Expressive Models for Monadic Constraint Programming Pieter Wuille - - PowerPoint PPT Presentation

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Expressive Models for Monadic Constraint Programming Pieter Wuille - - PowerPoint PPT Presentation

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Introduction The FD-MCP Language Translation process Evaluation Future work Expressive Models for Monadic Constraint Programming Pieter Wuille Tom Schrijvers ModRef10 St. Andrews, September 6, 2010 Introduction The FD-MCP Language

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Introduction The FD-MCP Language Translation process Evaluation Future work

Expressive Models for Monadic Constraint Programming

Pieter Wuille Tom Schrijvers ModRef’10

  • St. Andrews, September 6, 2010
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Introduction The FD-MCP Language Translation process Evaluation Future work

FD-MCP?

FD-MCP FD-MCP: is a CP system for Finite-Domain (FD) problems is a subsystem of MCP, a Haskell CP framework provides an EDSL for writing FD problems

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Introduction The FD-MCP Language Translation process Evaluation Future work

Why an EDSL for CP Modelling?

EDSL An EDSL (Embedded Domain Specific Language) is more than an API: includes abstraction and syntactic sugar still embedded in host language, and able to interact with it Advantages The result allows advantages of both: Concise notation Declarative syntax (not a sequence of function calls) Full language feature set Directly usable results

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Introduction The FD-MCP Language Translation process Evaluation Future work

Haskell and MCP

Haskell Haskell: Lazy, purely functional programming language Support for first-class and higher-order functions Uses monads to order stateful operations Supports user-defined operators and overloading through type classes MCP Framework for CP in Haskell Does not fix variable domain, solver backend, search strategy, . . .

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Structure

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Expressions: example

Example (x > 5 ∧ x < 10 ∧ x2 = 49) model = exists $ \x -> do

  • - request a variable x

x @> 5

  • - state that x>5

x @< 10

  • - state that x<10

x*x @= 49

  • - state that x*x=49

return x

  • - return x
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Expressions

Expressions Everything is written as expressions Constraints are equivalent to boolean expressions New variables are introduced by passing a function that takes an expression representing the new variable as argument, to exists

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Parameters: example

Example (x > 5 ∧ x < 10 ∧ x2 = p) model p = exists $ \x -> do

  • - request a variable x

x @> 5

  • - state that x>5

x @< 10

  • - state that x<10

x*x @= p

  • - state that x*x=p

return x

  • - return x
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Parameters

Problem classes are written as functions that take an expression as parameter Known values can be passed at runtime, to obtain a problem instance Model functions can be compiled as-is to C+

+ code

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Higher-order constructs: examples

Example (a + b + c + d = 10 ∧ a2 + b2 + c2 + d2 = 30) model = exists $ \arr -> do size arr @= 4 csum arr @= 10 csum (cmap (\x -> x*x) arr) @= 30 return arr

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Higher-order constructs

Higher-order constructs Use equivalents of typical higher-order functions as primitives:

cmap f [a1, a2, a3, . . .]: [f (a1), f (a2), f (a3), . . .] cfold f i [a1, a2, a3, . . .]: . . . f (f (f (i, a1), a2), a3) . . .

To build typical CP higher-order constructs on top of

forall c: fold (∧) true c csum c: cfold (+) 0 c count v c: cfold (p i → p + (i = v)) c . . .

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Monadic bind

Monadic bind Boolean expressions can be used as solver actions that enforce their truth Solver actions can be combined using monadic bind Haskell provides syntactic sugar for this These are equivalent: model = exists $ \x -> do x @> 5 x @< 10 model = exists (\x -> (x @> 5) @&& (x @< 10))

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Building of expression tree

Building of expression tree The EDSL: Haskell functions and operators Syntactic sugar for boolean, integer and array expressions Models are monadic actions that introduce variables and post boolean expressions Evaluate at runtime to an expression tree

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Building of expression tree

x+y+z @< z-y Less (Plus x (Plus y z)) (Minus z y)

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Expression tree simplifications

Simplifications Simple pattern matching on the tree Applies some mathematical identities Attempts to minimize variable references and tree nodes

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Expression tree simplifications: examples

X + 0 → X X - X → 0 X + X → 2*X (a + (b + X)) → (a+b) + X size [a] → 1 . . .

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Conversion to Constraint Network Graph

For optimization purposes: We need information about a constraint’s variables. We need information those variables’ constraints. . . . Syntax tree does not make this explicit So we: We merge identical leaf nodes together, resulting in a graph . . . or even whole identical subtrees (CSE) We turn higher-order constructs without flattening into subgraphs

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Conversion to Constraint Network Graph

x+y+x @< z-y

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Graph-based optimizations

Graph-based optimizations Certain subgraphs can be recognized and replaced: A fold that sums values can become a sum A fold that sums equalities against a constant can become a count A fold that sums expressions can become a sum of a map

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Mapping to solver-specific constraints

So far: What we have A graph representation of the problem (class) Possibly still parametrized Compact, not flattened Independent of the solver’s supported constraints Next: mapping to solver-specific constraints

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Mapping to solver-specific constraints

Annotation algorithm Try to write nodes in function of other nodes, absorbing edges Start with options that may produce simple results Work recursively, but eager (no backtracking) Store resulting information in annotations on nodes When all nodes are annotated, the remaining edges are translated to constraints

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Mapping to solver-specific constraints

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Introduction The FD-MCP Language Translation process Evaluation Future work

Mapping to solver-specific constraints

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Introduction The FD-MCP Language Translation process Evaluation Future work

Mapping to solver-specific constraints

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Introduction The FD-MCP Language Translation process Evaluation Future work

Mapping to solver-specific constraints

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Introduction The FD-MCP Language Translation process Evaluation Future work

Mapping to solver-specific constraints

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Mapping to solver-specific constraints

“Linear” is not only possible annotation: Supported annotations Sizes of array variables Constant values (integers, arrays, booleans) Conditionals . . .

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Evaluation

1e-05 0.0001 0.001 0.01 0.1 1 10 100 7 8 9 10 11 12 13 14 15 time (s) problem size Benchmark allinterval Original C++ MCP Gecode srch. MCP Gecode run. MCP Generated C++

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Evaluation

1e-05 0.0001 0.001 0.01 0.1 1 10 100 6 7 8 9 10 11 time (s) problem size Benchmark golombruler Original C++ MCP Gecode srch. MCP Gecode run. MCP Generated C++

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Evaluation

0.0001 0.001 0.01 0.1 1 10 100 10 15 20 25 30 35 time (s) problem size Benchmark partition Original C++ MCP Gecode srch. MCP Gecode run. MCP Generated C++

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Evaluation

1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100 200 300 400 500 600 700 800 900 1000 time (s) problem size Benchmark magicseries Original C++ MCP Gecode srch. MCP Gecode run. MCP Generated C++

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Future work

Future work Extend system to labelling and search Code generation for search Further optimizations More benchmarks

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The end

Any questions?