# System L

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System L is a family of syntax for a variety of variants of linear logic, inspired from classical calculi such as $\bar\lambda\mu\tilde\mu$-calculus. These, in turn, stem from the study of abstract machines for λ-calculus. In this realm, polarization and focalization are features that appear naturally. Positives are typically values, and negatives pattern-matches. Contraction and weakening are implicit.

We present here a system for explicitely polarized and focalized linear logic. Polarization classifies terms and types between positive and negative; focalization separates values from non-values.

## Definitions

Positive types: $P ::= 1 \mid P_1 \otimes P_2 \mid 0 \mid P_1 \oplus P_2 \mid \shpos N \mid \oc N$

Negative types: $N ::= \bot \mid N_1 \parr N_2 \mid \top \mid N_1 \with N_2 \mid \shneg P \mid \wn P$

Positive values: $v^+ ::= x^+ \mid () \mid (v_1^+, v_2^+) \mid inl(v^+) \mid inr(v^+) \mid \shpos t^- \mid \mu(\wn x^+).c$

Positive terms: $t^+ ::= v^+ \mid \mu x^-.c$

Negative terms: $t^- ::= x^- \mid \mu x^+.c \mid \mu().c \mid \mu(x^+, y^+).c \mid \mu [\cdot] \mid \mu[inl(x^+).c_1 \mid inr(y^+).c_2] \mid \mu(\shpos x^-).c \mid \wn v^+$

Commands: $c ::= \langle t^+ \mid t^- \rangle$

## Typing

There are as many typing sequents classes as there are terms classes. Typing of positive values corresponds to focalized sequents, and commands are cuts.

Positive values: sequents are of the form $\vdash \Gamma :: v^+ : P$. $\LabelRule{\rulename{ax}^+} \NulRule{\vdash x^+:P\orth :: x^+: P} \DisplayProof$ $\LabelRule{1} \NulRule{\vdash \ :: () : 1} \DisplayProof$ $\AxRule{\vdash \Gamma_1 :: v_1^+ : P_1} \AxRule{\vdash \Gamma_2 :: v_2^+ : P_2} \LabelRule{\rulename{\otimes}} \BinRule{\vdash\Gamma_1, \Gamma_2 :: (v_1^+, v_2^+) : P_1\otimes P_2} \DisplayProof$ $\AxRule{\vdash \Gamma :: v^+ : P_1} \LabelRule{\rulename{\oplus_1}} \UnaRule{\vdash\Gamma :: inl(v^+) : P_1\oplus P_2} \DisplayProof \qquad \AxRule{\vdash \Gamma :: v^+ : P_2} \LabelRule{\rulename{\oplus_2}} \UnaRule{\vdash\Gamma :: inr(v^+) : P_1\oplus P_2} \DisplayProof$ $\AxRule{\vdash \Gamma \mid t^- : N} \LabelRule{\shpos} \UnaRule{\vdash\Gamma :: \shpos t^- : \shpos N} \DisplayProof$ $\AxRule{c \vdash \wn\Gamma, x^+ : N} \LabelRule{\oc} \UnaRule{\vdash\wn\Gamma :: \mu(\wn x^+).c : \oc N} \DisplayProof$

Positive terms: sequents are of the form $\vdash\Gamma\mid t^+:P$. $\AxRule{\vdash \Gamma :: v^+ : P} \LabelRule{\rulename{foc}} \UnaRule{\vdash\Gamma \mid v^+ : P} \DisplayProof$ $\AxRule{c \vdash \Gamma, x^- : P} \LabelRule{\rulename{\mu^-}} \UnaRule{\vdash\Gamma \mid\mu x^-.c : P} \DisplayProof$

Negative terms: sequents are of the form $\vdash\Gamma\mid t^-:N$. $\LabelRule{\rulename{ax}^-} \NulRule{\vdash x^-:N\orth \mid x^-: N} \DisplayProof$ $\AxRule{c\vdash \Gamma, x^+: N} \LabelRule{\mu^+} \UnaRule{\vdash\Gamma \mid \mu x^+.c : N} \DisplayProof$ $\AxRule{c \vdash \Gamma} \LabelRule{\bot} \UnaRule{\vdash \Gamma \mid \mu().c : \bot} \DisplayProof$ $\AxRule{c\vdash \Gamma, x^+: N_1, y^+: N_2} \LabelRule{\rulename{\parr}} \UnaRule{\vdash\Gamma \mid \mu(x^+, y^+).c : N_1 \parr N_2} \DisplayProof$ $\LabelRule{\rulename{\top}} \NulRule{\vdash \Gamma \mid \mu[\cdot] : \top} \DisplayProof$ $\AxRule{c_1\vdash \Gamma, x^+:N_1} \AxRule{c_2\vdash \Gamma, y^+:N_2} \LabelRule{\rulename{\with}} \BinRule{\vdash\Gamma \mid \mu[inl(x^+).c_1 \mid inr(y^+).c_2] : N_1 \with N_2} \DisplayProof$ $\AxRule{c\vdash \Gamma, x^-: P} \LabelRule{\shneg} \UnaRule{\vdash\Gamma \mid \mu(\shpos x^-).c : \shneg P} \DisplayProof$ $\AxRule{\vdash \Gamma :: v^+ : P} \LabelRule{\wn} \UnaRule{\vdash\Gamma \mid \wn v^+ : \wn P} \DisplayProof$

Commands: $\AxRule{\vdash \Gamma \mid t^+ : P} \AxRule{\vdash \Delta \mid t^- : P\orth} \LabelRule{\rulename{cut}} \BinRule{\langle t^+ \mid t^-\rangle\vdash\Gamma, \Delta} \DisplayProof$ $\AxRule{c \vdash \Gamma} \LabelRule{\rulename{wkn}} \UnaRule{c \vdash\Gamma, x^+: \wn P} \DisplayProof$ $\AxRule{c \vdash \Gamma, x_1^+:\wn P, x_2^+:\wn P} \LabelRule{\rulename{ctr}} \UnaRule{c[x_1^+ := x^+, x_2^+ := x^+] \vdash\Gamma, x^+: \wn P} \DisplayProof$

## Reduction rules $\langle v^+ \mid \mu x^+.c \rangle \rightarrow c[ x^+ := v^+]$ $\langle \mu x^-.c \mid t^- \rangle \rightarrow c[x^- := t^-]$ $\langle () \mid \mu().c \rangle \rightarrow c$ $\langle (v_1^+, v_2^+) \mid \mu(x^+, y^+).c \rangle \rightarrow c[x^+ := v_1^+, y^+ := v_2^+]$ $\langle inl(v^+) \mid \mu[inl(x^+).c_1 \mid inr(y^+).c_2] \rangle \rightarrow c_1[x^+ := v^+]$ $\langle inr(v^+) \mid \mu[inl(x^+).c_1 \mid inr(y^+).c_2] \rangle \rightarrow c_2[y^+ := v^+]$ $\langle \shpos t^- \mid \mu(\shpos x^-).c \rangle \rightarrow c[x^- := t^-]$ $\langle \mu(\wn x^+).c \mid \wn v^+ \rangle \rightarrow c[x^+ := v^+]$