I am trying to sum all the numbers in a set in Alloy.
For instance, in the signature abc, I want the value to be the sum of a.value + b.value + c.value, which is 4+1+3=8.
However, if I use "+", it gives me the union set and not the sum.
PS. I know there is the "plus" (as I used it in sig sumab), but this only allows me to sum two values.
Thanks
open util/integer
sig a{value: Int}
{
value = 4
}
sig b{value: Int}
{
value = 1
}
sig c{value: Int}
{
value = 3
}
sig abc{value: set Int}
{
value = a.value + b.value + c.value
}
sig sumab{
value : Int
}
{
value = plus[a.value, b.value]
}
pred add{}
run add for 4 int, exactly 1 sumab, exactly 1 a, exactly 1 b, exactly 1 c, exactly 1 abc
Note: I wrote this in pseudo-code, it may help to get to an answer:
fun plusN [setInt : set de Int] : Int { // function "plusN" should take a set of integers "setInt", return an integer
if #setInt = 2 //if only two numbers in set, sum them
then plus[max setInt , min setInt]
else // if more than 2, use recursion
plusN [max setInt , plusN[{setInt - max setInt}]]
}
Note 2. The function sum may seem to be a good idea, but if I sum 1+1+1=1, the result will be 1 intead of 3 as the only number in the set is 1.
Alloy comes with a built-in sum function, so just do sum[value].
As mentioned in this answer, Alloy has a sum keyword (different than the sum function) which can be used.
The general format is:
sum e: <set> | <expression involving e>
Variable sig values
Here's a simple example that sums the number of sales during a day for a restaurant's three meals. In particular, note the use of the sum keyword in fun sum_sales.
one sig Restaurant { total_sales: Int }
abstract sig Meal { sales: Int }
one sig Breakfast, Lunch, Dinner in Meal {}
// Each meal only falls in one category
fact { disj[Breakfast, Lunch, Dinner] }
// Every meal is in some category
fact { Breakfast + Lunch + Dinner = Meal }
// Keep the numbers small because the max alloy int is 7
fact { all m: Meal | m.sales <= 4 }
// Negative sales don't make sense
fact { all m: Meal | m.sales >= 0 }
fun sum_sales: Int { sum m: Meal | m.sales }
fact { Restaurant.total_sales = sum_sales }
This works even when all meals have the same number of sales (1 + 1 + 1 = 3), as shown in this check.
check { (all m: Meal | m.sales = 1) => Restaurant.total_sales = 3 }
Here are some other ways to play around with the example.
check {
{
// One meal with three sales
#{ m: Meal | m.sales = 3 } = 1
// Two meals with one sale
#{ m: Meal | m.sales = 1 } = 2
} => Restaurant.total_sales = 5
}
run { Restaurant.total_sales = 5 }
Constant sig values
Another way you might want to use this is to have the value associated with each type of Meal be constant, but allow the number of Meals to vary. You can model this with a relation mapping each meal type to a number of sales as follows.
one sig Restaurant { total_sales: Int }
abstract sig Meal {}
lone sig Breakfast, Lunch, Dinner in Meal {}
// Each meal only falls in one category
fact { disj[Breakfast, Lunch, Dinner] }
// Every meal is in some category
fact { Breakfast + Lunch + Dinner = Meal }
fun meal_sales: Meal -> Int {
Breakfast -> 2
+ Lunch -> 2
+ Dinner -> 3
}
fun sum_sales: Int { sum m: Meal | m.meal_sales }
fact { Restaurant.total_sales = sum_sales }
check { #Meal = 3 => Restaurant.total_sales = 6 }
run { Restaurant.total_sales = 5 }
I'm trying to define an Alloy Fact, in the below specification, which would prevent adding a Person to an Airplane's set of passengers, unless there's enough capacity. Furthermore, I would like to add another Fact, which would disallow the removal of a Person from an Airplane's passenger set, unless the latter's cardinality exceeded zero.
Any help would be much appreciated.
Cheers,
Phiroc
PS Here's my first stab at the initial Fact:
a'.onboard <= a'.capacity implies
no p and add [a,a',p] and del [a',a'',p] implies a.onboard = a''.onboard
... According to Daniel Jackson, it's invalid because the condition before the implied keyword could potentially be false. Furthermore, it doesn't have the expected effect of preventing the number of passengers from exceeding the airplane capacity, when running the model. For instance, you sometimes get sets with cardinality 4, although capacity is 2, when you execute the model...
sig Person {}
sig Airplane {onboard: set Person, capacity: Int}
fact {
some a,a': Airplane | disj [a.onboard, a'.onboard]
}
pred show(a: Airplane) {
#a.onboard > 0
a.capacity = 2
}
run show for 10 but 2 Airplane
pred add (a, a': Airplane, p: Person) {
a'.onboard = a.onboard + p
}
pred del (a, a': Airplane, p: Person) {
a'.onboard = a.onboard - p
}
assert delUndoesAdd {
all a,a',a'': Airplane, p: Person |
no p and add [a,a',p] and del [a',a'',p] implies a.onboard = a''.onboard
}
assert addIdempotence {
all a,a',a'': Airplane, p: Person |
add [a,a',p] and add [a',a'',p] implies a'.onboard = a''.onboard
}
check delUndoesAdd for 10 but 3 Airplane
check addIdempotence for 3
I think I've found a solution.
sig Person {}
sig Airplane {onboard: set Person, capacity: Int}
fact {
some a,a': Airplane | disj [a.onboard, a'.onboard]
}
fact {
all a: Airplane | a.capacity = 3 and #a.onboard <= a.capacity
}
pred show(a: Airplane) {
}
run show for 10 but 2 Airplane
pred add (a, a': Airplane, p: Person) {
a'.onboard = a.onboard + p
}
pred del (a, a': Airplane, p: Person) {
a'.onboard = a.onboard - p
}
assert delUndoesAdd {
all a,a',a'': Airplane, p: Person |
no p and add [a,a',p] and del [a',a'',p] implies a.onboard = a''.onboard
}
assert addIdempotence {
all a,a',a'': Airplane, p: Person |
add [a,a',p] and add [a',a'',p] implies a'.onboard = a''.onboard
}
check delUndoesAdd for 10 but 3 Airplane
check addIdempotence for 3
I would like to solve the following lock challenge using Alloy.
My main issue is how to model the integers representing the digit keys.
I created a quick draft:
sig Digit, Position{}
sig Lock {
d: Digit one -> lone Position
}
run {} for exactly 1 Lock, exactly 3 Position, 10 Digit
In this context, could you please:
tell me if Alloy seems to you suitable to solve this kind of problem?
give me some pointers regarding the way I could model the key digits (without using Ints)?
Thank you.
My frame of this puzzle is:
enum Digit { N0,N1,N2,N3,N4,N5,N6,N7,N8,N9 }
one sig Code {a,b,c:Digit}
pred hint(h1,h2,h3:Digit, matched,wellPlaced:Int) {
matched = #(XXXX) // fix XXXX
wellPlaced = #(XXXX) // fix XXXX
}
fact {
hint[N6,N8,N2, 1,1]
hint[N6,N1,N4, 1,0]
hint[N2,N0,N6, 2,0]
hint[N7,N3,N8, 0,0]
hint[N7,N8,N0, 1,0]
}
run {}
A simple way to get started, you do not always need sig's. The solution found is probably not the intended solution but that is because the requirements are ambiguous, took a shortcut.
pred lock[ a,b,c : Int ] {
a=6 || b=8 || c= 2
a in 1+4 || b in 6+4 || c in 6+1
a in 0+6 || b in 2+6 || c in 2+0
a != 7 && b != 3 && c != 8
a = 7 || b=8 || c=0
}
run lock for 6 int
Look in the Text view for the answer.
upate we had a discussion on the Alloy list and I'd like to amend my solution with a more readable version:
let sq[a,b,c] = 0->a + 1->b + 2->c
let digit = { n : Int | n>=0 and n <10 }
fun correct[ lck : seq digit, a, b, c : digit ] : Int { # (Int.lck & (a+b+c)) }
fun wellPlaced[ lck : seq digit, a, b, c : digit ] : Int { # (lck & sq[a,b,c]) }
pred lock[ a, b, c : digit ] {
let lck = sq[a,b,c] {
1 = correct[ lck, 6,8,2] and 1 = wellPlaced[ lck, 6,8,2]
1 = correct[ lck, 6,1,4] and 0 = wellPlaced[ lck, 6,1,4]
2 = correct[ lck, 2,0,6] and 0 = wellPlaced[ lck, 2,0,6]
0 = correct[ lck, 7,3,8]
1 = correct[ lck, 7,8,0] and 0 = wellPlaced[ lck, 7,8,0]
}
}
run lock for 6 Int
When you think solve complete, let's examine whether the solution is generic.
Here is another lock.
If you can’t solve this in same form, your solution may not enough.
Hint1: (1,2,3) - Nothing is correct.
Hint2: (4,5,6) - Nothing is correct.
Hint3: (7,8,9) - One number is correct but wrong placed.
Hint4: (9,0,0) - All numbers are correct, with one well placed.
Yes, I think Alloy is suitable for this kind of problem.
Regarding digits, you don't need integers at all: in fact, it is a bit irrelevant for this particular purpose if they are digits or any set of 10 different identifiers (no arithmetic is performed with them). You can use singleton signatures to declare the digits, all extending signature Digit, which should be marked as abstract. Something like:
abstract sig Digit {}
one sig Zero, One, ..., Nine extends Digit {}
A similar strategy can be used to declare the three different positions of the lock. And btw since you have exactly one lock you can also declare Lock as singleton signature.
I like the Nomura solution on this page. I made a slight modification of the predicate and the fact to solve.
enum Digit { N0,N1,N2,N3,N4,N5,N6,N7,N8,N9 }
one sig Code {a,b,c: Digit}
pred hint(code: Code, d1,d2,d3: Digit, correct, wellPlaced:Int) {
correct = #((code.a + code.b + code.c)&(d1 + d2 + d3))
wellPlaced = #((0->code.a + 1->code.b + 2->code.c)&(0->d1 + 1->d2 + 2->d3))
}
fact {
some code: Code |
hint[code, N6,N8,N2, 1,1] and
hint[code, N6,N1,N4, 1,0] and
hint[code, N2,N0,N6, 2,0] and
hint[code, N7,N3,N8, 0,0] and
hint[code, N7,N8,N0, 1,0]
}
run {}
Update (2020-12-29):
The new puzzle presented by Nomura (https://stackoverflow.com/a/61022419/5005552) demonstrates a weakness in the original solution: it does not account for multiple uses of a digit within a code. A modification to the expression for "correct" fixes this. Intersect each guessed digit with the union of the digits from the passed code and sum them for the true cardinality. I encapsulated the matching in a function, which will return 0 or 1 for each digit.
enum Digit {N0,N1,N2,N3,N4,N5,N6,N7,N8,N9}
let sequence[a,b,c] = 0->a + 1->b + 2->c
one sig Code {c1, c2, c3: Digit}
fun match[code: Code, d: Digit]: Int { #((code.c1 + code.c2 + code.c3) & d) }
pred hint(code: Code, d1,d2,d3: Digit, correct, wellPlaced:Int) {
// The intersection of each guessed digit with the code (unordered) tells us
// whether any of the digits match each other and how many
correct = match[code,d1].plus[match[code,d2]].plus[match[code,d3]]
// The intersection of the sequences of digits (ordered) tells us whether
// any of the digits are correct AND in the right place in the sequence
wellPlaced = #(sequence[code.c1,code.c2,code.c3] & sequence[d1, d2, d3])
}
pred originalLock {
some code: Code |
hint[code, N6,N8,N2, 1,1] and
hint[code, N6,N1,N4, 1,0] and
hint[code, N2,N0,N6, 2,0] and
hint[code, N7,N3,N8, 0,0] and
hint[code, N7,N8,N0, 1,0]
}
pred newLock {
some code: Code |
hint[code, N1,N2,N3, 0,0] and
hint[code, N4,N5,N6, 0,0] and
hint[code, N7,N8,N9, 1,0] and
hint[code, N9,N0,N0, 3,1]
}
run originalLock
run newLock
run test {some code: Code | hint[code, N9,N0,N0, 3,1]}