Differentiable Logic Specification

Connect differentiable components with logical operators.

Install

pip install git+https://github.com/ZikangXiong/diff-spec.git

First Order Logic

First-order logic is a logic formalism to describe the behavior of a system. It contains basic logic operators such as and, or, not, and quantifiers like forall, and exists.

We can connect any differentiable components with logical operators, with the requirement that a component outputs a great-than-0 value representing true and a less-than-0 value representing false.

p1 = InFirstQuadrant()
p2 = XGreatThan(2.0)
p3 = XLessThan(3.0)
p4 = YLessThan(3.0)

# Define a formula
f = FOL(p1).forall() & FOL(p2).exists() & FOL(p3).forall() & FOL(p4).forall()
print(f)

# ((((∀ InFirstQuadrant) & (∃ XGreatThan(2.0))) & (∀ XLessThan(3.0))) & (∀ YLessThan(3.0)))

p1-4 can be any differentiable components, including neural networks. In the above example, p1 is a predicate that checks if the input is in the first quadrant. p2-4 are predicates that check if the input is greater than 2, less than 3, and less than 3 in the x and y-axis, respectively.

One can optimize a world of inputs to satisfy the formula with gradient.

Signal Temporal Logic

Signal temporal logic is a formal language to describe the behavior of a dynamical system. It is widely used in the formal verification of cyber-physical systems.

We can use STL to describe the behavior of a system. For example, we can use STL to describe repeatedly visit goal_1 and goal_2 in timestep 0 to 13.

# goal_1 is a rectangle area centered in [0, 0] with width and height 1
goal_1 = STL(RectReachPredicate(np.array([0, 0]), np.array([1, 1]), "goal_1"))
# goal_2 is a rectangle area centered in [2, 2] with width and height 1
goal_2 = STL(RectReachPredicate(np.array([2, 2]), np.array([1, 1]), "goal_2"))

# form is the formula goal_1 eventually in 0 to 5 and goal_2 eventually in 0 to 5
# and that always holds in 0 to 8
# In other words, the path will repeatedly visit goal_1 and goal_2 in 0 to 13
form = (goal_1.eventually(0, 5) & goal_2.eventually(0, 5)).always(0, 8)

We can synthesize a trace with gradient or mixed-integer programming.

Citation

If you find this repository useful in your research, considering to cite:

@misc{xiong2023diffspec,
      title={DiffSpec: A Differentiable Logic Specification Framework},
      url={https://github.com/ZikangXiong/diff-spec/},
      author={Zikang Xiong},
      year={2023},
}