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

1. CVIČENIE Z PREDMETU ÚVOD DO DEKLARATÍVNEHO PROGRAMOVANIA

http://ii.fmph.uniba.sk/cl/courses/1-AIN-505-udp/0809ls/ex/ex01.cl

Dátum: pondelok 16. 2. 2009, streda 18. 2. 2009

Odporúčaná verzia CL: 5.81.19

WWW: http://ii.fmph.uniba.sk/cl/courses?lang=sk

Kontakt na vyučujúcich: voda_at_fmph.uniba.sk, kluka_at_fmph.uniba.sk

A S S I G N M E N T S

Assignment 1. Your task is to use CL as a simple calculator for the evaluation of arithmetic expressions in the form of so-called queries. The window Query which appears at the end is designed for the input of queries. The result of a given query will be displayed in the window Results.

Examples of simple queries:

Addition. Query: $2 + 3 = x$ . CL-syntax is: $2+3 = x$ . Result: true for $x = 5$ .
Subtraction (modified). Query: $12 \dot{-} 5 = x$ . CL-syntax is: $12-5 = x$ . Result: true for $x = 7$ .
Multiplication. Query: $5 \cdot 3 = x$ . CL-syntax is: $5*3 = x$ . Result: true for $x = 15$ .
Integer division (modified). Query: $13 \div 3 = x$ . CL-syntax is: $13/3 = x$ . Result: true for $x = 4$ .
Remainder after integer division (modified). Query: $13 mod 8 = x$ . CL-syntax is: $13 mod 8 = x$ . Result: true for $x = 5$ .

Further apply these queries:

$(2 + 3) \cdot 5 \div 4 = x$
$19 \dot{-} 20 = x$
$2 \div 0 + 3 mod 0 = x$

The last two examples illustrate the following properties of arithmetic operators: $\forall x \forall y (x ≦ y \to x \dot{-} y = 0)$ , $\forall x x \div 0 = 0$ and $\forall x x mod 0 = 0$ .

It is possible to evaluate also compound queries (conjunctions of simple queries). E.g.

2 + 3 = x \land 4 \cdot 5 = y \land x + y \dot{-} 3 = z

CL-syntax is:

2+3 = x & 4*5 = y & x+y-3 = z

Result: true for $x = 5$ , $y = 20$ , and $z = 22$ .

Examples of syntactically incorrect queries:

$x = 2 + 3$ . The variable $x$ has not been assigned a value.
$2 + y = x$ . The variable $y$ has not been assigned a value.
$2 + 3 = x \land z = 4 \cdot x$ . The variable $z$ has not been assigned a value.

Assignment 2. Your next task is to use CL for the evaluation of arithmetic comparison relations.

Examples of simple queries:

Query: $2 + 3 = 1 + 4$ . CL-syntax: $2+3 = 1+4$ . Result: true.
Query: $2 + 3 \neq 1 + 4$ . CL-syntax: $2+3 != 1+4$ . Result: false.
Query: $2 + 3 ≦ 1 + 4$ . CL-syntax: $2+3 <= 1+4$ . Result: true.
Query: $2 + 3 < 1 + 4$ . CL-syntax: $2+3 < 1+4$ . Result: false.
Query: $2 + 3 > 1 + 4$ . CL-syntax: $2+3 > 1+4$ . Result: false.
Query: $2 + 3 ≧ 1 + 4$ . CL-syntax: $2+3 >= 1+4$ . Result: true.

Example of a compound query:

Query: $2 + 3 = x \land 1 + 4 = y \land x ≦ y \land x + 1 > y$ . Result: true.

Examples of syntactically incorrect queries:

$x ≦ 2 + 3$ . The variable $x$ has not been assigned a value.
$2 + 3 = x \land y ≧ 4 \cdot x$ . The variable $y$ has not been assigned a value.

Example. The following component contains a CL-definition (program) of a function $Square (x)$ which computes the square of the number $x$ . Try to evaluate this function for several arguments; e.g. execute the query $Square (5) = x$ .

Square (x) = x \cdot x

Note. CL identifier conventions:

Function identifier. An alpha-numeric string starting with a capital letter and containing lower case letters, digits, and underline symbols '_'. Example: $Fib$ , $Max_number_of_list$ , ${A1b2c}_{3}$ .
Variable identifier. A string of lower case letters possibly followed by one or two digits as indices. Example: $xx$ , $i_{1}$ , ${foo}_{99}$ .

Assignment 3. Your task is to write a definition of a function $F_{1} (x)$ computing the cube of the number $x$ , i.e. $\forall x F_{1} (x) = x \cdot x \cdot x$ .

Insert a new component below this assignment by clicking [Insert].

By a component we mean a remark, a definition or a theorem. In our case, the component is a definition of a function.

Example. We define functions by equations with conditions (so-called clauses or conditional equations). A definition of a function $Min (x, y)$ computing the minimum of the numbers $x$ and $y$ follows:

Min (x, y) = x \leftarrow x ≦ y

Min (x, y) = y \leftarrow x > y

Here is an example of a syntactically incorrect definition:

Min (x, y) = x \leftarrow x ≦ y

Min (a, b) = b \leftarrow a > b

The clauses must have identical arguments.

Assignment 4. Define a function $Max (x, y)$ computing the maximum of the numbers $x$ and $y$ .

Assignment 5. Define a function ${Max}_{3} (x, y, z)$ computing the maximum of the three numbers $x$ , $y$ , and $z$ . (You may use the previously defined function $Max$ ).

Assignment 6. (2 marks) Define a function ${Min}_{3} (x, y, z)$ computing the minimum of the three numbers $x$ , $y$ , and $z$ . (You may use the previously defined function $Min$ ).

Assignment 7. Define a function $Average (x, y, z)$ computing the average of the numbers $x$ , $y$ , and $z$ over natural numbers.

Assignment 8. Define a function $Sum_of_squared (x, y, z)$ computing the sum of the numbers $x$ , $y$ , and $z$ squared.

Assignment 9. Define a function $Median (x, y, z)$ computing the median (the middle number) of the numbers $x$ , $y$ , and $z$ .

Assignment 10. Save your solutions in the file 'ex01.cl' and exit the system CL (click on the Save & Exit button or press Alt + X).

Query:

Heap used: 232480 free: 133984492

Time used: 0:0:0:33