Formal Semantics and Automated Reasoning for Probabilistic Programs Modeling Epistemic and Aleatoric Uncertainty

Główne pojęcia

This work formalizes the syntax and semantics of a probabilistic programming language based on Hehner's predicative probabilistic programming, introducing Iverson brackets to separate relations from arithmetic. It establishes constructive semantics for probabilistic loops using fixed-point theorems, enriches the semantics to handle subdistributions and superdistributions, and mechanizes the theory in Isabelle/UTP for automated reasoning.

Streszczenie

The paper presents a probabilistic programming language called "probabilistic relations" based on Hehner's predicative probabilistic programming. The key contributions are:

1. Formalization of the syntax and semantics of the language, introducing Iverson brackets to clearly separate relations from arithmetic expressions.

2. Establishment of constructive semantics for probabilistic loops using Kleene's fixed-point theorem, defining the semantics as the least fixed point of a loop function. To simplify reasoning, a unique fixed-point theorem is also proven.

3. Enrichment of the semantic domain from distributions to subdistributions and superdistributions to enable the use of Kleene iterations for loop semantics.

4. Mechanization of the theory in Isabelle/UTP, an implementation of Unifying Theories of Programming in Isabelle/HOL, enabling automated reasoning about probabilistic programs using theorem proving.

The paper demonstrates the approach with six examples, including problems in robot localization, machine learning classification, and probabilistic loop termination analysis.

Statystyki

The paper does not contain any explicit numerical data or statistics. The focus is on the formal semantics and automated reasoning aspects of probabilistic programming.

Cytaty

"Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty."
"Probabilistic algorithms have been used in practice for a long time in autonomous robots, self-driving cars, and artificial intelligence."
"Robots are usually equipped with probabilistic control algorithms to deal with these uncertainties."

Kluczowe wnioski z

Probabilistic unifying relations for modelling epistemic and aleatoric uncertainty: semantics and automated reasoning with theorem proving

by Kangfeng Ye,... o arxiv.org 09-30-2024

https://arxiv.org/pdf/2303.09692.pdf

Głębsze pytania

How can the presented probabilistic programming language be extended to support nondeterminism and concurrency, enabling the verification of more complex probabilistic systems?

The presented probabilistic programming language, based on Hehner’s probabilistic predicative programming, can be extended to support nondeterminism and concurrency by introducing constructs that allow for the representation of these features within the existing framework.

Nondeterminism: To incorporate nondeterminism, we can introduce a nondeterministic choice operator that allows the program to select between multiple possible actions or states without a predetermined probability distribution. This can be achieved by defining a new construct that combines the existing probabilistic choice with nondeterministic choice, allowing for a richer set of behaviors. For instance, a construct could be defined as ( P \oplus Q ), where the execution of either ( P ) or ( Q ) occurs based on nondeterministic selection.

Concurrency: For concurrency, we can leverage the existing semantics of parallel composition in probabilistic programming. By defining a parallel composition operator that allows multiple probabilistic processes to execute simultaneously, we can model complex interactions between concurrent systems. This operator would need to account for the probabilistic outcomes of each concurrent process and how they affect the overall system state.

Verification Techniques: To verify these extended constructs, we can utilize existing theorem proving techniques and model checking approaches. The integration of nondeterminism and concurrency would require the development of new verification strategies that can handle the increased complexity of state spaces and transition systems. This could involve adapting the existing mechanization in Isabelle/UTP to accommodate the new constructs, ensuring that the verification process remains automated and scalable.

By implementing these extensions, the probabilistic programming language can effectively model and verify more complex systems that exhibit both nondeterministic and concurrent behaviors, thereby enhancing its applicability in real-world scenarios such as robotics and machine learning.

What are the limitations of the current approach in handling continuous probability distributions, and how can it be addressed?

The current approach primarily focuses on discrete probability distributions, which poses several limitations when dealing with continuous probability distributions:

Lack of Continuous Constructs: The existing framework does not provide explicit constructs for defining and manipulating continuous distributions, which are essential for modeling real-world phenomena that exhibit continuous variability. This absence limits the expressiveness of the probabilistic programming language in scenarios where continuous outcomes are prevalent.

Integration with Fixed-Point Theorems: The fixed-point theorems utilized in the current approach, such as Kleene’s fixed-point theorem, are primarily designed for discrete settings. Extending these theorems to accommodate continuous distributions requires a re-evaluation of the underlying mathematical framework, as the properties of continuous functions differ significantly from those of discrete functions.

Summation vs. Integration: The current approach relies on summation over discrete sets, which is not directly applicable to continuous distributions. This necessitates the development of integration techniques that can handle the continuous nature of probability distributions, ensuring that the semantics of probabilistic programs remain valid.

To address these limitations, the following strategies can be implemented:

Introduce Continuous Constructs: Extend the language to include constructs for defining continuous random variables and distributions, such as Gaussian or exponential distributions. This would allow for a more comprehensive representation of probabilistic behaviors.

Adapt Fixed-Point Theorems: Develop new fixed-point theorems or adapt existing ones to work within the context of continuous functions. This may involve leveraging concepts from measure theory and functional analysis to ensure that the semantics of continuous probabilistic programs are well-defined.

Utilize Integration Techniques: Implement integration techniques within the verification framework to replace summation, allowing for the evaluation of expected values and probabilities in continuous settings. This would enhance the language's capability to model and verify systems that operate under continuous uncertainty.
By addressing these limitations, the probabilistic programming language can become more robust and applicable to a wider range of scenarios, particularly in fields such as robotics and machine learning, where continuous distributions are often encountered.

Can the unique fixed-point theorem be generalized to a broader class of probabilistic programs beyond just loops?

Yes, the unique fixed-point theorem can be generalized to a broader class of probabilistic programs beyond just loops. The current application of the unique fixed-point theorem primarily focuses on the semantics of loops, where the theorem establishes a relationship between the least fixed point (lfp) and the greatest fixed point (gfp) of a function representing the loop's behavior. However, the underlying principles of fixed-point theory can be extended to encompass a wider range of constructs in probabilistic programming.

Generalization to Other Constructs: The unique fixed-point theorem can be applied to other constructs such as conditionals, sequential compositions, and even probabilistic choices. By defining appropriate functions that capture the semantics of these constructs, we can establish fixed-point relationships that allow for reasoning about their behavior in a probabilistic context.

Broader Class of Programs: The generalization can include programs that exhibit complex interactions, such as those involving both nondeterminism and concurrency. By extending the fixed-point framework to accommodate these interactions, we can derive meaningful properties about the overall behavior of the program, including termination, expected outcomes, and probabilistic guarantees.

Mathematical Foundations: The generalization relies on the mathematical foundations of complete lattices and Scott continuity, which are applicable to a wide range of functions beyond those representing loops. By ensuring that the functions defined for various constructs are monotonic and continuous, we can leverage the unique fixed-point theorem to derive insights into the behavior of these programs.

Mechanization in Theorem Provers: The mechanization of the generalized fixed-point theorem in theorem provers like Isabelle/UTP would facilitate automated reasoning about a broader class of probabilistic programs. This would enhance the verification capabilities of the framework, allowing for the analysis of more complex systems.

In conclusion, by generalizing the unique fixed-point theorem to encompass a broader class of probabilistic programs, we can enhance the expressiveness and verification capabilities of the probabilistic programming language, making it more applicable to real-world scenarios that involve diverse probabilistic behaviors.