This document discusses using relations and relational calculus to specify programs in a more natural way. It proposes enhancing data types with invariants to tame non-determinism and partiality in relational specifications. This allows inferring checkable domain and range predicates for relations to optimize execution. Bidirectional transformations are also specified relationally to maximize updatability.

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