TY - GEN
T1 - Proof Nets and the Linear Substitution Calculus
AU - Accattoli, Beniamino
N1 - Publisher Copyright:
© 2018, Springer Nature Switzerland AG.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - Since the very beginning of the theory of linear logic it is known how to represent the λ -calculus as linear logic proof nets. The two systems however have different granularities, in particular proof nets have an explicit notion of sharing—the exponentials—and a micro-step operational semantics, while the λ -calculus has no sharing and a small-step operational semantics. Here we show that the linear substitution calculus, a simple refinement of the λ -calculus with sharing, is isomorphic to proof nets at the operational level. Nonetheless, two different terms with sharing can still have the same proof nets representation—a further result is the characterisation of the equality induced by proof nets over terms with sharing. Finally, such a detailed analysis of the relationship between terms and proof nets, suggests a new, abstract notion of proof net, based on rewriting considerations and not necessarily of a graphical nature.
AB - Since the very beginning of the theory of linear logic it is known how to represent the λ -calculus as linear logic proof nets. The two systems however have different granularities, in particular proof nets have an explicit notion of sharing—the exponentials—and a micro-step operational semantics, while the λ -calculus has no sharing and a small-step operational semantics. Here we show that the linear substitution calculus, a simple refinement of the λ -calculus with sharing, is isomorphic to proof nets at the operational level. Nonetheless, two different terms with sharing can still have the same proof nets representation—a further result is the characterisation of the equality induced by proof nets over terms with sharing. Finally, such a detailed analysis of the relationship between terms and proof nets, suggests a new, abstract notion of proof net, based on rewriting considerations and not necessarily of a graphical nature.
U2 - 10.1007/978-3-030-02508-3_3
DO - 10.1007/978-3-030-02508-3_3
M3 - Conference contribution
AN - SCOPUS:85055411892
SN - 9783030025076
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 37
EP - 61
BT - Theoretical Aspects of Computing – ICTAC 2018 - 15th International Colloquium, 2018, Proceedings
A2 - Fischer, Bernd
A2 - Uustalu, Tarmo
A2 - Uustalu, Tarmo
PB - Springer Verlag
T2 - 15th International Colloquium on Theoretical Aspects of Computing, ICTAC 2018
Y2 - 16 October 2018 through 19 October 2018
ER -