# FO²-Solver

Published in ICDT 2017 and available on the Dagstuhl repository:

On the automated verification of web applications with embedded SQL.
Shachar Itzhaky, Tomer Kotek, Noam Rinetzky, Mooly Sagiv, Orr Tamir, Helmut Veith, Florian Zuleger.
International Conference on Database Theory

FO²-solver is a new finite satisfiability solver for FO², the two-variable fragment of first-order logic. FO²-solver receives as input a sentence in FO² and answers whether the sentence is satisfiable by a structure with a finite universe.

The basic principle of FO²-solver uses the bounded model property of FO². The solver computes an upper bound on the universe size of the smallest structure satisfying the input sentence, if any structure which satisfies the sentence exists. Then, the solver uses a SAT solver to search for a model up to the upper bound size. If such a model is found, the sentence is satisfiable. If no such model is found, the sentence is unsatisfiable. The algorithm that the solver uses is sound and complete. Practically, of course, the solver may timeout or run out of space. The solver employs some strategies to lower the upper bound on the universe size and improve performance. For more details, see Section 4 of arXiv:1610.02101. FO²-solver was implemented by Tomer Kotek, based on some basic infrastructure from Shachar Itzhaky‘s EPR-based Verification tool.

# The logic FO²

FO² contains all the first-order logic sentences which use only two-variables: x and y. Examples of FO² sentences are:

Φ0 = (∀ x U(x)) ∧ (∃ x ¬ U(x))
Φ1 = (∀ x ∃ y B1(x,y)) → (∃ y ∀ x B2(x,y))

Note that quantification over the same variable can be nested:

Φ2 = ∀ x ∃ y (U(x) → ∃ x B(x,y))

The sentences may use unary relation symbols (such U above) and binary relation symbols (such as B, B1, and B2 above). The sentences may not use function symbols. The equality symbol = is in the language. No other special symbols are allowed (e.g., one cannot assume to have the smaller than < relation on the universe elements). Relation symbols with arity larger than 2 cannot be used.

Additionally, constant symbols may be used. For example:

Φ3 = ∀ x ∀ y (¬(x=c) → B(c,y))
Φ4 = (c1=c2) ∧ B(c1,c2)

Here, c, c1, and c2 are constant symbols.

# Input format: SMT-LIBv2

The input format of FO²-solver is SMT-LIBv2. Here is how an smt2 file given as input to FO²-solver should be constructed.

## Declarations

`(declare-sort V 0)`

Then, the symbols in the sentence are declared.
A binary relation symbol is declared as follows:

`(declare-fun B (V V) Bool)`

A unary relation symbol is declared as follows:

`(declare-fun U (V) Bool)`

A constant symbol is declared as follows:

`(declare-const c V)`

The equality symbol does not need to be declared.

## The assert command

After the declarations are done comes the most important part of the smt2 file. Inside the assert command you may use exists, forall, and, or, =>, not, and atomic formulas.

For instance:

`(assert (exists ((x V)) (or (B x x) (exists ((y V)) (=> (B x y) (not (= x y)))))))`

## The check-sat command

Every smt2 should end with:

`(check-sat)`

The command (get-model) is ignored, but it is allowed to occur here for compatibility.

## Examples

The smt2 file of Φ0 = (∀ x U(x)) ∧ (∃ x ¬ U(x)):

```(declare-sort V 0)
(declare-fun U (V) Bool)
(assert (and (forall ((x V)) (U x)) (exists ((x V)) (not (U x)))))
(check-sat)```

The smt2 file of Φ1 = (∀ x ∃ y B1(x,y)) → (∃ y ∀ x B2(x,y)):

```(declare-sort V 0)
(declare-fun B1 (V V) Bool)
(declare-fun B2 (V V) Bool)
(assert
(=>
(forall ((x V)) (exists ((y V)) (B1 x y)))
(exists ((y V)) (forall ((x V)) (B2 x y)))))
(check-sat)```

The smt2 file of Φ2 = ∀ x ∃ y (U(x) → ∃ x B(x,y)):

```(declare-sort V 0)
(declare-fun U (V) Bool)
(declare-fun B (V V) Bool)
(assert (forall ((x V)) (exists ((y V))
(=> (U x) (exists ((x V)) (B x y))))))
(check-sat)```

The smt2 file of Φ3 = ∀ x ∀ y (¬(x=c) → B(c,y)):

```(declare-sort V 0)
(declare-const c V)
(declare-fun B (V V) Bool)
(assert (forall ((x V) (y V)) (=> (not (= x c)) (B c y))))
(check-sat)```

The smt2 file of Φ4 = (c1=c2) ∧ B(c1,c2):

```(declare-sort V 0)
(declare-const c1 V)
(declare-const c2 V)
(assert (and (not (= c1 c2)) (B c1 c2)))
(check-sat)```

## System requirements

FO²-solver is designed to work on Linux systems with equipped with Java SE 7.

## Setup

Edit the script sat_solver_caller.sh to point to a SAT solver installed on your system. Any SAT solver which follows DIMACS input/output requirements is compatible (see SAT Competition 2009 for details). FO²-solver has been tested with Glucose and Lingeling.

# Running the tool

## Commands

The basic command to run the solver is:

`\$ java -jar FO2solver-wrapper.jar filename.smt2`

If the –print-model option is used, then FO²-solver prints a satisfying model to stdout if it exists.

`\$ java -jar FO2solver-wrapper.jar filename.smt2 --print-model`

The –sat-solver-path option overrides the setting of the SAT solver path in sat_solver_caller.sh.

## Output

FO²-solver begins by testing that the SAT solver is configured properly. If the SAT solver is configured correctly, the following will be printed:

```------ Testing SAT solver ------
--------------------------------```

If there is any error message between these two lines, then the SAT solver is not configured correctly.

Next, FO²-solver will print an output of the form:

```Reading smt2 file
------------------------
Computing formula in normal form
------------------------
Max model size 200
------------------------
Searching for a model of size at most 1:
Preparing input to SAT solver
Running SAT solver
Searching for a model of size at most 2:
Preparing input to SAT solver
Running SAT solver
Searching for a model of size at most 3:
Preparing input to SAT solver
Running SAT solver
Searching for a model of size at most 12:
Preparing input to SAT solver
Running SAT solver
------------------------
SATISFIABLE by a model of size at most 12```

If the input sentence is unsatisfiable, then the last line will be

` UNSATISFIABLE`

If the –print-model option is used and the input sentence is satisfiable, a model will be printed. For example:

```--- Satisfying model ---
The universe is 0,...,11
The true atoms are as follows: (atoms which do not occur are false)
U0(5)
U1(2)
U1(1)
U1(4)
U2(6)
U2(3)
U3(0)
B(0,5)
B(1,6)
B(5,1)
B(6,0)
B(7,1)
B(8,0)
B(9,1)
A(3)
A(4)
A(2)
A(5)
A(6)
A(7)
A(10)
A(11)
A(9)```

The order of the atoms is arbitrary.
In this case, the satisfying model has universe {0,…,11}. The relations in the model are:

U0 = {5}
U1 = {1, 2, 4}
U2 = {3, 6}
U3 = {0}
A = {2, 3, 4, 5, 6, 7, 9, 10, 11}
B = {(0,5), (1,6), (5,1), (6,0), (7,1), (8,0), (9,1)}

## Benchmark

The benchmarks from Section 5 of arXiv:1610.02101 are available.

## LogicLounge in memoriam Helmut Veith

Will robots take away your job? In memory of Helmut Veith, this year’s Conference on Computer Aided Verification (CAV), which takes place in Toronto, will feature a LogicLounge on the effect of automation and artificial intelligence on our jobs.

## Marijana Lazić wins ZONTA mobility stipend

Marijana Lazić won the “ZONTA CLUB WIEN I – TU-Mobilitätsstipendium” for her research in the intersection of computer-aided verification and distributed computing theory.

## Questions answered by Leslie Lamport

Leslie Lamport is going to answer questions about a Mathematical View of Computer Systems in Informatikhörsaal on Tuesday at 6pm. Check the announcement at the VCLA website and do not forget to watch the lecture beforehand.

## FRIDA 2016 in Marocco

We had nice talks at the workshop on Formal Reasoning in Distributed Algorithms (FRIDA) this year. Look at the workshop web page.

## Laura Kovács joins FORSYTE as full professor

We welcome Laura Kovács, who joined the FORSYTE group as a full professor of computer science in April 2016.