QBF and DQBF to TPTP THF and SMT-LIB2 converters

This repository contains tools to convert QBF problems in QDIMACs format and DQBF problem in BUNSAT format to higher-order problems in TPTP THF format. Furthermore, coversation from QDIMACS to SMT-LIB2 is also possible.

Building the Tool

Both tools are written in Haskell. While there are multiple ways to build the tools, using the Stack build toolis strongly recommended by us and documented here. To install Stack follow the instructions on the Stack homepage.

The qbfTool folder contains the source code for the QBF to HOL/SMT-LIB converter, and the DQBF tool is in the dqbfTool folder. After the installation of Stack is complete, open a terminal and navigate to either folder. Then run then following commands, where {tool} is either qbfTool, or dqbfTool:

> stack setup
> stack build
> stack exec {tool}

The stack build command outputs the path the resulting binary is placed in. On Linux calling stack install will copy the binary to ~/.local/bin. Alternatively, the programs can be run with additional arguments ({args}) by calling

> stack exec {tool} -- {args}

Converting QBF Problems

The QDIMACS format is widely used to represent QBF problems in CNF. It is an extension of the DIMACS format used for propositional problems and QDIMACS is compatible with DIMACS. Every valid DIMACS file is also a valid QDIMACS file. Variables in the QDIMACS format are expressed by non-zero natural numbers and literals are expressed by non-zero integers. The variable index of a literal is the absolute value of the integer, the polarity is represented by the sign of the integer. The following is a small QDIMACS file:

p cnf 3 3
c an example problem
a 1 2
e 3
-1 -3 -2 0
-2 -3 -1 0
3 0

A formal grammar of the QDIMACS format can be found online. This grammar forbids empty clauses and files without any clauses. We found that BLOQQER sometimes does generate problems with empty clauses and completely empty problems. Hence, our tool tries to be very liberal when parsing QDIMACS problems. It especially ignores the metadata given in the first line.

The default behavior of qbfTool is to accept a QDIMACS problem on the standard input and to output a TPTP problem to standard output. Assuming a file test.qdimacs which contains the following QDIMACS problem:

p cnf 6 3
e 1 0
a 0
e 2 0
a 3 4 0
e 5 6 0
1 3 5 0
1 2 0
2 4 6 0
99 0

This problem is not a valid QDIMACS file. The meta values are wrong and a quantifier line contains an empty list of variables. Executing qbfTool on this problem results in the following output:

> cat test.qdimacs | stack exec qbfTool
thf(c,conjecture,(? [X1: $o,X2: $o,X3: $o]:(! [X4: $o,X5: $o]:(? [X6: $o,X7: $o]:((X1|X4|X6)&(X1|X3)&(X3|X5|X7)&(X2)))))).

This output is not a direct translation of the input problem. The qbfTool also applies some normalization steps. Those are:

• Consequitive lines containing the same quantifier are merged.
• Empty quantifiers are removed and the surrounding quantifiers are merged.
• Variables are renamed by the order they appear in the quantifier. This also assures that all variables are used without gap.
• Variables which are used in the matrix, but which do not appear in the quantifier prefix, are added to the outermost quantifier if the outermost quantifier is an existential quantifier. Otherwise, an existential quantifier containing those variables is added.
• If the input problem does not contain any clauses, the clause x∨¬x is added.
• If the input problem contains the empty clause, a clause x and a clause ¬x is added.

If the tool is called with the command line parameter -n, the output is a valid QDIMACS problem resulting from these transformations instead of a TPTP problem. For the QDIMACS problem given above this output is:

p cnf 7 4
e 1 2 3 0
a 4 5 0
e 6 7 0
1 4 6 0
1 3 0
3 5 7 0
2 0

As described above, free variables are implicitly existentially quantified in the input problem and explicitly existentially quantified in the output problem. Hence, the input QBF problem is satisfiable if the output problem is valid. A resolution prover will negate the input problem and show its unsatisfiability. Negating a problem in clause normal form and clausifying the negated problem requires repeated application of the distributive laws and is therefore an expensive operation. Furthermore, a prover is geared towards showing validity and will seldom deduce that an input problem is not valid. Hence, the -i command line parameter will instruct qbfTool to add a negation in front of the output problem. The resulting problem will be a theorem if an only if the input problem is unsatisfiable.

To convert the input probem into a SMT-LIB2 problem the -s option can be used. The resulting problem is in the UF logic and can be read by many SMT solvers.

Finally, the -o {filename} parameter can be used to instruct qbfTool to write the output into a file. If this parameter is used, qbfTool will output exactly one line to the standard output. This line will contain the number of variables and number of clause in the problem separated by a space.

Converting DQBF Problems

The dqbfTool generates HOL problems in the TPTP format from DQBF problems in the BUNSAT format. This format supports input problems which are not in CNF.

The following example of the BUNSAT format was taken from the BUNSAT web page. This page also contains a grammar of the format.

A x1 x2: E{x1} y1 : E{x2} y2: ((~y1) | y2) <--> (x1 ^ x2)

The dqbfTool behaves the same way as the qbfTool behaves, but does not apply any normalizations to the input problem. Hence, the -n command line parameter is not supported. The -i parameter adds a negation symbol to the output and the -o parameter writes the output into a file. If the -o parameter is given, the tool does not write any metadata to the standard output.