# Chip-level Programming

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CHIP (Constraint handling in the prologue) is a programming assembly language for constraint analysis compiled with programmers based on the ECRC's Prolog – a Prolog-based language. It was the first programming language that implemented constraints programming over finite domains. CHIP V5 was developed and distributed in 1993 at the Cosytec factory in Paris and uses the C-based language interface C++. The successful logging solution, ILOV Solver, is primarily a derivative from the CHIP version of the ECRC.

Chip-level programming is a way of solving problems that involve the use of constraints. The goal is to find an optimal solution for any given problem, and this article provides insight into how it works. Here's a brief explanation for those who are not familiar with this type of approach, so you know what to expect.

Many companies have used this method to solve complex problems involving different constraints. It can be applied in various areas such as logistics, manufacturing, transportation, and even finance. Using this approach, different solutions can be found depending on the specific needs at hand. If you're interested in learning more about chip-level programming or if you have questions regarding its implementation within your business model, feel free to contact us today! We'd love nothing more than to help our readers achieve their goals through the use of technology!

## Principles of Digital Computing

A constraint is a condition analogous to arithmetic relationships in algebra.

For this article, an arithmetical example is given by formula_1. The domain of discourse here is that of integers for which the constraints are valid - no negative numbers, no fractions.

Constraint handling rules are used when programming our solutions.

In developing a solution, we have to identify constraints and how they interact with arriving at an optimal solution. In other words, how constraints work together is crucial when applying them in the real world. There are different rules for constraint solving, which can be divided into two major groups: relational and arithmetic. An example of a relational constraint would be "less than or equal to." An arithmetic example could be "3 times greater than".

There are two main types of constraints in modern-day computing: Boolean and numerical. Boolean constraints are typically used to encode conditions that can be true or false, such as whether an edge is coloured red or green. Numerical constraints, which are used to encode arithmetic functions, evaluate inequalities and equalities for real numbers.

This article will focus mainly on Boolean Constraints or rules that can be expressed with a limited number of literals.

The idea is then to reduce these constraints into a more simple form.

Part 1:

Constraint handling rules are used when programming our solutions. There are two main types of constraints in modern-day computing: Boolean and numerical Constraints. Boolean constraints are typically used to encode conditions that can be true or false, such as whether an edge is coloured red or green. Numerical constraints, which are used to encode arithmetic functions, evaluate inequalities and equalities for real numbers.

In our example, we will focus mainly on Boolean Constraints or rules expressed with a limited number of literals. The idea is then to reduce these constraints into a more simple form.

Part 2:

To solve this problem, you first need to identify the constraints and interact. One of these is the relation between $x$ and $y$. If $x=0$, then $y$, which is supposed to be greater than $x$ by definition, will also be less than or equal to zero - making it strictly undefined.

Part 3:

This article was written to provide insight into the world of chip-level programming. Using this approach, different constraints can be handled so that any resulting solution is optimal. However, for the benefit of readers who are not familiar with this type of programming, here's a brief explanation of what it entails so you get an idea of what to expect.

Part 4:

The idea is then to reduce these constraints into a more simple form. This can be done by encoding the constraints in a programming language, formalizing the problem's mathematical structure, and implementing an algorithm that computes an optimal solution. The most popular problem-solving paradigm in this area of study is Constraint Logic Programming (CLP), a branch of Declarative Programming.

Part 5:

Most programming languages can now generate an optimal solution for this problem. At the end of the article, we will use Prolog as our programming language to solve the ILOV solver problem.

Part 6:

The example problem uses Boolean Constraints. For this type of constraint, the two literals are $x=0$ and $y > x$. This means that if $y=0$, then $x$ must be strictly equal to $0$. Another example would be $\textrm{true} \lor \textrm{false}$. The logical OR expresses the idea "and" in the digital logic analyzer. $\textrm{true}\lor\textrm{false}$ is true when one or both of the variables on either side of $$\lor$$ are true. In our case, we will use a Prolog programming language to generate an optimal solution for our ILOV solver problem.

Part 7:

The most popular problem-solving paradigm in this area of study is Constraint Logic Programming (CLP), a branch of Declarative Programming. This approach uses rules and facts to represent problems and their solutions, respectively. For our ILOV solver example, we will encode the previously stated constraints as two Boolean clauses that can be used as decision variables for clp(FD). These constraints are called functional dependencies because each clause represents a constraint between different functions.

One thing worth mentioning here is that Prolog supports arithmetic operations such as $+$, $-$, $\times$, $\div$, and $\mod$. The variable $x$ mentioned above would not be continuous but discrete. This means that this variable might take on negative and positive values.

Part 8:

The final problem can now be formalized as a CLP (FD), the whole program using the clauses we built previously. The Prolog structure for our ILOV solver is shown below:

CLP(FD):-

Variables: X Y Z Q W S T U V C I K L M N O P R F D E.

Variables are used to encode decision variables in Prolog programs. In this case, $X$, $Y$, $Z$, $Q$, $W$, and $S$ represent the constraints we described above. These five variables will be as input for clp(FD).

The next step is to encode the constraint relations mentioned above in Prolog's binary infix notation. This can be done by appending the names of these relations along with their domains and ranges inside angle brackets. In Prolog, a binary relation between sets $A$ and $B$ takes on the form $\langle A, B\rangle$. In our example, we have six infix notation constructions: We have $$\geqslant_Z\langleZ, S\rangle$$, which means that for all values of Z and S, Z is greater than or equal to S. We also have $$\leqslant_I\langleW, Crangle$$ which means that for all values of W and C, W is less than or equal to C. We have $$\subseteq_Y\langleY, A\rangle$$ which means that for all values of Y and A, Y is a subset of A. Similarly, we have $$\sqsubseteq_X\langleX, B\rangle$$, which means that X is a subset of B for all values of X and B. Finally, we have $$\sqsubseteq_Q\langleR=L-S+1:N0:, R\rangle$$ which states that for all possible combinations of the variables R = L - S + 1 : N 0 , R Q.

To call clp(FD), we first need to move all our clauses into the body of the program (i.e., outside any goals). After this, we can invoke Prolog's \+" operator, which takes two arguments: The first is the goal that needs to be satisfied, and the second is a list of clauses that can be used to satisfy this goal. In our case, the goal is $\geqslant_Z\langleZ, S\rangle$ with variables Z and S as arguments for each clause in the list.

The final step is to runProlog() with this argument set. This will return a solution if one exists or an empty list otherwise. A possible outcome would be [[$X\geqslant_Y\langleY,A\rangle$], [$Y \in/ A$, $X=0$]] if there are no solutions.

Part 9:

This solution is not optimal because it satisfies the constraints for both X and Y at the same time. If we now change our argument to find a solution for only one of the variables (e.g., $Z=counter$, $W=num/cards$) , we get this result: [[$Z=$counter$, W="card" <=> S="suit" = 'hearts'='clubs'='diamonds'='spades'"$, T=$S"], [Q=$T-1$], [$Z=0$,$W=0$]] This is a better solution because it automatically satisfies the constraint for Y. These steps can be automated by creating a solver that takes these arguments and then assigns values to$X,Y,Z$. The goal of this solver can be represented as \+clp(FD). It will return two lists: The first is the list of all variables and their inferred constraints, and the second is the one-step solution sequence given above. Both these lists are returned in Prolog format. While we do not need such an elaborate operating system (and may never need it!), having such "hybrid" embedded systems on board will pave great ways for future projects which involve designing ad-hoc reasoning techniques. Part 10: An exciting way to solve this problem is by creating a hybrid system with Prolog assembly code. Here, the Prolog clauses are combined with predicate definitions in Python, which means that you can continue programming in your favourite machine language while using Prolog as your inference engine! We will not go through all the details of implementing our example in Python because there are already many excellent tutorials on learning about Prolog available online. Instead, we have provided an example for solving the given problem and relevant machine code snippets. We will now discuss how to write the Python code for clpFD. First note that we need six functions: clp() , prolog() , findSolver() , read_prolog_file() , solver() and clpfd() . The clp function contains all of your logic programming code in separate clauses, represented as lists. Each element of the list represents a single clause, where each item in that clause consists of three sub-items: head, body, and tail. A convenient way to represent these clauses is by using a Python dictionary. In our case, the head is the propositional formula created from your constraints using the prolog() function. The body is a list of clauses. Each clause consists of two elements: a predicate to be called and a list of arguments for that predicate in data lines (and additional side conditions). For example, we used this clause: ["_Z=$x$, "_W=\"card\" <=> S=\"suit\"=[' +str(prolog.replace("$X ", "").split(' = ')[0]).upper()+='hearts'='clubs'='diamonds'='spades']'"],

The first element in this clause executes "_Z=$x$;" if global variable "_x$" is defined (in Python, global variables are represented by the "global" keyword). The second element calls Prolog's string manipulation predicate replace() with "_W=\"card\" <=> S=\"suit\"=[' +str(prolog.replace("$X ", "").split(' = ')[0]).upper()+='hearts'='clubs'='diamonds'='spades']'" as an argument that replaces variable "_x$" with its string value. The tail of the clauses is a list containing all of the additional constraints for each clause. In our case, we used this tail: ["="], which means that any of these clauses can be satisfied if given a constraint that holds two atoms equal to each other. The second required function, prolog(), contains our formula's actual representation using Prolog syntax. Thus, it uses Prolog's notation for representing relational predicates (e.g., "<=>", "=") and calls Python's replace() predicate to handle variables in this representation (i.e., "_x$" is replaced by "_Z\$"). This function should be called with the list of clauses that you get from clp(). The findSolver function extracts all variable values from your clause list and returns them as a separate list along with the callable predicate list from Prolog. You can now use these two lists as arguments to the solver(), which will return a scheme that will finally lead to the solution. You can now use this scheme to look up the values of your variables and visualize them as desired.

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