Undecidability of context-sensitive data-independence analysis

ABSTRACT

A number of program-analysis problems can be tackled by transforming them into certain kinds of graph-reachability problems in labeled directed graphs. The edge labels can be used to filter out paths that are not interest: a path P from vertex s to vertex t only counts as a“valid connection” between s and t if the word spelled out by P is in a certain language. Often the languages used for such filtering purposes are languages of matching parantheses. In some cases, the matched-parenthesis condition is used to filter out paths with mismatched calls and returns. This leads to so-called “context-sensitive” program analyses, such as context-sensitive interprocedural slicing and context-sensitive interprocedural dataflow analysis. In other cases, the matched-parenthesis condition is used to capture a graph-theoretic analog of McCarthy's rules: “car (cons(x,y)) = x” and “cdr(cons(x,y)) =y”. That is, in the code fragment c = cons(a,b); d = car(c); the fact that there is a “structure-transmitted data-dependence” from a to d, but not from b to d, is captured in a graph by (1) using a vertex for each variable, (2)an edge from vertex i to vertex j when i is used on the right-hand side of an assignment to j, (3) parentheses that match as the labels on the edges that run from a to c and c to d, and (4) parentheses that do not match as the labels on the edges that run from a to c and c to d. However, structure-transmitted data-dependence analysis is context-insensitive, because there are no constraints that filter out paths with mismatched calls and returns. Thus, a natural question is whether these two kinds of uses of parentheses can be combined to create a context-sensitive analysis for structure-transmitted data-dependences. This article answers the question in the negative: in general, the problem of contextsensitive, structure-transmitted data-dependence analysis is undecidable. The results imply that in general, both context-sensitive set-based analysis and ∞ -CFA (when data constructors and selectors and selectors are taken into account) are also undecidable.

REFERENCES

• 1: Alexander Aiken , Brian R. Murphy, Implementing regular tree expressions, Proceedings of the 5th ACM conference on Functional programming languages and computer architecture, p.427-447, June 1991, Cambridge, Massachusetts, United States
• 2: Alex Aiken , Brian Murphy, Static type inference in a dynamically typed language, Proceedings of the 18th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.279-290, January 21-23, 1991, Orlando, Florida, United States [doi>10.1145/99583.99621]
• 3: Alexander Aiken , Edward L. Wimmers, Type inclusion constraints and type inference, Proceedings of the conference on Functional programming languages and computer architecture, p.31-41, June 09-11, 1993, Copenhagen, Denmark [doi>10.1145/165180.165188]
• 4: D. Callahan, The program summary graph and flow-sensitive interprocedual data flow analysis, Proceedings of the ACM SIGPLAN 1988 conference on Programming Language design and Implementation, p.47-56, June 20-24, 1988, Atlanta, Georgia, United States
• 5: K. D. Cooper , K. Kennedy, Interprocedural side-effect analysis in linear time, Proceedings of the ACM SIGPLAN 1988 conference on Programming Language design and Implementation, p.57-66, June 20-24, 1988, Atlanta, Georgia, United States
• 6: K. D. Cooper , K. Kennedy, Fast interprocedual alias analysis, Proceedings of the 16th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.49-59, January 11-13, 1989, Austin, Texas, United States [doi>10.1145/75277.75282]
• 7: Maryam Emami , Rakesh Ghiya , Laurie J. Hendren, Context-sensitive interprocedural points-to analysis in the presence of function pointers, Proceedings of the ACM SIGPLAN 1994 conference on Programming language design and implementation, p.242-256, June 20-24, 1994, Orlando, Florida, United States
• 8: M.A. A. Harrison , M. A. Harrison, Introduction to Formal Language Theory, Addison-Wesley Longman Publishing Co., Inc., Boston, MA, 1978
• 9: Matthew S. Hecht, Flow Analysis of Computer Programs, Elsevier Science Inc., New York, NY, 1977
• 10: Nevin Heintze, Set Based Analysis of ML Programs (Extended Abstract), Carnegie Mellon University, Pittsburgh, PA, 1993
• 11: HEINTZE, N. AND JAFFAR, J. 1991. A decision procedure for a class of set constraints. Tech. Rep. CMU-CS-91-110. School of Computer Science, Carnegie-Mellon Univ., Pittsburgh, PA.
• 12: Nevin Heintze , David McAllester, Linear-time subtransitive control flow analysis, Proceedings of the ACM SIGPLAN 1997 conference on Programming language design and implementation, p.261-272, June 16-18, 1997, Las Vegas, Nevada, United States
• 13: John E. Hopcroft , Jeffrey D. Ullman, Introduction To Automata Theory, Languages, And Computation, Addison-Wesley Longman Publishing Co., Inc., Boston, MA, 1990
• 14: Susan Horwitz , Thomas Reps , David Binkley, Interprocedural slicing using dependence graphs, ACM Transactions on Programming Languages and Systems (TOPLAS), v.12 n.1, p.26-60, Jan. 1990 [doi>10.1145/77606.77608]
• 15: Susan Horwitz , Thomas Reps , Mooly Sagiv, Demand interprocedural dataflow analysis, Proceedings of the 3rd ACM SIGSOFT symposium on Foundations of software engineering, p.104-115, October 12-15, 1995, Washington, D.C., United States
• 16: Suresh Jagannathan , Stephen Weeks, A unified treatment of flow analysis in higher-order languages, Proceedings of the 22nd ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.393-407, January 23-25, 1995, San Francisco, California, United States [doi>10.1145/199448.199536]
• 17: JONES, N. D. AND MUCHNICK, S. S. 1981. Flow analysis and optimization of Lisp-like structures. In Prograamm Flow Analysis: Theory and Applications, S. S. Muchnick and N. D. Jones, Eds. Prentice-Hall, 102-131.
• 18: Uday P. Khedker , Dhananjay M. Dhamdhere, A generalized theory of bit vector data flow analysis, ACM Transactions on Programming Languages and Systems (TOPLAS), v.16 n.5, p.1472-1511, Sept. 1994 [doi>10.1145/186025.186043]
• 19: Lawrence T. Kou, On Live-Dead Analysis for Global Data Flow Problems, Journal of the ACM (JACM), v.24 n.3, p.473-483, July 1977 [doi>10.1145/322017.322027]
• 20: David L. Kuck, Structure of Computers and Computations, John Wiley & Sons, Inc., New York, NY, 1978
• 21: D. J. Kuck , R. H. Kuhn , D. A. Padua , B. Leasure , M. Wolfe, Dependence graphs and compiler optimizations, Proceedings of the 8th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.207-218, January 26-28, 1981, Williamsburg, Virginia [doi>10.1145/567532.567555]
• 22: Harry R. Lewis , Christos H. Papadimitriou , Christos Papadimitriou, Elements of the Theory of Computation, Prentice Hall PTR, Upper Saddle River, NJ, 1997
• 23: MCCARTHY, J. 1963. A basis for a mathematical theory of computation. In Computer Programming and Formal Systems. North-Holland, 33-70.
• 24: David Melski , Thomas Reps, Interconvertibility of a class of set constraints and context-free-language reachability, Theoretical Computer Science, v.248 n.1-2, p.29-98, October 2000 [doi>10.1016/S0304-3975(00)00049-9 ]
• 25: Flemming Nielson , Hanne Riis Nielson, Infinitary control flow analysis: a collecting semantics for closure analysis, Proceedings of the 24th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.332-345, January 15-17, 1997, Paris, France [doi>10.1145/263699.263745]
• 26: RAMALINGAM, G. 1999. Context-sensitive synchronization-sensitive analysis is undecidable. Res. Rep. TC 21493. IBM T. J. Watson Research Center, Yorktown Heights, NY.
• 27: Thomas Reps, Shape analysis as a generalized path problem, Proceedings of the 1995 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulation, p.1-11, June 21-23, 1995, La Jolla, California, United States [doi>10.1145/215465.215466]
• 28: REPS, T. 1996. On the sequential nature of interprocedural program-analysis problems. Acta Inf. 33, 739-757.
• 29: REPS, T. 1998. Program analysis via graph-reachability. Inf. Softw. Tech. 40, 11-12, 701- 726.
• 30: Thomas Reps , Susan Horwitz , Mooly Sagiv, Precise interprocedural dataflow analysis via graph reachability, Proceedings of the 22nd ACM SIGPLAN-SIGACT symposium on Principles of programming languages, p.49-61, January 23-25, 1995, San Francisco, California, United States [doi>10.1145/199448.199462]
• 31: Thomas Reps , Susan Horwitz , Mooly Sagiv , Genevieve Rosay, Speeding up slicing, Proceedings of the 2nd ACM SIGSOFT symposium on Foundations of software engineering, p.11-20, December 06-09, 1994, New Orleans, Louisiana, United States
• 32: REYNOLDS, J. C. 1968. Automatic computation of data set definitions. In Information Processing 68: Proceedings of the IFIP Congress 68. North-Holland, 456-461.
• 33: Mooly Sagiv , Thomas Reps , Susan Horwitz, Precise interprocedural dataflow analysis with applications to constant propagation, Theoretical Computer Science, v.167 n.1-2, p.131-170, Oct. 30, 1996
• 34: SHARIR, M. AND PNUELI, A. 1981. Two approaches to interprocedural dataflow analysis. In Program Flow Analysis: Theory and Applications, S. S. Muchnick and N. D. Jones, Eds. Prentice-Hall.
• 35: O. Shivers, Control flow analysis in scheme, Proceedings of the ACM SIGPLAN 1988 conference on Programming Language design and Implementation, p.164-174, June 20-24, 1988, Atlanta, Georgia, United States
• 36: Robert P. Wilson , Monica S. Lam, Efficient context-sensitive pointer analysis for C programs, Proceedings of the ACM SIGPLAN 1995 conference on Programming language design and implementation, p.1-12, June 18-21, 1995, La Jolla, California, United States
• 37: Mihalis Yannakakis, Graph-theoretic methods in database theory, Proceedings of the ninth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems, p.230-242, April 02-04, 1990, Nashville, Tennessee, United States [doi>10.1145/298514.298576]

INDEX TERMS

Primary Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.4 Processors
          Subjects: Compilers

Additional Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.4 Processors
          Subjects: Optimization

  F. Theory of Computation
  F.4 MATHEMATICAL LOGIC AND FORMAL LANGUAGES
      F.4.1 Mathematical Logic
          Subjects: Computability theory
      F.4.3 Formal Languages
          Subjects: Decision problems

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.2 Graph Theory
          Subjects: Path and circuit problems

General Terms:
Languages, Theory

Keywords:
∞ , context-sensitive program-analysis, control-flow analysis, dependence analysis, graph-reachability problem, linear matche-parenthesis language, set constraints, set-based analysis, structure-transmitted data-dependence