Intrinsically typed linear logic in Coq

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In this post, I will present an encoding of linear logic (linear \(\lambda\)-calculus) using an intrinsically typed syntax using Coq proof assistant. Dependent types are a great way to enforce, statically, invariants on any sort of data structure, like in our case, a syntax tree.

Before diving into the technical details of the Coq code, I believe that we should spent some time on linear logic and, specially, on how it is related to classic / intuitionistic logic.

Prelude: Intuitionistic logic - the logic of true truth

Computer scientists had a good understanding of classical logics, since it pervasive through several areas like artificial intelligence, digital circuit design and programming language semantics.

The syntax of classical logic propositions is presented below.

\[ P ::= V \,\mid\, P \land P \,\mid\, P \to P \,\mid\, P \lor P \]

In introductory courses on discrete mathematics, logics is usually presented using the Tarsky semantics, in which, each connective is denoted by its “truth-table”. Here, we will represent each boolean function by its semantics using natural deduction. In natural deduction, we specify the semantics of formulas using judgments. We will follow common practice and let \(\Gamma \vdash P\) denote that formulat \(P\) can be deduced from the sequence of formulas \(\Gamma\) using the rules of natural deduction, which are specified next.

The first rule is identity, which specifies the rather obvious fact that a formula \(P\) is provable from itself. \[ \dfrac{}{P \vdash P}(id) \] The next set of rules are known as structural, since they show how we can manipulate the sequence of assumptions in a proof. \[ \begin{array}{ccc} \dfrac{\Gamma,\Delta \vdash P}{\Delta,\Gamma \vdash P}(exchange) & \dfrac{\Gamma, A, A \vdash B}{\Gamma, A \vdash B}(contraction) & \dfrac{\Gamma \vdash B}{\Gamma,A \vdash B}(weakening) \end{array} \] The rule exchange specified that provability is closed under assumptions permutations, contraction allows the duplication of a certain hypothesis and weakening allows us to discard some formula from the assumptions. Most presentations of natural deduction didn’t mention such structual rules because they can be avoided if we consider a set of assumptions instead of a sequence. But, the true reason behind linear logic only become apparent if we use sequences.

For connectives, we have two kind of rules: introduction and elimination. Introduction rules allow us to conclude a formula having a certain connective, while elimination allow us to use a formula with a certain connective. The first connective we consider is conjunction, whose rules are shown below. \[ \begin{array}{cc} \dfrac{\Gamma \vdash A\,\,\,\,\,\,\,\,\,\,\Delta \vdash B}{\Gamma,\Delta \vdash A \land B}(\land_I) & \dfrac{\Gamma \vdash A \land B \,\,\,\,\,\,\,\,\,\,\Delta,A,B\vdash C}{\Gamma,\Delta \vdash C}(\land_E) \end{array} \] The first rule says that if we have a proof of \(A\) and another for \(B\) we can build a proof for \(A\land B\). The conjunction elimination rule specifies that if we have a proof of \(A\land B\) we can use both \(A\) and \(B\) as additional assumptions for proving another formula, \(C\).

For disjunction, we have two introduction rules, since we can prove \(A \lor B\) from a proof of \(A\) or from a proof of \(B\).

\[ \begin{array}{cc} \dfrac{\Gamma\vdash A}{\Gamma\vdash A \lor B}(\lor_{IL}) & \dfrac{\Gamma\vdash B}{\Gamma\vdash A \lor B}(\lor_{IR}) \end{array} \]

The elimination specifies that we can conclude a formula \(C\) from \(A \lor B\) if we can prove \(C\) from \(A\) and from \(B\).

\[ \dfrac{\Gamma\vdash A\lor B \,\,\,\,\,\,\,\,\,\, \Delta,A \vdash C\,\,\,\,\,\,\,\,\,\, \Delta, B \vdash C} {\Gamma,\Delta\vdash A \lor B}(\lor_{IL}) \]

The last set of rules is for implication. Implication elimination rule (also known as modus ponens) reflects our intuitive understanding of deduction. We can deduce \(B\) from proofs of \(A \to B\) and \(A\). Implication introduction specifies that we can deduce \(A\to B\) from a proof of \(B\) which uses \(A\) as an additional assumption. \[ \begin{array}{cc} \dfrac{\Gamma,A \vdash B}{\Gamma \vdash A \to B}(\to_I) & \dfrac{\Gamma \vdash A \to B \,\,\,\,\,\,\,\,\,\, \Delta\vdash A}{\Gamma,\Delta \vdash B}(\to_E) \end{array} \]

The previous rules for the propositional logic form what is called intuitionistic logic, in which the semantics of formulas is based on evidence,instead of relying on a vacuously notion of truth, like classical logic and its axiom of excluded middle. Next, we present linear logic which restrict the usage of structural rules aiming to control assumptions as resources.

Linear logic: the logic of food

Traditional logics deal with truth and it (in principle) is forever. If something is true, according to classical (and intuitionistic logic) it will remain true forever. As an example of how such view of validity can be problematic, consider the following sentences:

  • If eat a cake then I’ll become full.
  • I have a cake

Using traditional logics, these can be represented by the following formula, where proposition \(A\) represents “I have a cake” and \(B\) denotes “I’m full”:

  • \(A \to B\)
  • \(B\)

Note that, using natural deduction, we can prove the following \[ {A\to B, A}\vdash A \land B \] the formula in the conclusion means that you have a cake and is full. So far, so good. But, what if you have only one cake? It is not the case that you are full, after eat it, and still have it: it is cheating!

The main idea of linear logic is that it imposes a strict control on the usage of hypothesis. In such logic, it is not possible to reuse assumptions as you wish.