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CounterTV -- Counterexample-Guided Correlation Algorithm for Translation Validation

Table of contents

  1. Getting started
    1. Ideal machine configuration
    2. Setup
    3. Running equivalence checker for an example function
  2. Step-by-Step instructions for running the benchmarks
    1. Directory Structure
    2. Reproducing the Table 2 results
    3. Running a particular benchmark category with either Best-First or Depth-First Strategy
    4. Reproducing the discovered bug in diet libc library
    5. Running equivalence checker on custom code
  3. HOWTO: Interpret the input and output files
    1. Interpreting the input CFG (.tfg and .etfg) files
    2. Interpreting the output product-CFG (.proof) file
    3. Interpreting the output log (.eqlog) file
  4. CounterTV code walkthrough
  5. Archived output (.eqlog and .proof) files

Getting started

Ideal machine configuration

An ideal machine for running this artifact would have at least:

  • 8 physical CPUs
  • 32 GiB of RAM
  • 60 GiB disk space
  • Broadband connection


The artifact is packaged as a Docker application. Installation of Docker is covered here. Follow these steps for building and running the equivalence checker based on CounterTV:

  1. Install Docker Engine and set it up. Make sure you are able to run the hello-world example.
  2. Go to the top-level directory of the source tree and build the Docker image. Note that internet connectivity is required in this step.
    docker build -t oopsla20-ctv .
    This process can take a while depending upon your internet connection bandwidth.
  3. Run a container with the built image.
    docker run -it oopsla20-ctv
  4. (Inside the container) Build the artifact and install the equivalence checker.
    make install

Equivalence checker is now ready for use. Different make targets are provided for reproducing the individual results of the paper. For example, use make tableresults for reproducing the full table 2 results. Other targets are discussed in later sections.

Running equivalence checker for an example function

  1. To test the above set-up, you may run the equivalence checker on the example function shown in Fig 1 of the paper.
  2. The source code for the example is present at path: superopt-tests/paper_ex.
    1. The two implementations of function nestedLoop in paper_ex_src.c and paper_ex_dst.c respectively are compared for equivalence.
  3. Execute make run_paper_ex to run this unit test.
  4. After executing this command, the source code will be compiled to generate unoptimized LLVM IR and optimized x86 assembly. Further, the CFGs will be constructed for the unoptimized LLVM IR and the optimized assembly and given as input to equivalence checker.
    1. The paper_ex_src.c version is compiled to unoptimized LLVM IR and paper_ex_dst.c is compiled to optimized x86 assembly.
  5. The equivalence checker using CounterTV will then try to construct a product-CFG that proves bisimulation.
  6. The appearance of message /home/user/artifact/superopt-tests/build/paper_ex/eqlogs/nestedLoop.gcc.BFS.eqlog passed - [5.39] would confirm a successful run (5.39 may not match).
    1. There will be other messages as well but they can be safely ignored.
    2. The number in square brackets is the time taken (in seconds) for constructing the product-CFG. It is 5.39 seconds in the above case.
    3. The .eqlog file listed above contains log of this run and other useful information.
  7. The whole procedure including step 4 and 5 should take close to 2 minutes.

Instructions for interpreting the results (the .eqlog file and .proof file) are discussed in a later section

Step-by-Step instructions for running the benchmarks

Directory Structure

Source code

The benchmarks are in superopt-tests directory. Two benchmark suites are available: TSVC and LORE loop nests.

TSVC suite

  • TSVC_prior_work directory contains the 28 TSVC benchmark functions which have already been demonstrated by SPA.
  • TSVC_new directory contains the TSVC benchmark functions for which CounterTV is the first to automatically generate equivalence proofs (they were not demonstrated by SPA).
  • The TSVC benchmark functions are compiled using recent versions of production compilers, namely, GCC-8, Clang/LLVM-11, and ICC-18.0.3 with -O3 -msse4.2 compiler flags (highest optimization levels) to generate optimized x86-32 binaries.

LORE Loop Nests

  • For LORE loop nests, we use one representative pattern for a set of structurally-similar program/transformation pairs, irrespective of the compiler that generated it. The space of additional transformations performed in this category (that are not covered by TSVC functions) include loop splitting, loop fusion for bounded number of iterations, loop unswitching, and summarization of loop with small and constant bounds.
  • We consider total 16 different loop nest patterns. For each of these 16 patterns, we test two variations: one where the loop bodies involve a memory write contained in LORE_mem_write directory , and another where at least one of the loop bodies does not involve a memory write contained in LORE_no_mem_write directory.
  • Among the 16 variations that involve a memory write in the loop bodies, the compilers produce non-bisimilar transformations for five of them. Thus we show results for 11 loop nest patterns where loop bodies have memory writes, and 16 loop nest patterns where the loop bodies do not have memory writes. Further, for each loop nest variation, we test across two different unroll factors (μ = 4 and μ = 8). The patterns with unroll factor 8 are due to compilations generated by LLVM-11 or by GCC-8 with the appropriate pragma switch.

Input, Output and intermediate files

The input files viz. unoptimized LLVM IR and optimized assembly code generated by the compiler; the intermediate .tfg and .etfg files; and the output .eqlog and .proof files are present in the superopt-tests/build/BMC directory, where BMC represents the benchmark (one of TSVC_prior_work, TSVC_new, LORE_mem_write, LORE_no_mem_write).

There is a structure in naming and location of these files. To give an example, for a source file foo.c of BMC benchmark suite:

  • The unoptimized LLVM IR is stored at the path superopt-tests/build/BMC/foo.bc.eqchecker.O0.
  • The optimized assembly code generated by compiler CC is stored at the path superopt-tests/build/BMC/foo.CC.eqchecker.O3.i386.
  • The intermediate .etfg file corresponding to the unoptimized IR is stored at the path superopt-tests/build/BMC/foo.bc.eqchecker.O0.ll.ALL.etfg.
  • The intermediate .tfg file corresponding to the optimized assembly code is stored at the path superopt-tests/build/BMC/foo.CC.eqchecker.O3.i386.ALL.tfg.
  • The output .eqlog and .proof files corresponding to the function bar, compiler CC and correlation search strategy BFS are generated at the path superopt-tests/build/BMC/eqlogs/bar.CC.BFS.eqlog and superopt-tests/build/BMC/eqlogs/bar.CC.BFS.proof respectively.
  • The output .eqlog and .proof files corresponding to the function foo, compiler CC and correlation search strategy DFS are generated at the path superopt-tests/build/BMC/eqlogs/foo.CC.DFS.eqlog and superopt-tests/build/BMC/eqlogs/foo.CC.DFS.proof respectively.

The format of the intermediate input files and output files is discussed in a later section.

Reproducing Table 2 results

To reproduce the table 2 results, run make tableresults from /home/user/artifact/ directory of the docker container. The equivalence checker tool is run for all four benchmark categories with both BFS and DFS strategy to reproduce the results. The benchmarks runs in parallel if multiple CPUs are available to the docker container.

The output is produced in the form of a CSV file at path /home/user/artifact/table2.csv. This output CSV is generated by combining the statistics from the generated .eqlog files.

Time taken for running the benchmarks

  • It takes around 3 CPU-hours to run the TSVC_prior_work in BFS strategy.
  • It takes around 11.5 CPU-hours to run the TSVC_new in BFS strategy.
  • It takes around 2.5 CPU-hours to run the LORE_mem_write in BFS strategy.
  • It takes around 4 CPU-hours to run the LORE_no_mem_write in BFS strategy.
  • With a time limit of 5 hours and memory limit of 12 GB, it takes around 200 CPU-hours to run all benchmarks with DFS strategy.

Description of the output CSV file

The CSV file mimics the structure of Table 2 in the paper. The top row of the CSV lists the benchmarks with compiler/unroll-factor variations and subsequent rows list properties and results for each of these configurations.

The following list summarizes the properties.

  1. total-fns and failing-fns: The total number and failing functions in the corresponding benchmark category. Note that the total (and failing) count may differ from table 2 because the functions known to fail are not run in certain configurations and thus do not appear in data extracted from the run. The number of passing functions (total-fns - failing-fns), however, should match the Table 2 numbers with exception of caveat mentioned at the end of this section.
  2. avg-ALOC and max-ALOC: The average and maximum assembly lines of code in the optimized assembly. This number is calculated by counting instructions in our internal representation and might be off by 2 or 4 because no-op instructions are ignored in our representation.
  3. avg-product-CFG-nodes/avg-product-CFG-edges and max-product-CFG-nodes/max-product-CFG-edges: The average and maximum number of nodes and edges in the product-CFG generated by CounterTV.
  4. avg-total-CEs-node and avg-gen-CEs-node: The average number of counterexamples per final product-CFG node ("Avg # of total CEs/node" in paper) and the average number of counterexamples that were generated (not propagated) per node through SMT queries ("Avg # of gen. CEs/node" in paper).
  5. BFS-avg-eqtime: The average time taken (in seconds) to generate equivalence proof using the best-first search algorithm in each benchmark category ("Avg equivalence time" in paper).
  6. BFS-avg-paths-enum, BFS-avg-paths-pruned, BFS-avg-paths-expanded: Statistics for the best-first search (BFS) algorithm: the number of correlation possibilities that were created (paths enumerated) before the complete product-CFG was found, the number of correlation possibilities that were remaining after pruning (paths pruned) and the number of correlation possibilities which were actually expanded further (paths expanded).
  7. DFS-avg-paths-enum, DFS-avg-paths-expanded: Corresponding statistics for backtracking-based depth-first strategy (DFS) with static heuristic where counterexample-guided pruning and ranking is omitted.
  8. avg-paths-expanded-DFS-by-BFS: The ratio of the average number of paths expanded in DFS with-respect-to the average number of paths expanded in BFS.
  9. DFS-mem-timeout-reached: The DFS strategy runs out of either time or memory resources for some of the benchmarks, this statistic count those cases.

It is important to note the final numbers may differ slightly from the paper numbers because the algorithm is counterexample driven which might come out different due to various reasons. However, the difference is expected to be small.

Note 0: Mistake in reporting ALOC for "LORE_mem_write u=8"

The maximum ALOC for "LORE Loop Nest | All loops have memory write | μ = 8" was incorrectly reported as 31 while the actual number is 55.

Note 1: Bug in original implementation

We found a bug in our implementation after paper submission. The submitted implementation is able to successfully prove equivalence for 185 TSVC function-compiler pairs out of the total 189 function-compiler pairs mentioned in Table 2. There are now 2 additional functions for which equivalence could not be established in the time limit of 5 hours. All the results demonstrated for LORE loop nests still hold. Further, the time required to prove equivalence has increased due to the patch required to fix the bug. Needless to say, other statistics will also be affected by this change. We will update these numbers in the next paper revision.

Note 2: Absence of Intel C Compiler (ICC)

The Intel C Compiler is not packaged with this artifact, instead pre-compiled binaries are provided (see icc_binaries directory). These binaries are automatically copied to appropriate paths when required and the final results include the data from ICC runs.

Running a particular benchmark category with either Best-First or Depth-First Strategy

The top level Makefile provides targets for running individual benchmarks with either BFS or DFS correlation search strategy.

For each benchmark,strategy pair, use the target template oopsla_<benchmark>_<strategy> for running it where <benchmark> is one of

  • tsvc_prior
  • tsvc_new
  • lore_mem_write
  • lore_no_mem_write

and <strategy> can be:

  • bfs for best-first search
  • dfs for depth-first search.

As an example, execute

make oopsla_tsvc_prior_bfs

for running TSVC_prior_work benchmark functions with best-first search strategy and

make oopsla_lore_no_mem_write_dfs

for running LORE_no_mem_write benchmark functions with depth-first search.

Reproducing the discovered bug in diet libc library

The diet libc bugs mentioned in Experiments section and discussed in Appendix A.2 can be reproduced by running make run_dietlibc. Each of these benchmarks is expected to fail i.e. the result will be "FAILED". The C code for the corresponding function pairs, OpenBSD and buggy Diet libc, can be found in directory superopt-tests/dietlibc.

Running equivalence checker on custom code

The equivalence checker can be made to run on custom code by editing the paper_ex_src.c,paper_ex_dst.c files in superopt-tests/paper_ex. The following steps describe how to achieve it.

  • Add the function in both paper_ex_src.c and paper_ex_dst.c. The function's name and signature must be same in both files.
    • The paper_ex_src.c implementation will be compiled to unoptimized LLVM bitcode while the paper_ex_dst.c version will be compiled to optimized x86 assembly.
    • Note that adding identical implementation will allow equivalence checking of same program's compilation.
  • The name of the added function should be appended to paper_func file.
    • The equivalence checker is run only for functions named in paper_func.
  • Run make run_paper_ex from base directory of the artifact (/home/user/artifact)

The result of the run will appear in the form of message /home/user/artifact/superopt-tests/build/paper_ex/eqlogs/<function_name>.gcc.BFS.eqlog <passed/FAILED> - [<time taken>]. A "passed" output will indicate a successful run.

HOWTO: Interpret the input and output files

Interpreting the input CFG (.tfg and .etfg) files

The .tfg and .etfg files are control flow graph (CFG) representations of x86 binary and LLVM bitcode respectively as required by the equivalence checker. Both files have a hierarchical structure and share common attributes. Headers in the files are prefixed by =. For example, =Node, =Edge etc.

Important headers (in order as they appear) are explained as follows.

  1. =FunctionName: Identifies name of the C function.

  2. =TFG: Beginning of control flow graph representation for the C function previously identified by =FunctionName.

  3. =Nodes: List of PCs (program points/locations) in this CFG.

    1. The PC identifier is encoded as a 3-tuple: <part0>%<part1>%<part2>.
    2. Start PC is L0%0%1 and exit PC is E0%0%1.
  4. =Edge: A CFG edge is represented as an SP-graph with the following children fields:

    1. =Edge.EdgeCond: The edge condition in expr format (explained below).
    2. =Edge.StateTo: Compressed transfer function for this edge. State variables or domain of machine state includes: list of all LLVM variables, x86 registers and memory state variables. The compressed representation only contains mapping for state variables which were modified over this edge. Further, the transfer function:
      1. Has 0 or more (statevar, expr) tuple as children, and
      2. Each tuple is formatted as: =<state-variable name> followed by expr for this state-variable.
  5. =Input: List of arguments to the C function. Each argument gets a separate =Input header.

  6. =Output: The values returned by the function. Includes:

    1. the return value/register (if the return type is non-void)
    2. heap
    3. global symbols modified by this function
    4. the return address.
  7. =Symbol-map: A list of symbols that are used in the function. Contains additional information such as name (as it appears in LLVM's symbol map), size, alignment and "constness" (0 == non-const, 1 == const) separated by ':'.

  8. =memlabel_map.<numeric_id>: The alias analysis associates a potentially points-to set with each memory operation's target address and stores it in a mapping. This is dump for the same.

  9. =Locs: Locs are set of state variables which are considered for alias analysis, invariant inference and other analyses. This includes LLVM variables, x86 registers, and memory elements whose address is a known (symbolic or numeric) constant and memories (segmented into heap, global variables and stack). Locs are associated with a unique numeric identifier.

  10. =Liveness: Result of liveness-analysis. Lists locs (represented with locid) live at each PC.

  11. =sprel_maps: Result of available expressions analysis.

  12. =String-contents: Contents of read-only (RO) symbols from .rodata section of ELF which are referenced in this function.

  13. =Nextpc-map: Map of targets for each call instruction.

  14. =TFGdone: TFG end string.

Structures of common fields

  1. Expression or expr: Expressions involving state variables are represented as DAGs. The format is:

    <expr_id> : <op> : <type>

    where <op> can be a state element or an SMT-like operator referencing other expressions using <expr_id>s. E.g. bvadd(123, input.src.llvm-%0) is represented as:

    1 : 123 : BV:32
    2 : input.src.llvm-%0 : BV:32
    3 : bvadd(1, 2) : BV:32
  2. Predicate or pred: A predicate encodes (precond) => (lhs == rhs) in the following structure:

    <this field can be safely ignored>
    <precondition (represented as an SP-graph) for this predicate>

    For example, the following predicate unconditionally asserts that state variable input.src.llvm-%1 and input.dst.exreg.0.5 have same value.

    1 : input.src.llvm-%1 : BV:32
    1 : input.dst.exreg.0.5 : BV:32
  3. Pathset representation or the SP-graph: The SP-graph is a recursively defined structure.

    1. epsilon (empty edge) is (epsilon)
    2. Singular edge is <from_pc> => <to_pc>
    3. A parallel combination is ( <sp_graph> + <sp_graph> )
    4. A serial combination is ( <sp_graph> * <sp_graph> )

Interpreting the output product-CFG (.proof) file

The .proof is structured similar to .tfg file with same heading style. The equivalence witness is a correlation graph (or product-CFG) and the .proof is essentially a dump of this structure.

  • A correlation graph node is a tuple of input graphs' nodes.
  • A correlation graph edge is a tuple of input graphs' pathsets.
  • Each correlation graph node has a set of inductive invariants which are provable at that program point.

Important bits (in the order as they appear) are:

  1. =FunctionName: Name of the corresponding C function.
  2. =result: Will be 1 if equivalence check succeeded.
  3. =corr_graph: Beginning of dump of the correlation graph. This is followed by =src_tfg (CFG for LLVM bitcode i.e. C) and =dst_tfg (CFG for x86 assembly i.e. A) dumps.
  4. =eqcheck (and =eqcheck_info): Dump of meta data information, can be safely ignored.
  5. =Nodes: List of correlation graph PC identifiers formatted as <src_cfg_pc_id>_<x86_cfg_pc_id>.
  6. =Edges: List of correlation graph edges as 2-tuple of from PC and to PC.
  7. =Edge list: Each entry in the list describes a correlation graph edge. Constituent pathsets from C and A are listed under =Edge.src_edge_composition and =Edge.dst_edge_composition respectively.
  8. =Invariant state list: Each entry in the list describes set of invariants at each PC of the correlation graph. The most interesting part of invariant state is the set of predicates (pred):
    1. A predicate is conveniently preceded by a line which lists the PC where it was proved and its "type" (which can be arr, bv among others). The format of this line is =pc <PC> invariant_state_eqclass <id> type <type> pred <pred_num>. Example: =pc Lif.then%2%200003_L11%1%200003 invariant_state_eqclass 0 type arr pred 0
    2. The predicate is structured as explained in previous section.
    3. Full example of a predicate which asserts that memory states are equal at PC Lif.then%2%200003_L11%1%200003:
      =pc Lif.then%2%200003_L11%1%200003 invariant_state_eqclass 0 type arr pred 0
      1 : 1 : BOOL
      1 : input.src.llvm-mem : ARRAY[BV:32 -> BV:8]
      2 : input.dst.mem : ARRAY[BV:32 -> BV:8]
      3 : memlabel-mem-symbol.70.0-symbol.77.0-symbol.78.1-heap : MEMLABEL
      4 : memmasks_are_equal(1, 2, 3) : BOOL
      The operator memmasks_are_equal returns true iff its first two memory operands have same state in the regions specified by the third operand. In this particular example, the memory state variables input.src.llvm-mem (i.e. LLVM input memory) and input.dst.mem (i.e. x86 input memory) are asserted to be equal in the global symbol memory regions and heap.

Interpreting the output log (.eqlog) file

Given the CFGs for LLVM bit code and x86 assembly, the equivalence checker tool generates a log file (.eqlog) while constructing the product-CFG. This .eqlog file contains the input CFGs, the function name, SMT solver timeout, global timeout, path of the input .tfg and .etfg files and path of the output .proof file. Further, it captures the input and output for high level functions shown in Fig 7 of the paper. This includes:

  • The initial product-CFG returned by initProductCFG() function.
  • PCpair returned by findIncompleteNode() function for the constructed partial product-CFG.
  • FullPathSet in A in SP-graph notation as returned by getNextPathsetRPO() function and the associated nexthop PC.
  • The FullPathSets in C enumerated by getCandCorrelations() function in SP-graph representation. For each enumerated pathset, the delta, μ (mu), from PC (nC), anchor node to-PC (wC), number of paths in the pathset, number of edges, and minimum and maximum path length are captured in the log file.
  • The outcome of CEsSatisfyCorrelCriterion and InvRelatesHeapAtEachNode checks for every pair of the above chosen pathset in A and enumerated pathset in C.
  • The output of computeRank() function for pathset pairs which satisfy both the above checks.
  • The number of new partial product-CFGs enumerated by expandProductCFG() function and the size of the frontier after adding these new product-CFGs (probable correlations).
  • The partial product-CFG chosen by function removeMostPromising().

It also captures whether the tool is able to construct the final product-CFG that proves bisimulation or the tool reached a memory or time threshold. In both the cases, the statistics/counters as shown in table 2 of paper are printed at the end of the file.

CounterTV code walkthrough

This section intends to provide a quick walkthrough of the implementation of the CounterTV algorithm for anyone who wants to reuse or extend the algorithm. The source code is present in superopt directory.

  1. The top-level function genEqProof is present in file tools/eq_main.cpp which takes as input the CFGs for unoptimized LLVM IR and optimized x86 assembly.
  2. The function bestFirstSearch() is implemented as find_correlation() in file lib/eq/correlate.cpp. It tries to construct a product-CFG which proves bisimulation across input CFGs. It first initializes the frontier with the product-CFG containing the start pcpair of the input CFGs.
  3. The best-first search strategy is implemented as choose_most_promising_correlation_entry() function and it picks the most promising product-CFG from the frontier based on the ranking strategy described in fig. 6 in the paper. The class bv_rank_val_t implements the rank for the live variables in C and A respectively. The choose_most_promising_correlation_entry function needs to modified to implement a different ranking strategy than described in the paper.
  4. The function cg_check_new_cg_edge() in file lib/eq/corr_graph.cpp implements the checkCriterionForEdges check and inferInvariantsAndCounterExample(). If the checkCriterionForEdges check fails or heap states are not equal after inferInvariantsAndCounterExample(), the cg_check_new_cg_edge() function returns a nullptr thus discarding the current product-CFG.
  5. If both the above check passes, calculate_rank_bveqclass_at_pc() function in file lib/eq/corr_graph.cpp calculates the rank for the current product-CFG as described in fig 6 in the paper. This function needs to be modified to alter the rank calculation metric.
  6. The get_next_dst_edge_composition_to_correlate() function in file lib/eq/correlate.cpp implements the findIncompleteNode() and getNextPathsetRPO() functions and returns the fullPathset in A to be correlated next. If no incomplete node is found, the current product-CFG is returned as the final product-CFG.
  7. The get_src_unrolled_paths() function in file lib/eq/correlate.cpp implements the getCandCorrelations() function and returns the fullpathsets in C for given fullpathset in A and unroll factor.
  8. The above two functions needs to modified to alter the path enumeration strategy in A and C respectively.
  9. The function corr_graph_prune_and_add_correlations_to_pc() in file lib /eq/correlate.cpp implements the Counterexample-guided Pruning and Ranking as described in section 3.9 in paper. The function reports at the end the total number of new product-CFGs added to the frontier that passes the counterexample-guided pruning and ranking. This function needs to be modified to alter the pruning and ranking strategy.

Archived output (.eqlog and .proof) files

We have included the generated .eqlog and .proof files for all benchmark categories and BFS search strategy at the path archived-results/oopsla-results.tar.xz. The summary table from these output files is present at the path archived-results/table2.csv. The numbers generated from make tableresults can be matched against this.


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