependence Analysis Time % Time-usage comparison (ms) Library (preprocessing) helloworld check compiler sample crypto derby mpegaudio xml mushroom btree startup sunflow compress parser scimark Our Algorithm TAL CFL A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 32

78 Data-dependence Analysis Time % Time % Time-usage comparison (ms) Library (preprocessing) helloworld check compiler sample crypto derby mpegaudio xml mushroom btree startup sunflow compress parser scimark Our Algorithm TAL CFL Client helloworld check compiler sample crypto derby mpegaudio xml mushroom btree startup sunflow compress parser scimark Our Algorithm TAL CFL A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 32

79 Data-dependence Analysis Time % Time-usage comparison (ms) Library (preprocessing) Time % 0 helloworld check compiler sample crypto derby mpegaudio xml mushroom btree startup sunflow compress parser scimark Our Algorithm TAL CFL Client helloworld check compiler sample crypto derby mpegaudio xml mushroom btree startup sunflow compress parser scimark Our Algorithm TAL A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 32

80 Data-dependence Analysis Space % Space-usage comparison (MB) Library (preprocessing) helloworld check compiler sample crypto derby mpegaudio xml mushroom btree startup sunflow compress parser scimark Our Algorithm TAL CFL A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 33

81 Data-dependence Analysis Space % Space-usage comparison (MB) Library (preprocessing) helloworld check compiler sample crypto derby mpegaudio xml mushroom btree startup sunflow compress parser scimark Our Algorithm TAL CFL A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 33

82 Thank you! Questions? A. Pavlogiannis New Algorithms for Static Analysis via Dyck Reachability 34