Logic Gates: Realize OR gate using ONLY XOR gates

Constructing an OR Gate Using Only XOR Gates

Explore the fascinating world of logic gates by learning how to synthesize a fundamental OR gate using only XOR gates, demonstrating the versatility of Boolean algebra.

Logic gates are the fundamental building blocks of digital circuits. While basic gates like AND, OR, and NOT are commonly understood, it's often possible to construct one type of gate using combinations of others. This article delves into a specific, intriguing challenge: realizing an OR gate using only XOR gates. This exercise not only deepens understanding of Boolean logic but also showcases the power of gate equivalence and transformation.

Understanding XOR and OR Gate Logic

Before we can synthesize an OR gate from XOR gates, it's crucial to recall the truth tables and Boolean expressions for both.

An OR gate outputs true (1) if at least one of its inputs is true. Its Boolean expression for inputs A and B is A + B.

An XOR gate (Exclusive OR) outputs true (1) if its inputs are different. Its Boolean expression is A ⊕ B or A'B + AB' (where ' denotes NOT).

Let's compare their truth tables:

OR Gate Truth Table:
| A | B | A + B |
|---|---|-------|
| 0 | 0 |   0   |
| 0 | 1 |   1   |
| 1 | 0 |   1   |
| 1 | 1 |   1   |

XOR Gate Truth Table:
| A | B | A ⊕ B |
|---|---|-------|
| 0 | 0 |   0   |
| 0 | 1 |   1   |
| 1 | 0 |   1   |
| 1 | 1 |   0   |

Truth tables for OR and XOR gates.

Notice the key difference: when both inputs are 1, the OR gate outputs 1, while the XOR gate outputs 0. This is the specific case we need to address to transform an XOR-based circuit into an OR gate.

The Transformation: XOR to OR

The Boolean expression for an OR gate is A + B. We know that A + B can also be expressed as A ⊕ B + AB. This identity is key to our transformation. Since we are restricted to using only XOR gates, we need to find a way to generate the AB term (AND operation) and then combine it with A ⊕ B using another XOR gate, all while only using XOR gates.

Consider the identity: A + B = (A ⊕ B) ⊕ (AB). This means if we can generate A ⊕ B and AB using only XOR gates, we can then XOR them together to get A + B.

However, generating AB (AND gate) using only XOR gates is impossible. An AND gate requires a 0 output for 0,1 and 1,0 inputs, and a 1 for 1,1. An XOR gate outputs 1 for 0,1 and 1,0, and 0 for 1,1. They are fundamentally different.

This problem statement implies a common misconception or a trick question. It is not possible to realize an OR gate using only XOR gates without additional components like NOT gates or constants (0 or 1). If we are allowed to use constants, the problem becomes solvable.

Let's assume the problem implies using XOR gates and potentially a constant input (like '0' or '1') if necessary, or that the question is designed to highlight the limitations. A more common approach to realize an OR gate using other gates often involves NAND or NOR gates, which are functionally complete.

However, if we interpret "using ONLY XOR gates" as meaning the primary logic elements are XOR gates, and we can use the inputs themselves in creative ways, we can explore a common identity: A + B = (A ⊕ B) ⊕ (A AND B). The challenge remains how to get A AND B from XOR gates. It's generally accepted that XOR gates alone are not functionally complete, meaning you cannot build all other gates from them. You need at least one other type of gate (like AND, OR, or NOT) to achieve functional completeness.

If the question implies a scenario where we can use the inputs A and B directly, and also the constant 0 or 1, then we can construct an OR gate. For example, A OR B is equivalent to (A XOR B) XOR (A AND B). If we could somehow derive A AND B from XOR gates, we'd be set. But we can't.

Let's re-evaluate the premise. A common way to realize an OR gate using XOR and AND gates is A + B = (A ⊕ B) ⊕ (A ⋅ B). If the constraint is strictly "ONLY XOR gates" and no other gates or constants, then it's impossible. If the constraint is interpreted as "using XOR gates as the primary logic, and perhaps a constant input", then we can proceed.

Let's consider a common identity for OR: A + B = A ⊕ B ⊕ (A ⋅ B). This still requires an AND gate.

Another approach for OR is A + B = NOT(NOT A AND NOT B). This requires NOT and AND gates.

Given the strict constraint, the answer is that it's not possible. However, if the question is a common riddle or a simplified problem, it might be hinting at a specific configuration that appears to use only XORs but implicitly relies on something else, or it's a trick.

Let's consider the possibility of a misunderstanding of the problem or a common simplification. If we are allowed to use a NOT gate, then A OR B = NOT(A XOR B) XOR (A AND B). This is still not just XORs.

Conclusion for strict interpretation: It is generally accepted that XOR gates alone are not functionally complete. You cannot construct an OR gate (or an AND gate, or a NOT gate) using only XOR gates. You need at least one other type of gate (e.g., a NOT gate) to achieve functional completeness with XOR. For example, XOR and AND gates together are functionally complete. XOR and NOT gates together are also functionally complete.

If the question implies a scenario where we can use the inputs A and B, and also the constant '0' or '1' as inputs to the XOR gates, then it's still not directly possible to get an OR gate from only XOR gates.

Let's assume the question is a common puzzle that has a known (though perhaps non-obvious) solution, possibly involving a constant input or a specific interpretation. A common identity is A OR B = (A XOR B) XOR (A AND B). The challenge is to get A AND B from XORs. This is the sticking point.

However, if we consider A OR B = A XOR B XOR (A AND B), and we know that A AND B = NOT(A XOR B) AND (A OR B), this becomes circular.

Let's consider the identity: A OR B = (A XOR B) XOR (A AND B). If we had a way to generate A AND B using only XOR gates, we could then XOR it with A XOR B to get A OR B. But generating A AND B from only XOR gates is impossible.

Therefore, under the strict interpretation of "ONLY XOR gates", it is not possible to realize an OR gate. This is a fundamental concept in Boolean algebra and functional completeness. XOR gates are not functionally complete on their own.

To illustrate why this is the case, consider the properties of an XOR gate. An XOR gate always outputs 0 when both inputs are the same (0,0 or 1,1). An OR gate, however, outputs 1 when both inputs are 1. There is no combination of XOR gates that can produce a 1 when both inputs are 1 and 0 when both inputs are 0, while also producing 1 for 0,1 and 1,0 (which XOR already does). The 1,1 -> 1 case for OR is fundamentally different from 1,1 -> 0 for XOR, and XOR gates cannot bridge this difference on their own.

Functional Completeness and Gate Sets

A set of logic gates is considered 'functionally complete' if it can be used to construct any Boolean function. Examples of functionally complete sets include:

• {AND, OR, NOT}
• {NAND} (NAND gates alone are functionally complete)
• {NOR} (NOR gates alone are functionally complete)
• {XOR, AND} (XOR and AND together are functionally complete)
• {XOR, NOT} (XOR and NOT together are functionally complete)

As established, {XOR} alone is not a functionally complete set. This means there are certain Boolean functions (like the OR function) that cannot be implemented using only XOR gates.

graph TD
    A[Functionally Complete Sets] --> B{Can build any Boolean function}
    B --> C1{NAND Gate}
    B --> C2{NOR Gate}
    B --> C3{AND, OR, NOT}
    B --> C4{XOR, AND}
    B --> C5{XOR, NOT}
    A --> D{Not Functionally Complete Sets}
    D --> E1{XOR Gate (alone)}
    D --> E2{AND Gate (alone)}
    D --> E3{OR Gate (alone)}
    E1 -- Cannot build --> F[OR Gate]

Functional completeness of various logic gate sets.