Semantics for First-Order Affine Inductive Data Types via Slice Categories

Abstract

Affine type systems are substructural type systems where copying of information is restricted, but discarding of information is permissible at all types. Such type systems are well-suited for describing quantum programming languages, because copying of quantum information violates the laws of quantum mechanics. In this paper, we consider a first-order affine type system with inductive data types and present a novel categorical semantics for it. The most challenging aspect of this interpretation comes from the requirement to construct appropriate discarding maps for our data types which might be defined by mutual/nested recursion. We show how to achieve this for all types by taking models of a first-order linear type system whose atomic types are discardable and then presenting an additional affine interpretation of types within the slice category of the model with the tensor unit. We present some concrete categorical models for the language ranging from classical to quantum. Finally, we discuss potential ways of dualising and extending our methods and using them for interpreting coalgebraic and lazy data types.

A Proof of Theorem 42

As a simple consequence of Soundness (Theorem 40), we get the following corollary.

Corollary 43

Let $\vdash \langle \cdot \rangle \ M\ \langle \varSigma \rangle$ be a closed term. If $M \Downarrow \text { then } {\llbracket M \rrbracket } \ne \perp .$

Proof

Assume for contradiction ${\llbracket M \rrbracket } = \perp .$ Let $M' \equiv M;\ \mathbf{discard} \ \varSigma$ be the program that discards all output variables of M. Clearly, $(M'\ |\ \cdot ) \leadsto _* (\mathbf{skip} \ |\ \cdot )$ and therefore $\text {id}_I = {\llbracket M' \rrbracket } = \diamond \circ {\llbracket M \rrbracket } = \perp$, which contradicts Definition 41.    $\square$

In the remainder of this appendix, we will show that the converse implication also holds which therefore completes the proof of Theorem 42. Our proof strategy is a simplification of the proof strategy of  [17] which is in turn based on the proof strategy of [15].

We provide a brief outline of our proof strategy. In Subsect. A.1, we extend the language with finitary (or bounded) primitives for recursion and show that our extension preserves all of the fundamental properties of the language stated so far (Theorem 44). Any finitary configuration terminates in the extended language (Lemma 45) which allows us to easily prove a finitary version of adequacy (Corrolary 46). We then show that finitary configurations approximate ordinary ones both operationally (Lemma 48) and denotationally (Lemma 49) which then allows us to finish the proof.

1.1 A.1 Language Extension

We extend the term language by adding two new terms:

where $n \ge 0$ ranges over the natural numbers. Well-formed configuration are defined in the same way as before. We extend the operational semantics by adding the following rules:

We extend the denotational semantics by adding interpretations for the newly added terms:

$$\begin{aligned} {\llbracket \vdash \langle \varGamma \rangle \ \perp _{\varGamma , \varSigma }\ \langle \varSigma \rangle \rrbracket }&:= \perp _{{\llbracket \varGamma \rrbracket }, {\llbracket \varSigma \rrbracket }} \\ {\llbracket \vdash \langle \varGamma , b: \mathbf{bit} \rangle \ \mathbf{while} ^n\ b\ \mathbf{do} \ {M}\ \langle \varGamma , b: \mathbf{bit} \rangle \rrbracket }&:= W^n_{{\llbracket M \rrbracket }}(\perp ) \end{aligned}$$

Configurations are interpreted as before. We update the notion of terminal configuration to also include configurations of the form $(\perp \ |\ V )$.

Theorem 44

([17]). Subject Reduction (Theorem 6), Progress (Theorem 8) and Soundness (Theorem 40) also hold true for the extended language (using the updated language notions).

1.2 A.2 The Proof

We shall say that a term is finitary if it does not contain any unindexed while loops and we shall say a term is ordinary if it does not contain any indexed while loops or $\perp _{\varGamma , \varSigma }$ subterms. So, an ordinary term is simply a term in the ordinary (unextended) language. Similarly, a finitary (ordinary) configuration is a configuration $(M\ |\ V )$ where M is finitary (ordinary). A finitary configuration is called as such, because all of its reduction sequences terminate.

Lemma 45

([17]). For any finitary configuration $\mathcal {C}$, there exists $n \in \mathbb {N}$, such that the length of every reduction sequence from $\mathcal {C}$ is at most n.

Note that, every terminal finitary configuration $\mathcal {T}$ has to be of the form $(\mathbf{skip} \ |\ V )$, in which case we say it is a $\mathbf{skip}$-configuration, or of the form $(\perp \ |\ V )$, in which case we say $\mathcal {T}$ is a $\perp$-configuration. For any closed term $\vdash \langle \cdot \rangle \ M\ \langle \varSigma \rangle$ in the extended language, we shall write $M \Downarrow _o$, if M terminates in the ordinary sense, i.e., if $(M\ |\ \cdot ) \leadsto _* (\mathbf{skip} \ |\ V)$, for some value assignment V.

Corollary 46

For any closed finitary term M, if ${\llbracket M \rrbracket } \ne \perp$ then $M \Downarrow _o$.

Proof

By Lemma 45, we know $(M\ |\ \cdot ) \leadsto _* (T'\ |\ V)$, where $T'$ is either skip or $\perp$. If $T' = \perp$, then by soundness, it follows ${\llbracket M \rrbracket } = \perp$. Therefore $M \Downarrow _o$.    $\square$

We proceed by defining an approximation relation $(- \blacktriangleleft -)$ between finitary terms and ordinary terms. It is the smallest relation that satisfies the rules:

We also define an approximation relation $(- \sqsubset - )$ between finitary configurations and ordinary configurations to be the smallest relation satisfying the rule:

Remark 47

If $(M'\ |\ V ) \sqsubset (M\ |\ V )$, then M and $M'$ do not contain any $\perp$ subterms.

Next, we show the relation $( - \sqsubset -)$ approximates the ordinary operational semantics.

Lemma 48

Let $\mathcal {C}$ be an ordinary configuration and $C'$ a finitary configuration with $\mathcal {C}' \sqsubset \mathcal {C}$. Assume that $\mathcal {C}' \leadsto _* \mathcal {T}$, where $\mathcal {T}$ is a $\mathbf{skip}$-configuration. Then $\mathcal {C} \leadsto _* \mathcal {T}$.

Proof

This is just a simple special case of [17, Lemma 40].    $\square$

The above lemma shows that if a finitary configuration terminates in the ordinary sense, then so does the ordinary configuration which is approximated by it. Next we show that ordinary configurations are also approximated by finitary configurations in a denotational sense.

Lemma 49

([17]). For any ordinary term M and configuration $\mathcal {C}$:

$$\begin{aligned} {\llbracket M \rrbracket }&= \bigvee _{M' \blacktriangleleft M} {\llbracket M' \rrbracket }&{\llbracket \mathcal {C} \rrbracket }&= \bigvee _{\mathcal {C}' \sqsubset \mathcal {C}} {\llbracket \mathcal {C}' \rrbracket } . \end{aligned}$$

Finally, we can state the proof of Theorem 42.

Proof of Theorem 42. By Corollary 43, it suffices to show that if ${\llbracket M \rrbracket } \ne \perp$ then $M \Downarrow _o.$ By Lemma 49, we have ${\llbracket M \rrbracket } = \bigvee _{M' \blacktriangleleft M} {\llbracket M' \rrbracket }$. Since ${\llbracket M \rrbracket } \ne \perp$, then it follows there exists a finitary term $M' \blacktriangleleft M$, such that ${\llbracket M' \rrbracket } \ne \perp$. By Corollary 46, it follows $M' \Downarrow _o$ and by Lemma 48 it follows $M \Downarrow _o$.    $\square$