Functional programs as compressed data

Abstract

We propose an application of programming language techniques to lossless data compression, where tree data are compressed as functional programs that generate them. This “functional programs as compressed data” approach has several advantages. First, it follows from the standard argument of Kolmogorov complexity that the size of compressed data can be optimal up to an additive constant.

Secondly, a compression algorithm is clean: it is just a sequence of β-expansions (i.e., the inverse of β-reductions) for λ-terms. Thirdly, one can use program verification and transformation techniques (higher-order model checking, in particular) to apply certain operations on data without decompression. In this article, we present algorithms for data compression and manipulation based on the approach, and prove their correctness. We also report preliminary experiments on prototype data compression/transformation systems.

Appendix: Proofs for Sect. 2

In this section, we shall prove the following lemma, which corresponds to the “only if” direction of Theorem 1.

Lemma 11

If ∅⊢M:o, then there exists a tree T such that $M\longrightarrow ^{*}_{\beta}T$.

As sketched in the proof of Theorem 1, the lemma follows from more or less standard properties of intersection types [2, 50]. Nevertheless, we provide a self-contained proof of Lemma 11 below to familiarize readers with the type-based transformation techniques used in Sect. 4.2.

We prove the lemma by using the strong normalization of the simply-typed λ-calculus. To this end, we define a type-based transformation relation Γ⊢M:τ⟹N, which transforms a well-typed λ-term M (in the intersection type system) into a corresponding term N of the simply-typed λ-calculus. It is defined by the following extension of typing rules.

Here, we assume that the simply-typed λ-calculus has a base type ⋆(≠o), inhabited by ( ). The idea of the translation to the simply-typed λ-calculus is to replicate each function argument according to its type. Thus, we replicate a formal parameter x to $x_{\tau_{1}},\ldots,x_{\tau_{n}}$ in rule STr-Abs above, and accordingly duplicate an actual parameter M ₂ to N _2,1,…,N _2,n in rule STr-App. The dummy formal (actual, resp.) parameter x _⋆ (( ), resp.) is added in STr-AbsT (STr-AppT, resp.) to enforce that the transformation preserves the “shape” of a term. Note that without the dummy parameter, an application M ₁ M ₂ would be transformed to a term of arbitrary shape (when n=0 in STr-AppT).

We first show that the result of the transformation is simply-typed. We write Γ⊢ _ST N:κ for the standard type judgment relation for the simply-typed λ-calculus, where the syntax of types are generated by

$$\kappa::= \mathtt {o}\mid \star \mid\kappa_1 \rightarrow \kappa_2. $$

We define the translation ⋅^† from intersection types to simple types by:

$$\begin{aligned} &{ \mathtt {o}}^\dagger = \mathtt {o}\qquad {(\top \rightarrow \tau)}^\dagger = \star \rightarrow {\tau}^\dagger \\ &{(\tau_1\land\cdots\land\tau_k \rightarrow \tau)}^\dagger = {\tau_1}^\dagger \rightarrow \cdots \rightarrow {\tau_k}^\dagger \rightarrow {\tau}^\dagger \mbox{(if $k\geq1$)}. \end{aligned}$$

We extend the translation to type environments by:

$${\varGamma}^\dagger = \{x_{\tau} \mathbin {:}{\tau}^\dagger \mid x\mathbin {:}\tau\in \varGamma\}. $$

We assume that $x_{\tau} = x'_{\tau'}$ if and only if x=x′ and τ=τ′ so that Γ ^† is a simple type environment.

Lemma 12

If Γ⊢M:κ⟹N, then Γ ^†⊢ _ST N:κ ^†.

Proof

This follows by straightforward induction on the derivation of Γ⊢M:κ⟹N. □

Lemma 13

Suppose $x\not\in \mathit {dom}(\varGamma)$. If Γ,x:τ ₁,…,x:τ _k ⊢M:τ⟹N and Γ⊢K:τ _i ⟹N _i for each i∈{1,…,k}, then $\varGamma \vdash [K/x]M:\tau \Longrightarrow [N_{1}/x_{\tau_{1}},\ldots,N_{k}/x_{\tau_{k}}]N$.

Proof

This follows by induction on the structure of M.

• Case M=x: In this case, $N= x_{\tau_{i}}$ and τ=τ _i for some i∈{1,…,k}. The result follows immediately from Γ⊢K:τ _i ⟹N _i .

• Case M=y(≠x): In this case, we have y:τ∈Γ and N=y _τ . Thus, [K/x]M=y and $[N_{1}/x_{\tau_{1}},\ldots,N_{k}/x_{\tau_{k}}]N=y_{\tau}$. By using STr-Var, we have $\varGamma \vdash [K/x]M:\tau \Longrightarrow [N_{1}/x_{\tau_{1}},\ldots,N_{k}/x_{\tau_{k}}]N$ as required.

• Case M=a: The result follows immediately, as M=N=a and τ=o ^Σ(a)→o.

• Case M=λy.M ₀: We can assume that y≠x without loss of generality. By the assumption, we have:

  $$\begin{array}{l} \varGamma,y\mathbin {:}\tau'_1,\ldots,y\mathbin {:}\tau'_\ell \vdash M_0:\tau' \Longrightarrow N_0\\ (\ell\geq1\land N = \lambda y_{\tau'_1}.\cdots\lambda y_{\tau'_\ell }.N_0) \lor(\ell=0\land N = \lambda y_{\star }.N_0)\\ \tau= \tau'_1\land\cdots\land\tau'_\ell \rightarrow \tau' \end{array} $$

  By the induction hypothesis, we have $\varGamma,y\mathbin {:}\tau_{1},\ldots,y\mathbin {:}\tau_{\ell} \vdash [K/x]M_{0}:\tau' \Longrightarrow [N_{1}/x_{\tau_{1}},\ldots, N_{k}/x_{\tau_{k}}]N_{0}$. By applying STr-Abs or STr-AbsT, we obtain Γ⊢[K/x]M:τ⟹N as required.

• Case M=M ₁ M ₂: By the assumption, we have:

  $$\begin{array}{l} \varGamma \vdash M_1:\tau'_1\land\cdots\land\tau'_\ell \rightarrow \tau \Longrightarrow N_1'\\ \varGamma \vdash M_2:\tau'_i \Longrightarrow N_{2,i} \mbox{ for each $i\in \{1,\ldots,\ell\}$}\\ (\ell\geq1 \land N = N'_1\, N_{2,1} \cdots N_{2,k})\lor(\ell=0\land N = N'_1\,(\,)) \end{array} $$

  By the induction hypothesis, we have:

  $$\begin{array}{l} \varGamma \vdash [K/x]M_1:\tau_1\land\cdots \land\tau_k\rightarrow \tau \Longrightarrow [N_1/x_{\tau_1},\ldots,N_k/x_{\tau_k}]N'_1\\ \varGamma \vdash [K/x]M_2:\tau_i \Longrightarrow [N_1/x_{\tau_1},\ldots,N_k/x_{\tau_k}]N_{2,i} \mbox{ for each $i\in \{1,\ldots,k\}$}\\ \end{array} $$

  By applying STr-App or STr-AppT, we obtain $\varGamma \vdash [K/x]M:\tau \Longrightarrow [N_{1}/x_{\tau _{1}},\ldots, N_{k}/x_{\tau_{k}}]N$ as required. □

The following is a special case of Lemma 13 above, where k=0.

Corollary 1

If $x\not\in \mathit {dom}(\varGamma)$ and Γ⊢M:τ⟹N, then Γ⊢[K/x]M:τ⟹N holds for any K.

We are now ready to show the main lemmas (Lemmas 14 and 15 below), which say that if Γ⊢M:τ⟹N, then reductions of M and N can be simulated by each other.

Lemma 14

If Γ⊢M:τ⟹N and M⟶ _β M′, then there exists N′ such that Γ⊢M′:τ⟹N′ with $N\longrightarrow ^{*}_{\beta}N'$.

Proof

This follows by easy induction on the derivation of Γ⊢M:τ⟹N. We omit details as the proof is similar to that of Lemma 5 in Sect. 4.2.1. □

Lemma 15

If Γ⊢M:τ⟹N and N⟶ _β N′, then there exist M′ and N″ such that Γ⊢M′:τ⟹N″ with $M\longrightarrow ^{*}_{\beta}M'$ and $N'\longrightarrow ^{*}_{\beta}N''$.

Proof

The proof proceeds by induction on derivation of ∅⊢M:τ⟹N, with case analysis on the last rule used. By the assumption N⟶ _β N′, the last rule cannot be ST-Var or ST-Const.

• Case STr-Abs: In this case, we have:

  $$\begin{array}{l} \varGamma,x\mathbin {:}\tau_1,\ldots,x\mathbin {:}\tau_k \vdash M_1\mathbin {:}\tau'\Longrightarrow N_1\\ M = \lambda x.M_1\qquad N = \lambda x_{\tau_1}.\cdots\lambda x_{\tau _k}.N_1\qquad N' = \lambda x_{\tau_1}.\cdots\lambda x_{\tau_k}.N_1'\\ N_1 \longrightarrow _\beta N_1'\\ \tau= \tau_1\land\cdots\land\tau_k\rightarrow \tau' \end{array} $$

  By the induction hypothesis, we have $M_{1}'$ and $N_{1}''$ such that $\varGamma,x\mathbin {:}\tau_{1},\ldots,x\mathbin {:}\tau_{k} \vdash M_{1}'\mathbin {:}\tau'\Longrightarrow N_{1}''$ with $M_{1}\longrightarrow ^{*}_{\beta}M_{1}'$ and $N_{1}'\longrightarrow ^{*}_{\beta}N_{1}''$. Thus, the required properties hold for $M'=\lambda x.M_{1}'$ and $N'' = \lambda x_{\tau_{1}}.\cdots\lambda x_{\tau_{k}}.N_{1}'$.

• Case STr-AbsT: Similar to the case STr-Abs.

• Case STr-App: In this case, we have:

  $$\begin{array}{l} \varGamma \vdash M_1:\tau_1\land\cdots\land\tau_k\rightarrow \tau \Longrightarrow N_1\quad (k\geq1)\\ \varGamma \vdash M_2: \tau_i\Longrightarrow N_{2,i} \mbox{ for each $i\in \{1,\ldots ,k\}$}\\ M = M_1M_2\qquad N = N_1\, N_{2,1}\cdots N_{2,k}\\ \end{array} $$

  By the assumption $N\longrightarrow ^{*}_{\beta}N'$, there are three cases to consider:

  • (1) $N_{1}\longrightarrow ^{*}_{\beta}N_{1}'$ with $N'=N_{1}'\, N_{2,1}\cdots N_{2,k}$.

  • (2) $N_{2,j}\longrightarrow ^{*}_{\beta}N_{2,j}$ with $N'=N_{1}\,N_{2,1}\cdots N_{2,j-1}\,N'_{2,j}\,N_{2,j+1}\cdots N_{2,k}$.

  • (3) $N_{1} = \lambda x.N_{1}'$ with $N' = ([N_{2,1}/x]N_{1}') N_{2,2}\cdots N_{2,k}$.

  The result for case (1) follows immediately from the induction hypothesis. In case (2), by the induction hypothesis, we have $\varGamma \vdash M'_{2}:\tau_{j}\Longrightarrow N''_{2,j}$ with $M_{2}\longrightarrow ^{*}_{\beta}M_{2}'$ and $N'_{2,j}\longrightarrow ^{*}_{\beta}N''_{2,j}$ for some $M'_{2}$ and $N''_{2,j}$. By Lemma 14, for each i∈{1,…,k}∖{j}, there exists $N''_{2,i}$ such that $\varGamma \vdash M'_{2}:\tau_{i}\Longrightarrow N''_{2,i}$ and $N_{2,i}\longrightarrow ^{*}_{\beta}N''_{2,i}$. Let $M' = M_{1}M'_{2}$ and $N''=N_{1} N''_{2,1}\cdots N''_{2,k}$. Then we have Γ⊢M′:τ⟹N″ with $M\longrightarrow ^{*}_{\beta}M'$ and $N'\longrightarrow ^{*}_{\beta}N''$ as required.

  In case (3), by the transformation rules, Γ⊢M ₁:τ ₁∧⋯∧τ _k →τ⟹N ₁ must have been derived from STr-Abs, so that we have:

  $$\begin{array}{l} M_1 = \lambda y.M_3\\ \varGamma,y\mathbin {:}\tau_1,\ldots,y\mathbin {:}\tau_k \vdash M_3:\tau \Longrightarrow N_3\\ (\lambda x.N_1') = (\lambda y_{\tau_1}.\cdots\lambda y_{\tau_k}.N_3) \end{array} $$

  Let M′=[M ₂/x]M ₃ and $N''=[N_{2,1}/y_{\tau_{1}},\ldots ,N_{2,k}/y_{\tau_{k}}]N_{3}$. Then we have $M\longrightarrow ^{*}_{\beta}M'$ and $N'\longrightarrow ^{*}_{\beta}N''$. Furthermore, by Lemma 13, we have Γ⊢M′:τ⟹N″ as required.

• Case STr-AppT: In this case, we have:

  $$\begin{array}{l} \varGamma \vdash M_1:\top \Longrightarrow N_1\\ M = M_1M_2\qquad N = N_1\, (\,)\\ \end{array} $$

  By the assumption $N\longrightarrow ^{*}_{\beta}N'$, there are two cases to consider:

  • (1) $N_{1}\longrightarrow ^{*}_{\beta}N_{1}'$ with $N'=N_{1}'\, (\,)$.

  • (2) $N_{1} = \lambda x.N_{1}'$ with $N' = [(\,)/x]N_{1}'$.

  The result for case (1) follows immediately from the induction hypothesis. In case (2), Γ⊢M ₁:⊤τ⟹N ₁ must have been derived from STr-AbsT, so that we have:

  $$\begin{array}{l} M_1 = \lambda x.M_3\qquad \varGamma \vdash M_3: \tau \Longrightarrow N_1'\qquad x\not\in \mathit {dom}(\varGamma) \end{array} $$

  By Lemma 12, we have ${\varGamma}^{\dagger} \vdash _{\mathtt {ST}}N_{1}'$, so that x does not occur in $N_{1}'$. Thus, $N' = [(\,)/x]N_{1}' = N_{1}'$. Let M′ be [M ₂/x]M ₃ and N″ be N′. Then we have $M\longrightarrow ^{*}_{\beta}M'$ and $N'\longrightarrow ^{*}_{\beta}N''$. Furthermore, by Lemma 1 we have Γ⊢M′:τ⟹N″ as required. □

Lemma 16

If ∅⊢M:o⟹N and N is a β-normal form, then M is a tree and M=N.

Proof

The proof proceeds by induction on the structure of N. By Lemma 12, we have ∅⊢N:o. Since N does not contain any β-redex, N must be of the form a  N ₁ ⋯ N _k , where k may be 0. By the transformation rules, we have:

$$\begin{array}{l} M = a\,M_{1}\,\cdots\,M_{k}\\ \varSigma(a)=k\\ \emptyset \vdash M_i:\mathtt {o}\Longrightarrow N_i \mbox{ for each $i\in \{1,\ldots,k\}$} \end{array} $$

By the induction hypothesis, M _i is a tree and M _i =N _i . Thus, M is also a tree and M=N as required. □

We are now ready to prove Lemma 11.

Proof of Lemma 11

If ∅⊢M:o, then by the transformation rules, there exists N such that ∅⊢M:o⟹N. By Lemma 12, we have ∅⊢ _ST N:o. By the strong normalization property of the simply-typed λ-calculus, there exists a β-normal form N′ such that $N\longrightarrow ^{*}_{\beta}N'$. By Lemma 15, there exists M′ such that $M\longrightarrow ^{*}_{\beta}M'$ and ∅⊢M′:o⟹N′. By Lemma 16, M′ is a tree. □