A Linear Logic Semantics for CHR

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A Linear Logic Semantics for CHR

• Hariolf Betz, Thom Frühwirth
• University of Ulm

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Overview

• CHR Roundup
• Constraints and Rules
• What are the limits of the classical declarative
  semantics?
• Introduction to Linear Logic
• Why choose linear logic?
• A Linear Logic Semantics for CHR
• Soundness and Completeness

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CHR overview

• constraint handling rules
• developed in the 1990s
• originally intended for the implementation of
  constraint solvers
• nowadays increasingly used as a general-purpose
  concurrent programming language

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CHR Constraints

• We distinguish two disjoint sets of constraints
• CHR constraints
• defined by CHR program (P)
• built-in constraints
• predefined/hard-wired
• defined by constraint theory (CT)

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CHR Rules
E ltgt C G
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Classical Declarative Semantics
E ltgt G
E ltgt G1 ? G2
E ltgt C G1 ? G2
E ltgt C G
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Example Coin Throw Simulator

• throw(Coin) ltgt Coinhead.
• throw(Coin) ltgt Cointail.

• Requires a fair selection strategy

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Coin Simulator Declarative Semantics

• throw(Coin) ltgt Coinhead.
• throw(Coin) ltgt Cointail.

• Classical logic cannot represent change
• Classical logic cannot represent multiplicities

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Linear Logic

• Introduced by Jean-Yves Girard in 1987
• Resource-oriented logic
• one can be prohibited from re-using the same
  hypotheses several times
• Logic of actions
• Logic of resources

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Linear Implication

• lollipop
• consuming ... yields ...

• For one euro, we can buy a pizza.

• Classical vs. Linear Implication

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With Conjunction

• with
• choose from ... and ...

• For one euro, we can buy a pizza or a cake

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Times Conjunction

• times
• both ... and ...

• For one euro, we can buy a pizza and an ice
  cream

• A pizza is two euros

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! Bang Modality

• bang
• of course ...
• For one euro, we can buy a cup of coffee with
  unlimited refills

• used to represent
• stable facts
• unlimited resources

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Why use linear logic for CHR?

• Similarity between CHR operational semantics and
  linear logic calculus
• Change and dynamics
• Multiplicities
• Seamless embedding of classical logic into linear
  logic

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Linear Logic Declarative Semantics

• Recursively defined function

• E ltgt C G

• Programs semantics is ?-conjunction of rules
  logical semantics

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Back to Coin Throwing Problem

• throw(Coin) ltgt Coinhead.
• throw(Coin) ltgt Cointail.

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Soundness and Completeness

• Soundness Let P be a CHR program and S0 be an
  initial state. If S0 has a derivation with a
  final state SnL, then

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Soundness and completeness

• Completeness If S0 and Sn are states such that
• then S0 has a derivation with a state S?
  such that

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Conclusion

• The classical declarative semantics of CHR aims
  at constraint solvers
• When CHR is used as a general-purpose concurrent
  programming language some limitations arise with
  the classical declarative semantics
• Representation of redundant resources
• Representation of dynamics and change
• Misleading semantics
• The linear logic represents more aspects of the
  operational semantics

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Future work

• Linear-logic based program analysis
• Integration of linear and classical semantics
• Linear-logic based constraint systems
• Find a restaurant in Sitges where the following
  holds