Talk @ RAMiCS 2012

1. Relations as Executable Specifications
Taming Partiality and Non-determinism Using Invariants
Nuno Macedo
Hugo Pacheco
Alcino Cunha
HASLab — High Assurance Software Laboratory
INESC TEC & Universidade do Minho, Braga, Portugal
RAMiCS 13
September 20, 2012, Cambridge

2. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Introduction
• Relational calculus provides a more natural way to specify
programs;
• Many programs are partial and non-deterministic;
• A point-free (PF) version has been used in a variety of
computer science areas;
• Such specifications are not amenable for execution.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
2 / 21

3. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Motivating Example
(id
id)◦ ◦ (head◦
length◦ ) : Nat → [Nat]
• Calculates a list with length n and the same n at its head;
• Not total nor functional;
• Very inefficient given a naive semantics;
• head◦ could be generating all lists by increasing length and
never reach n.
(id4id)
(id4id)
Nuno Macedo, Hugo Pacheco, Alcino Cunha
head 4length
(head 4length )
3 / 21

4. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Motivating Example
• We can predict the behavior of partial expressions by
calculating the exact domain and range;
• They can also be used to narrow non-deterministic executions.
●{(l, l0 ) 2 [Nat] ⇥ [Nat]|head l = length l0 ^ l = l0 }
(id4id)
●{l 2 [Nat]|head l = length l}
Nuno Macedo, Hugo Pacheco, Alcino Cunha
(id4id)
head 4length
(head 4length )
●{n 2 Nat|n 6= 0}
4 / 21

5. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Taming Partiality and Non-determinism
• We propose a PF relational framework where data-types are
enhanced with invariants;
• The simplicity of the PF calculus allows us to develop
practical type-inference and type-checking algorithms;
• Those invariants can then used to run the specifications more
efficiently;
• Bidirectional transformations are our application scenario.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
5 / 21

6. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
PF Relational Calculus
Identity
Top
Bottom
Converse
Composition
Intersection
Union
Split
Projections
Either
Injections
Constants
Conditional
id : A → A
:A→B
⊥:A→B
·◦ : (A → B) → (B → A)
· ◦ · : (B → C ) → (A → B) → (A → C )
· ∩ · : (A → B) → (A → B) → (A → B)
· ∪ · : (A → B) → (A → B) → (A → B)
· · : (A → B) → (A → C ) → (A → B × C )
π1 : A × B → A and π2 : A × B → B
· · : (B → A) → (C → A) → (B + C → A)
i1 : A → (A + B) and i2 : B → (A + B)
! : A → 1 and · : B → (A → B)
· ? :(A → A) → (A → A + A)
• The simple equational rules are harnessed in a strategic
rewrite system that simplifies expressions.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
6 / 21

7. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Membership Semantics
a id a
a R◦ b
b S ◦R a
b R ∩S a
b R ∪S a
(b, c) R S a
a π1 (a, b)
b π2 (a, b)
a R S (Left b)
a R S (Right c)
(Left a )
i1 a
(Left a )
i2 b
(Right b ) i1 a
(Right b ) i2 b
b
a
b ⊥ a
1 ! a
b b a
Nuno Macedo, Hugo Pacheco, Alcino Cunha
=a≡a
=b R a
= ∃ c. b S c
=b R a∧b
=b R a∨b
=b R a∧c
=a≡a
=b≡b
=a R b
=a S c
=a≡b
= False
= False
=a≡b
= True
= False
= True
=b≡b
∧c R a
S a
S a
S a
7 / 21

8. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Execution Semantics
| id |
| R◦ |
|S ◦R|
|R ∩S|
|R ∪S|
|R S|
| π1 |
| π2 |
|R S|
|R S|
| i1 |
| i2 |
| |
|⊥|
|!|
|b|
a
b
a
a
a
a
(a, b)
(a, b)
(Left b)
(Right c)
a
b
a
a
a
a
Nuno Macedo, Hugo Pacheco, Alcino Cunha
= {a}
= {a | a ← A, a R ◦ b }
= {b | c ← | R | a, b ← | S | c }
= {b | b ← | R | a, b S a}
= |R| a ∪ |S| a
= {(b, c) | b ← | R | a, c ← | S | a}
= {a}
= {b }
= |R| b
= |S| c
= {Left a}
= {Right b }
=B
= {}
= {1}
= {b }
8 / 21

9. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Predicates as Coreflexives
• Domain δR and range ρR are predicates that can be defined
as coreflexives;
• Coreflexives Φ : A → A are relations smaller than the identity;
• If a satisfies the predicate Φ then a Φ a;
• For pairs A × B related by R : A → B, the invariant is lifted as
[R] : A × B → A × B;
• Invariants on coproducts are simply the coproduct Φ + Ψ;
• R : Φ → Ψ denotes δR = Φ and ρR = Ψ.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
9 / 21

10. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Inferring Checkable Invariants
• Domain and range can be directly computed as
δR = R ◦ ◦ R ∩ id and ρR = R ◦ R ◦ ∩ id;
• May result in inefficient membership tests (due to
composition);
• By expanding the definition, we define an equivalent definition
with most compositions removed;
• Others will fall in the special case
b (f ◦ ◦ R ◦ g ) a ≡ (f b) R (g a);
• The rewriting system further simplifies the formula and issues
a warning if problematic expressions remain.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
10 / 21

11. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Example
(id
id)◦ ◦ (head◦
length◦ ) : Nat → [Nat]
ρ((id id)◦ ◦ (head◦ length◦ ))
={Range definition}
ρ((id id)◦ ◦ ρ(head◦ length◦ ))
={Range definition}
ρ((id id)◦ ◦ [length◦ ◦ head])
={Range definition}
δ([length◦ ◦ head] ◦ (id id))
={Domain definition, Simplifications : PF Laws}
(head◦ ◦ length) ∩ id
(id id)◦ ◦(length◦ head◦ ):inN ◦(⊥+id)◦outN → (head◦ ◦length)∩id
Nuno Macedo, Hugo Pacheco, Alcino Cunha
11 / 21

12. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Optimizing Non-deterministic Executions
• After determining the domain and range, they can be
propagated down to primitives,
• This reduces the generation of irrelevant intermediate values;
| id : Φ → Ψ | a = | Ψ | a
| π1 : [U ] → Ψ | (a, b) = | Ψ | a
| R ◦ S : Φ → Ψ | a = {b | c ← | S : Φ → δR | a,
b ← | R : ρS → Ψ | c }
| R ∩ S : Φ → Ψ | a = {b | b ← | R : Φ ∩ δS → ρ(Ψ ◦ S ◦ a) | a}
Nuno Macedo, Hugo Pacheco, Alcino Cunha
12 / 21

13. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Recursive Relations with Invariants
• We support the well-know recursion patterns of
catamorphisms (folds) and anamorphisms (unfolds);
• Execution is not problematic, as it is performed by unfolding
their definitions;
• However, there is no known normal form for invariants over
recursive types;
• The rewrite system tries to simplify the generic domain/range
expression.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
13 / 21

14. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Recursive Relations: Example
unzip : [A × B ] → [A] × [B ]
ρunzip
={Range definition}
(unzip ◦ unzip◦ ) ∩ id
={Simplifications : unzip is functional, Liftify : range of unzip is a product}
◦
[π2 ◦ unzip ◦ unzip◦ ◦ π1 ]
={Catamorphism fusion : π1 ◦ g = nil (cons ◦ (π1 × id)) ◦ F π1 }
[(
|nil (cons ◦ (π1 × id))| ◦ (
) |nil (cons ◦ (π2 × id))| ◦ ]
)
={Definitions : map}
[(map π1 ) ◦ (map π2 )◦ ]
={Simplifications : map converse, map fusion (see below )}
◦
[map (π1 ◦ π2 )]
={Simplifications : PF Laws}
[map ]
unzip : id → [map
Nuno Macedo, Hugo Pacheco, Alcino Cunha
]
14 / 21

15. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Bidirectional Transformations
• Bidirectional transformations are transformations between
data-structures that may be applied in both directions;
• Defining the transformations individually is expensive and
error-prone;
• BXs should be inferred a single specification;
• We propose a solution using the relational calculus.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
15 / 21

16. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Bidirectional Transformations
• Lenses are one of the most famous BX frameworks;
• A lens S
V between sources S and more abstract views V
consists of transformations Get : S → V and Put : V × S → S;
• It is said to be well-behaved if:
• Get ◦ Put ⊆ π1 (acceptability);
• Put ◦ (Get id) ⊆ id (stability);
• Usually there are multiple valid Puts, but existing frameworks
are deterministic.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
16 / 21

17. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Bidirectional Transformations
• In principle, it is possible to lift any functional expression to a
well-behaved lens;
• Existing frameworks either:
• Have maximum updatability but
disregard some operators;
• Refine the type-system to allow
operators with smaller updatability;
• Support any operator but disregard
updatability;
?
?
?
• We refine the type-system and
guarantee maximum updatability.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
17 / 21

18. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Generic Non-deterministic Lenses
• Using relational calculus we can define a generic Put that is
the largest relation that satisfies the properties;
• A transformation get : A → B can be lifted to a well-behaved
non-deterministic lens get : δget
Put = (π2
ρget, with
(get◦ ◦ π1 )) ◦ [get◦ ] ?
• Emerges naturally from the lens laws:
• [get◦ ] ? (v , s) tests if v was changed, returning either the
original s (acceptability), or any source s such that s = get v
(stability);
• Maximum updatability: δPut = ρget × δget.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
18 / 21

19. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Generic Non-deterministic Lenses
• Type-checking over δget and ρget directly could be
undecidable and due to the central role of the converse, Put
can not be directly executed;
• Both these issues can be addressed by the optimizations
already presented;
• Forward transformations are functional so problematic cases
are very limited:
• Regarding type-checking, only particular ranges of splits are
possibly undecidable;
• The backward transformation can be efficiently executed;
• Recursive expressions are also supported as they preserve the
functionality of their algebras.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
19 / 21

20. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Generic Non-deterministic Lenses: Example
π1
id : id
◦
[π1 ]
◦
• The range is [π1 ] : A × (A × B) → A × (A × B), so Put only
takes views (a, (b, c)) where a ≡ b;
◦
id)◦ will run, and π1 could
generate all pairs until reaching (a, c);
• When the view is updated, (π1
• With our optimization, the output of π1 is restricted to have c
in the second element.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
20 / 21

21. Introduction
Relational Calculus
Optimizations
Recursion
Bidirectional Transformations
Conclusions
Conclusions
• We have presented mechanisms for the efficient execution PF
relational expressions over data-types with invariants;
• Regarding BX, we identify an open problem in the
composition of lenses;
• By modeling lenses in this framework we were able to
implement an expressive PF BX language with maximum
updatability;
• Researching possible normal forms for invariants over recursive
types;
• Exploring mechanisms for a better control of the
non-determinism through user-defined quality measures.
Nuno Macedo, Hugo Pacheco, Alcino Cunha
21 / 21