Bridging Proofs and Programs: A New Theory

In the world of computer science and programming, there are two main tasks that people often work on: creating programs and proving that these programs work as intended. Over time, researchers have noticed that there’s a strong connection between these two tasks. In simple terms, you can think of proofs as programs that connect what we assume (our inputs) to what we want to show (our conclusions). Likewise, programs can be seen as proofs that demonstrate certain claims made by types. This leads us to a concept known as the Curry-Howard correspondence, which suggests that proofs and programs relate closely to each other.

#The Challenge

Despite the interesting relationship between proofs and programs, there are some challenges. Proofs mainly focus on validating claims, while programs are all about performing tasks and transforming information. For instance, some programming languages might not have clear matches in logic, yet they still impact our world due to their computational abilities.

To address this gap, we introduce a new type theory that divides typing rules into two levels: one for logic and another for programs. This separation allows us to refine our understanding of proofs and programs, leading to a more accurate Reflection of their uses.

#Overview of the Two-Level Type Theory

This new type theory combines two essential components: linearity and dependency. We establish a system where typing rules fall into separate categories based on their function-one for logical reasoning and another for practical programming.

#Logical Level

In the logical level, the focus is on forming propositions (or types) and their proofs (the terms that carry these types). This level is similar to standard dependent type theory, which means it also enjoys strong properties that make reasoning easier.

#Program Level

On the other hand, the program level introduces rules that are more concerned with computational resources. This level allows us to include considerations about how resources are used when writing programs. To put it simply, it’s like creating a set of rules that ensures programs can be run efficiently without wasting resources.

#Key Features

#Irrelevancy

One important concept we introduce is "irrelevancy." Essentially, we can say that proofs and types found within programs can be erased without affecting how the program operates. In more straightforward terms, you can think of this as being able to discard certain details while keeping the primary function intact.

#Operational Semantics

We developed a specific way to evaluate these programs, called heap-based operational semantics. This method ensures that programs always make progress during execution and manage memory cleanly. In simpler terms, the programs created with this theory will run smoothly without getting stuck or running out of memory.

#Reflection

The separation of proof and program levels means that programs can be reflected back into the logical level. This valuable feature allows programmers to prove properties about their programs using a unified language.

#Contributions Made by the New Theory

We are proud to present the following contributions through this new type theory:

1. Designing a Two-Level System: We have created a type system that divides rules into logical and program levels, allowing for more precise characterizations of both proofs and programs.

2. Studying Properties of Each Level: We have explored the properties of both levels, showing that the logical level has strong characteristics that make it suitable for reasoning tasks.

3. Building a Sound Erasure Procedure and Heap Semantics: Our type-directed approach helps us to remove unnecessary information, ensuring that extracted programs run without issues.

4. Formal Verification: All the concepts we introduced have been formalized and verified in a well-known proof assistant called Coq.

5. Implementation: We developed a compiler in OCaml that translates our new type theory programs into C, demonstrating its practical application.

#Structure of the Two-Level Type Theory

The syntax of our theory is relatively straightforward. The typing rules can be categorized into two judgments: logical and program. The logical judgment determines whether a term has a certain type based on a logical context, while the program judgment does so based on both logical and program contexts.

#Logical and Program Typing

Unlike traditional theories, our system uses two distinct types: linear and non-linear. A linear type means that it must be used exactly once, while a non-linear type can be used multiple times. This distinction helps prevent certain types of errors.

#Program Context

The program context is a sequence that keeps track of variables, their types, and sorts. Here, careful management is vital, as it helps ensure that resources are used properly and that no unnecessary duplication occurs.

#Key Rules and Behavior

#Typing Rules at the Logical Level

In the logical level, we set out rules for how to form types and their corresponding terms. These rules might seem redundant, but they’re crucial for understanding how logical proofs interrelate with programs.

#Program Typing Rules

When it comes to typing programs, we do not have rules like those at the logical level. Instead, the typing rules focus on ensuring that the context of the program is well-formed. It’s essential to understand how we can control the usage of resources in our programs.

#Erasure Procedure

We developed an erasure procedure that removes unnecessary type annotations and irrelevant terms from programs. The important takeaway here is that the extracted programs maintain their computational properties, ensuring they will not get stuck during execution.

#Operational Semantics of the Theory

Moving forward, we focus on the dynamic behavior of our type theory. We define two operational semantics to govern the logical and program levels.

#Logical Reductions

In the logical level, reductions behave in a standard way, allowing various strategies for evaluating terms. This flexibility ensures that we can verify equalities effectively.

#Program Reductions

On the program level, we utilize a call-by-value approach. This means that the evaluation happens eagerly, ensuring that resources are consumed correctly.

#Meta-Theoretical Properties

The next step is to examine the meta-theoretical properties of the new type theory. This analysis helps us understand how the logical and program levels behave.

#Logical Theories

At the logical level, we show that reductions do not depend on a fixed evaluation order. We prove that terms maintain their validity throughout the reduction process, making reasoning easier.

#Program Theories

In this section, we explore how the program context can be managed independently. We also introduce the concept of reflection, allowing us to transfer well-typed programs into the logical level for further analysis.

#Extensions and Practical Applications

As we move forward, we explore various extensions to the core language and how they can be practically applied.

#Propositional Equality

We introduce a method to express equality between terms. This concept allows us to reason about two terms being equal, enabling more thorough checks in our programs.

#Subset Types

The concept of subset types allows us to refine programs based on specific properties. This helps us create more precise types that encapsulate constraints on what can be included within them.

#Inductive Types and Concurrency

We provide support for user-defined inductive types, which enables the creation of complex data structures. Additionally, we introduce dependent session types that facilitate communication between concurrent processes, allowing developers to create more sophisticated applications.

#Summary of the New Type Theory

In summary, we present a two-level dependent type theory that provides a comprehensive framework for understanding and producing proofs and programs. By effectively managing logical and program levels, we can ensure that resources are used appropriately while maintaining clarity in reasoning.

#Future Directions

We aim to explore further extensions that could enhance our type theory. Our goal is to bridge the gap between theorem proving and practical programming, creating a more cohesive framework that benefits both fields. This includes investigating multiparty communication and verifying cryptographic protocols, ultimately striving to create a versatile tool for developers.

With this new understanding and approach, we invite readers to think about how type theory and programming can work together seamlessly to advance technology and software development in the future.