Clausal Language (ver. 5.81.19, by P.J. Voda, J. Komara, J. Kluka)

6. CVIČENIE Z PREDMETU ÚVOD DO DEKLARATÍVNEHO PROGRAMOVANIA

http://ii.fmph.uniba.sk/cl/courses/1-AIN-505-udp/0809ls/ex/ex06.cl

Dátum: 23. a 25. 3. 2009

Odporúčaná verzia CL: 5.81.19

WWW stránka predmetu: http://ii.fmph.uniba.sk/cl/courses/1-AIN-505-udp/0809ls/?lang=sk

Scroll down to “ASSIGNMENTS”.

Preskočte nasledujúce komponenty po nadpis „ÚLOHY“.

C V I Č E N I E

I. Sorting Lists

Assignment 1. Define a predicate $Ord (x)$ which holds if $x$ is an ordered list (in ascending order):

Ord (x) \leftrightarrow \forall y \forall a \forall b \forall z (x = y \oplus (a, b, z) \to a ≦ b)

Definition. A list can be sorted by a function $Sort (x)$ satisfying:

Ord Sort (x)

x \sim Sort (x)

In the next assignment, you will implement one of the simplest methods of sorting—the insertion sort.

Assignment 2. Implement the insertion sort in the following steps:

Define a function $Ins (a, x)$ satisfying:
$Ord (x) \to Ord Ins (a, x)$ $Ord (x) \to a, x \sim Ins (a, x)$
Using the function $Ins$ define the function $Isort (x)$ which orders the list $x$ and satisfies:
$Ord Isort (x)$ $x \sim Isort (x)$

Do not use other auxiliary functions (nor the predicate $x &CLin; y$ ).

Assignment 3. Implement the merge sort in the following steps:

Define first a function $Msplit (x)$ satisfying:
$\exists y \exists z (Msplit (x) = y, z \land L (x) = L (y) + L (z) \land (L (y) = L (z) \lor L (y) = L (z) + 1))$ $\exists y \exists z (Msplit (x) = y, z \land \forall i (2 \cdot i < L (x) \to {(x)}_{2 \cdot i} = {(y)}_{i}) \land \forall i (2 \cdot i + 1 < L (x) \to {(x)}_{2 \cdot i + 1} = {(z)}_{i}))$
Define next a function $Merge (x, y)$ satisfying:
$Ord (x) \land Ord (y) \to Ord Merge (x, y)$ $Ord (x) \land Ord (y) \to x \oplus y \sim Merge (x, y)$
Finally, using $Msplit$ and $Merge$ define the function $Msort (x)$ which orders the list $x$ :
$Ord Msort (x)$ $x \sim Msort (x)$

Do not use other auxiliary functions (nor the predicate $x &CLin; y$ ).

Assignment 4. Implement the quicksort in the following steps:

Define a function $Qsplit (p, x)$ satisfying:
$\exists y \exists z (Qsplit (p, x) = y, z \land \forall a (a &CLin; y \to a ≦ p) \land \forall a (a &CLin; z \to a > p) \land x \sim y \oplus z)$
Use the function $Qsplit$ in the definition of the function $Qsort (x)$ which orders the list $x$ :
$Ord Qsort (x)$ $x \sim Qsort (x)$

Do not use other auxiliary functions (nor the predicate $x &CLin; y$ ).

II. Sets as Lists

Assignment 5. Define a predicate $Set (x)$ which is true if $x$ is a strictly ordered list:

Set (x) \leftrightarrow \forall y \forall a \forall b \forall z (x = y \oplus (a, b, z) \to a < b)

In the following assignments, you will implement set relations and operations. Use the representation of finite sets by strictly ordered lists.

Assignment 6. Define a predicate $Empty (x)$ satisfying:

Set (x) \to Empty (x) \leftrightarrow \forall a a \notin x

Assignment 7. Define a predicate $a \in x$ ( $Member (a, x)$ ) satisfying:

Set (x) \to a \in x \leftrightarrow a &CLin; x

Assignment 8. Define a function $x \cup {a}$ ( $Insert (x, a)$ ) satisfying:

Set (x) \to Set (x \cup {a})

Set (x) \to \forall b (b \in x \cup {a} \leftrightarrow b = a \lor b \in x)

Caution. The function $Insert$ slightly differs from the function $Ins$ from Part I of this file.

Assignment 9. Define a function $Ln2set (x)$ satisfying:

Set Ln2set (x)

\forall z (z \in Ln2set (x) \leftrightarrow z &CLin; x)

Assignment 10. Define a predicate $x \subseteq y$ ( $Subset (x, y)$ ) satisfying:

Set (x) \land Set (y) \to x \subseteq y \leftrightarrow \forall a (a \in x \to a \in y)

Assignment 11. Define a predicate $x \equiv y$ ( $Eqset (x, y)$ ) satisfying:

Set (x) \land Set (y) \to x \equiv y \leftrightarrow \forall a (a \in x \leftrightarrow a \in y)

Assignment 12. Define a function $x \ {a}$ ( $Delete (x, a)$ ) satisfying:

Set (x) \to Set (x \ {a})

Set (x) \to \forall b (b \in x \ {a} \leftrightarrow b \neq a \land b \in x)

Assignment 13. Define a function $x \cup y$ ( $Union (x, y)$ ) satisfying:

Set (x) \land Set (y) \to Set (x \cup y)

Set (x) \land Set (y) \to \forall a (a \in x \cup y \leftrightarrow a \in x \lor a \in y)

Assignment 14. Define a function $x \cap y$ ( $Inter (x, y)$ ) satisfying:

Set (x) \land Set (y) \to Set (x \cap y)

Set (x) \land Set (y) \to \forall a (a \in x \cap y \leftrightarrow a \in x \land a \in y)

Assignment 15. Define a function $x \ y$ ( $Diff (x, y)$ ) satisfying:

Set (x) \land Set (y) \to Set (x \ y)

Set (x) \land Set (y) \to \forall a (a \in x \ y \leftrightarrow a \in x \land \neg a \in y)

Assignment 16. Define a function $x ∸ y$ ( $Sdiff (x, y)$ ) satisfying:

Set (x) \land Set (y) \to Set (x ∸ y)

Set (x) \land Set (y) \to \forall a (a \in x ∸ y \leftrightarrow a \in x \land \neg a \in y \lor \neg a \in x \land a \in y)

Note that the function also satisfies:

Set (x) \land Set (y) \to x ∸ y = x \ y \cup (y \ x)

Assignment 17. Define a predicate $Disjoint (x, y)$ satisfying:

Set (x) \land Set (y) \to Disjoint (x, y) \leftrightarrow \neg \exists a (a \in x \land a \in y)

The predicate also satisfies:

Set (x) \land Set (y) \to Disjoint (x, y) \leftrightarrow x \cap y = 0

Set (x) \land Set (y) \to Disjoint (x, y) \leftrightarrow x \cup y = x ∸ y