DRAT-trim

DRAT-trim

DRAT-trim is a satisfiability proof checking and trimming utility designed to validate proofs for all known satisfiability solving and preprocessing techniques. DRAT-trim can also emit trimmed formulas, optimized proofs, and TraceCheck+ dependency graphs. A system description can be found below.

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About

RAT Checks

DRAT-trim is an improvement over its predecessor DRUP-trim because it allows for a stronger form of redundancy, RAT. This check permits all known techniques including extended resolution, blocked clause addition, bounded variable addition, extended learning, and many more.

Deletion Information

DRAT-trim is based on the DRAT (Deletion Resolution Asymmetric Tautology) proof format which includes lemma deletion instructions that can greatly reduce proof checking time.

Backward Checking

DRAT-trim employs backward checking (Goldberg and Novikov 2003) of clausal proofs to minimize dependencies between clauses. Clauses are marked during conflict analysis and only marked clauses need to be checked.

Core-first Unit Propagation

The unit propagation routine of DRAT-trim has been modified to give preference to clauses marked as necessary, which further reduces the number of "necessary" clauses.

Trimmed Formulas

Trimmed formulas can optionally be emitted from DRAT-trim. These are ordered subsets of the original formula where unnecessary clauses have been omitted. One application of this output is preprocessing for Minimal Unsatisfiable Subset (MUS) extractors.

Optmized Proofs

DRAT-trim can optionally produce optimized proofs conataining a subset of the input lemmas and extra deletion information gained during backward checking.

Dependency Graphs

DRAT-trim also produces a dependency graph in the TraceCheck+ format. This format is a derivative of the TraceCheck resolution graph format, but allows for stronger dependencies.

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Download DRAT-trim

Last updated November 10, 2016.

Usage

DRAT-trim can be compiled with gcc/g++ with command:
gcc -O2 -o drat-trim drat-trim.c

A DIMACS CNF formula and DRAT/DRUP proof are mandatory arguments, but there several options for controlling behavior and emitting additional results.
./drat-trim FORMULA PROOF [options]

Option Description
-h Prints a command line option summary.
-c FILENAME Emits a trimmed formula in DIMACS CNF format.
-l FILENAME Emits an optimized DRAT proof.
-r FILENAME Emits a TraceCheck+ dependency graph.
-t NUMBER Enforce a time limit (in seconds). Default 20000.
-f Forward checking for proof of unsatisfiability.
-S Forward checking for proof of satisfiability.
-u Do not use core-first unit propagation.
-p Ignore deletion information.
-v Verbose output.
-x Only check lemmas for RUP.

Publications

Wetzler, N., Heule, M.J.H., and Hunt, Jr., W.A.: DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs. In: Theory and Applications of Satisfiability Testing (SAT) Volume 8561 of LNCS, Springer (2014) 422-429 [pdf]

Heule, M.J.H., Hunt, Jr., W.A., and Wetzler, N.: Trimming while checking clausal proofs. In: Formal Methods in Computer-Aided Design (FMCAD), IEEE (2013) 181-188 [pdf]

Heule, M.J.H., Hunt, Jr., W.A., and Wetzler, N.: Verifying refutations with extended resolution. In: International Conference on Automated Deduction (CADE). Volume 7898 of LNAI, Springer (2013) 345-359 [pdf]

Examples

Example 1. Consider below an input formula in DIMACS format (left), a DRAT proof with no deletion information (middle) and a DRAT proof with deletion information (right). Spaces and empty lines are added to the examples to improve readability. Deletion information is not required to include in proofs. However, adding deletion information to proofs can significantly lower the costs to check proofs by DRAT-trim.

p cnf 4 8 
 1  2 -3 0 
-1 -2  3 0 
 2  3 -4 0 
-2 -3  4 0 
-1 -3 -4 0 
 1  3  4 0 
-1  2  4 0 
 1 -2 -4 0 

 -1  0
  2  0
     0






        -1 0
d -1 -2  3 0
d -1 -3 -4 0
d -1  2  4 0
         2 0
d  1  2 -3 0
d  2  3 -4 0
           0


Example 2. In contrast to the DRUP format, DRAT allows the introduction of new variables. Consider the formula below in DIMACS format (left). The first six clauses in this formula can be replaced by five new clauses using a technique called bounded variable addition. To express this technique in DRAT, first add the new clauses with the literal of the new variable in the first position of the clause. Afterwards, remove the (original) clauses that have become redundant after adding the new ones. Addition of the new clauses and removal of some original clauses is shown in the DRAT proof below (right). The last two lines of the DRAT proof are added to make it a refutation.

p cnf 5 8 
    1  4 0  
    1  5 0 
    2  4 0 
    2  5 0
    3  4 0 
    3  5 0
-1 -2 -3 0
   -4 -5 0 






   6 1 0
   6 2 0
   6 3 0
  -6 4 0
  -6 5 0
d  1 4 0
d  2 4 0
d  3 4 0
d  1 5 0
d  2 5 0
d  3 5 0
     6 0
       0

Binary Format

Mapping DIMACS Literals to Unsigned Integers

The first step of the binary encoding is mapping literals in the DIMACS format to unsigned integers. The following mapping function is used: map(l) := (l > 0)? 2*l : -2*l + 1. The mapping for some DIMACS literals are shown below.

     DIMACS literals   unsigned integers
                      
                 -63     127
                 129     258   
               -8191   16383
               -8193   16387

Variable-Byte Encoding of Unsigned Integers

Assume that 'w0, ..., wi' are 7-bit words, 'w1' to 'wi' all non zero and the unsigned number 'x' can be represented as

    x = w0 + 2^7*w1 + 2^14*w2 + 2^(7*i)*wi

The variable-byte encoding of DRAT (also used in AIGER) is the sequence of i bytes b0, ... bi:

    1w0, 1w1, 1w2, ..., 0wi 

The MSB of a byte in this sequence signals whether this byte is the last byte in the sequence, or whether there are still more bytes to follow. Here are some examples:

    unsigned integer   byte sequence of encoding (in hexadecimal)
 
                   x   b0 b1 b2 b3 b4
                      
                   0   00
                   1   01
    2^7-1    =   127   7f
    2^7      =   128   80 01
    2^8  + 2 =   258   82 02
    2^14 - 1 = 16383   ff 7f
    2^14 + 3 = 16387   83 80 01
    2^28 - 1           ff ff ff 7f
    2^28 + 7           87 80 80 80 01

Bringing it together

In the binary DRAT format, each clause consists of at least two bytes. The first byte expresses whether the lemma is added (character 'a' or 61 in hexadecimal) or deleted (character 'd' or 64 in hexadecimal). The last byte of each lemma is the zero byte (00 in hexadecimal). In between these two bytes, the literals of the lemma are shown as unsigned integers in the variable-byte encoding.

In the example below, the plain DRAT format requires 26 bytes (including the new line characters and excluding the redundant spaces in the second lemma), while the binary DRAT format of the same proof requires only 12 bytes. Emiting proofs in the binary format reduces the size on disk by approximatedly a factor of three compared to the conventional (plain) DRAT format.

          plain DRAT      binary DRAT (in hexadecimal)

       d -63 -8193 0      64 7f 83 80 01 00 61 82 02 ff 7f 00 
         129 -8191 0

Contact Us

Marijn J.H. Heule --- firstname at cs dot utexas dot edu

Warren A. Hunt, Jr. --- lastname at cs dot utexas dot edu

Nathan Wetzler --- firstinitial nospace lastname at cs dot utexas dot edu