Table of Contents

• About
• Installation
• Usage
  • Python Operator API
    • Propositional logic (using python syntax)
    • Modal Logic (using python syntax)
  • String based API
    • Propositional logic (parse api)
    • Modal Logic (parser api)
  • Boolean Evaluation
  • Quantitative Evaluate (Signal Temporal Logic)
  • Utilities
• Similar Projects
• Citing

About

Python library for working with Metric Temporal Logic (MTL). Metric Temporal Logic is an extension of Linear Temporal Logic (LTL) for specifying properties over time series (See Alur). Some practical examples are given in the usage.

Installation

If you just need to use metric-temporal-logic, you can just run:

$ pip install metric-temporal-logic

For developers, note that this project uses the poetry python package/dependency management tool. Please familarize yourself with it and then run:

$ poetry install

Usage

To begin, we import mtl.

import mtl

There are two APIs for interacting with the mtl module. Namely, one can specify the MTL expression using:

1. Python Operators.
2. Strings + The parse API.

We begin with the Python Operator API:

Python Operator API

Propositional logic (using python syntax)

a, b = mtl.parse('a'), mtl.parse('b')
phi0 = ~a
phi1 = a & b
phi2 = a | b
phi3 = a ^ b
phi4 = a.iff(b)
phi5 = a.implies(b)

Modal Logic (using python syntax)

a, b = mtl.parse('a'), mtl.parse('b')

# Eventually `a` will hold.
phi1 = a.eventually()

# `a & b` will always hold.
phi2 = (a & b).always()

# `a` until `b`
phi3 = a.until(b)

# `a` weak until `b`
phi4 = a.weak_until(b)

# Whenever `a` holds, then `b` holds in the next two time units.
phi5 = (a.implies(b.eventually(lo=0, hi=2))).always()

# We also support timed until.
phi6 = a.timed_until(b, lo=0, hi=2)

# `a` holds in two time steps.
phi7 = a >> 2

String based API

Propositional logic (parse api)

# - Lowercase strings denote atomic predicates.
phi0 = mtl.parse('atomicpred')

# - infix operators need to be surrounded by parens.
phi1 = mtl.parse('((a & b & c) | d | e)')
phi2 = mtl.parse('(a -> b) & (~a -> c)')
phi3 = mtl.parse('(a -> b -> c)')
phi4 = mtl.parse('(a <-> b <-> c)')
phi5 = mtl.parse('(x ^ y ^ z)')

# - Unary operators (negation)
phi6 = mtl.parse('~a')
phi7 = mtl.parse('~(a)')

Modal Logic (parser api)

# Eventually `x` will hold.
phi1 = mtl.parse('F x')

# `x & y` will always hold.
phi2 = mtl.parse('G(x & y)')

# `x` holds until `y` holds. 
# Note that since `U` is binary, it requires parens.
phi3 = mtl.parse('(x U y)')

# Weak until (`y` never has to hold).
phi4 = mtl.parse('(x W y)')

# Whenever `x` holds, then `y` holds in the next two time units.
phi5 = mtl.parse('G(x -> F[0, 2] y)')

# We also support timed until.
phi6 = mtl.parse('(a U[0, 2] b)')

# Finally, if time is discretized, we also support the next operator.
# Thus, LTL can also be modeled.
# `a` holds in two time steps.
phi7 = mtl.parse('XX a')

Quantitative Evaluate (Signal Temporal Logic)

Given a property phi, one can evaluate if a timeseries satisifies phi. Time Series can either be defined using a dictionary mapping atomic predicate names to lists of (time, val) pairs or using the DiscreteSignals API (used internally).

There are two types of evaluation. One uses the boolean semantics of MTL and the other uses Signal Temporal Logic like semantics.

# Assumes piece wise constant interpolation.
data = {
    'a': [(0, 100), (1, -1), (3, -2)],
    'b': [(0, 20), (0.2, 2), (4, -10)]
}

phi = mtl.parse('F(a | b)')
print(phi(data))
# output: 100

# Evaluate at t=3
print(phi(data, time=3))
# output: 2

# Evaluate with discrete time
phi = mtl.parse('X b')
print(phi(data, dt=0.2))
# output: 2

Boolean Evaluation

To Boolean semantics can be thought of as a special case of the quantitative semantics where True ↦ 1 and False ↦ -1. This conversion happens automatically using the quantitative=False flag.

# Assumes piece wise constant interpolation.
data = {
    'a': [(0, True), (1, False), (3, False)],
    'b': [(0, False), (0.2, True), (4, False)]
}

phi = mtl.parse('F(a | b)')
print(phi(data, quantitative=False))
# output: True

phi = mtl.parse('F(a | b)')
print(phi(data))
# output: True

# Note, quantitative parameter defaults to False

# Evaluate at t=3. 
print(phi(data, time=3, quantitative=False))
# output: False

# Compute sliding satisifaction.
print(phi(data, time=None, quantitative=False))
# output: [(0, True), (0.2, True), (4, False)]

# Evaluate with discrete time
phi = mtl.parse('X b')
print(phi(data, dt=0.2, quantitative=False))
# output: True

Utilities

import mtl
from mtl import utils

print(utils.scope(mtl.parse('XX a'), dt=0.1))
# output: 0.2

print(utils.discretize(mtl.parse('F[0, 0.2] a'), dt=0.1))
# output: (a | X a | XX a)

Similar Projects

Feel free to open up a pull-request to add other similar projects. This library was written to meet some of my unique needs, for example I wanted the AST to be immutable and wanted the library to just handle manipulating MTL. Many other similar projects exist with different goals.

1. https://github.com/doganulus/python-monitors
2. https://github.com/STLInspector/STLInspector

Citing

@misc{pyMTL,
  author       = {Marcell Vazquez-Chanlatte},
  title        = {mvcisback/py-metric-temporal-logic: v0.1.1},
  month        = jan,
  year         = 2019,
  doi          = {10.5281/zenodo.2548862},
  url          = {https://doi.org/10.5281/zenodo.2548862}
}