When you have ever been answerable for managing advanced enterprise logic, you know the way nested if-else statements could be a jungle: painful to navigate and simple to get misplaced. In the case of mission-critical duties, for instance formal verification or satisfiability, many builders attain for stylish instruments comparable to automated theorem provers or SMT solvers. Though highly effective, these approaches will be overkill and a headache to implement. What if all you want is a straightforward, clear guidelines engine?
The important thing thought for constructing such a light-weight engine depends on an idea that we have been taught to be insightful however impractical: fact tables. Exponential progress, their deadly flaw, makes them unfit for real-world issues. So we have been informed.
A easy statement adjustments every little thing: In virtually all sensible circumstances, the “impossibly giant” fact desk is definitely not dense with info; it’s in reality a sparse matrix in disguise.
This reframing makes the reality tables each conceptually clear and computationally tractable.
This text exhibits you flip this perception into a light-weight and highly effective guidelines engine. We’ll information you thru all the required steps to construct the engine from scratch. Alternatively, you need to use our open-source library vector-logic to begin constructing purposes on day one. This tutorial provides you with all the required particulars to grasp what’s below the hood.
Whereas all of the theoretical background and mathematical particulars will be present in our analysis paper on the State Algebra [1], right here, we give attention to the hands-on utility. Let’s roll up our sleeves and begin constructing!
A Fast Refresher on Logic 101
Fact Tables
We’ll begin with a fast refresher: logical formulation are expressions which might be constructed from Boolean variables and logical connectors like AND, OR, and NOT. In a real-world context, Boolean variables will be regarded as representing occasions (e.g. “the espresso cup is full”, which is true if the cup is definitely full and false whether it is empty). For instance, the system (f = (x_1 vee x_2)) is true if (x_1) is true, (x_2) is true, or each are. We are able to use this framework to construct a complete brute-force map of each attainable actuality — the reality desk.
Utilizing 1 for “true” and 0 for “false”, the desk for (x_1 vee x_2) appears like this:
[ begin{Bmatrix}
x_1 & x_2 & x_1 vee x_2 hline
0 & 0 & 0
0 & 1 & 1
1 & 0 & 1
1 & 1 & 1
end{Bmatrix} ]
The whole lot we have to carry out logical inference is encoded within the fact desk. Let’s see it in motion.
Logical Inference
Think about a traditional instance of the transitivity of implication. Suppose we all know that… By the best way, every little thing we are saying “we all know” is named a premise. Suppose we now have two premises:
- If (x_1) is true, then (x_2) should be true ((x_1 to x_2))
- If (x_2) is true, then (x_3) should be true ((x_2 to x_3))
It’s simple to guess the conclusion: “If (x_1) is true, then (x_3) should be true” ((x_1 to x_3)). Nevertheless, we may give a proper proof utilizing fact tables. Let’s first label our formulation:
[begin{align*}
& f_1 = (x_1 to x_2) && text{premise 1}
& f_2 = (x_2 to x_3) && text{premise 2}
& f_3 = (x_1 to x_3) && text{conclusion}
end{align*}]
Step one is to construct a fact desk protecting all mixtures of the three base variables (x_1), (x_2), and (x_3):
[begin{align*}
begin{Bmatrix}
x_1 & x_2 & x_3 & f_1 & f_2 & f_3 hline
0 & 0 & 0 & 1 & 1 & 1
0 & 0 & 1 & 1 & 1 & 1
0 & 1 & 0 & 1 & 0 & 1
0 & 1 & 1 & 1 & 1 & 1
1 & 0 & 0 & 0 & 1 & 0
1 & 0 & 1 & 0 & 1 & 1
1 & 1 & 0 & 1 & 0 & 0
1 & 1 & 1 & 1 & 1 & 1
end{Bmatrix}
end{align*}]
This desk accommodates eight rows, one for every task of fact values to the bottom variables. The variables (f_1), (f_2) and (f_3) are derived, as we compute their values immediately from the (x)-variables.
Discover how giant the desk is, even for this easy case!
The following step is to let our premises, represented by (f_1) and (f_2), act as a filter on actuality. We’re provided that they’re each true. Due to this fact, any row the place both (f_1) or (f_2) is fake represents an not possible situation which must be discarded.
After making use of this filter, we’re left with a a lot smaller desk:
[begin{align*}
begin{Bmatrix}
x_1 & x_2 & x_3 & f_1 & f_2 & f_3 hline
0 & 0 & 0 & 1 & 1 & 1
0 & 0 & 1 & 1 & 1 & 1
0 & 1 & 1 & 1 & 1 & 1
1 & 1 & 1 & 1 & 1 & 1
end{Bmatrix}
end{align*}]
And right here we’re: In each remaining legitimate situation, (f_3) is true. We’ve got confirmed that (f_3) logically follows from (or is entailed by) (f_1) and (f_2).
A chic and intuitive technique certainly. So, why don’t we use it for advanced methods? The reply lies in easy maths: With solely three variables, we had (2^3=8) rows. With 20 variables, we’d have over one million. Take 200, and the variety of rows would exceed the variety of atoms within the photo voltaic system. However wait, our article doesn’t finish right here. We are able to repair that.
The Sparse Illustration
The important thing thought for addressing exponentially rising fact tables lies in a compact illustration enabling lossless compression.
Past simply compressing the reality tables, we are going to want an environment friendly approach to carry out logical inference. We’ll obtain this by introducing “state vectors” — which symbolize units of states (fact desk rows) — and adopting set concept operations like union and intersection to control them.
The Compressed Fact Desk
First, we return to system (f = (x_1 to x_2)). Let’s establish the rows that make the system true. We use the image (sim) to symbolize the correspondence between the system and a desk of its “legitimate” fact assignments. In our instance of (f) for implication, we write:
[begin{align*}
fquadsimquad
begin{Bmatrix}
x_1 & x_2 hline
0 & 0
0 & 1
1 & 1
end{Bmatrix}
end{align*}]
Observe that we dropped the row ((1, 0)) because it invalidates (f). What would occur to this desk, if we now prolonged it to contain a 3rd variable (x_3), that (f) doesn’t depend upon? The traditional strategy would double the dimensions of the reality desk to account for (x_3) being 0 or 1, though it doesn’t add any new details about (f):
[begin{align*}
fquadsimquad
begin{Bmatrix}
x_1 & x_2 & x_3 hline
0 & 0 & 0
0 & 0 & 1
0 & 1 & 0
0 & 1 & 1
1 & 1 & 0
1 & 1 & 1
end{Bmatrix}
end{align*}]
What a waste! Uninformative columns could possibly be compressed, and, for this objective, we introduce a touch (–) as a “wildcard” image. You’ll be able to consider it as a logical Schrödinger’s cat: the variable exists in a superposition of each 0 and 1 till a constraint or a measurement (within the context of studying, we name it “proof”) forces it right into a particular state, eradicating one of many prospects.
Now, we will symbolize (f) throughout a universe of (n) variables with none bloat:
[begin{align*}
fquadsimquad
begin{Bmatrix}
x_1 & x_2 & x_3 & ldots & x_n hline
0 & 0 & – & ldots & –
0 & 1 & – &ldots & –
1 & 1 & – &ldots & –
end{Bmatrix}
end{align*}]
We are able to generalise this by postulating that any row containing dashes is equal to the set of a number of rows obtained by all attainable substitutions of 0s and 1s within the locations of dashes. For instance (attempt it with pencil and paper!):
[begin{align*}
begin{Bmatrix}
x_1 & x_2 & x_3 hline
– & 0 & –
– & 1 & 1
end{Bmatrix} =
begin{Bmatrix}
x_1 & x_2 & x_3 hline
0 & 0 & 0
0 & 0 & 1
1 & 0 & 0
1 & 0 & 1
0 & 1 & 1
1 & 1 & 1
end{Bmatrix}
end{align*}]
That is the essence of sparse illustration. Let’s introduce a couple of definitions for primary operations: We name changing dashes with 0s and 1s enlargement. The reverse course of, during which we spot patterns to introduce dashes, is named discount. The best type of discount, changing two rows with one, is named atomic discount.
An Algebra of States
Now, let’s give these concepts some construction.
- A state is a single, full task of fact values to all variables — one row in a completely expanded fact desk (e.g. ((0, 1, 1))).
- A state vector is a set of states (consider it as a subset of the reality desk). A logical system can now be thought of as a state vector containing all of the states that make it true. Particular circumstances are an empty state vector (0) and a vector containing all (2^n) attainable states, which we name a trivial vector and denote as (mathbf{t}). (As we’ll see, this corresponds to a t-object with all wildcards.)
- A row in a state vector’s compact illustration (e.g. ((0, -, 1) )) is named a t-object. It’s our basic constructing block — a sample that may symbolize one or many states.
Conceptually, shifting the main focus from tables to units is an important step. Keep in mind how we carried out inference utilizing the reality desk technique: we used premises (f_1) and (f_2) as a filter, holding solely the rows the place each premises have been true. This operation, when it comes to the language of set concept, is an intersection.
Every premise corresponds to a state vector (the set of states that fulfill the premise). The state vector for our mixed data is the intersection of those premise vectors. This operation is on the core of the brand new mannequin.
For friendlier notation, we introduce some “syntax sugar” by mapping set operations to easy arithmetic operations:
- Set Union ((cup)) (rightarrow) Addition ((+))
- Set Intersection ((cap)) (rightarrow) Multiplication ((*))
The properties of those operations (associativity, commutativity, and distributivity) permit us to make use of high-school algebra notation for advanced expressions with set operations:
[
begin{align*}
& (Acup B) cap (Ccup D) = (Acap C) cup (Acap D) cup (Bcap C) cup (Bcap D)
& rightarrow
& (A+B)cdot(C+D) = A,C + A,D + B,C + B,D
end{align*}
]
Let’s take a break and see the place we’re. We’ve laid a robust basis for the brand new framework. Fact tables are actually represented sparsely, and we reinterpret them as units (state vectors). We additionally established that logical inference will be achieved by multiplying the state vectors.
We’re practically there. However earlier than we will apply this concept to develop an environment friendly inference algorithm, we want yet another ingredient. Let’s take a more in-depth have a look at operations on t-objects.
The Engine Room: Operations on T-Objects
We are actually able to go to the subsequent part — creating an algebraic engine to control state vectors effectively. The elemental constructing block of our building is the t-object — our compact, wildcard-powered illustration of a single row in a state vector.
Observe that to explain a row, we solely must know the positions of 0s and 1s. We denote a t-object as (mathbf{t}^alpha_beta), the place (alpha) is the set of indices the place the variable is 1, and (beta) is the set of indices the place it’s 0. For example:
[
begin{Bmatrix}
x_1 & x_2 & x_3 & x_4 hline
1 & 0 & – & 1
end{Bmatrix} = mathbf{t}_2^{14}
]
A t-object consisting of all of the dashes (mathbf{t} = { -;; – ldots -}) represents the beforehand talked about trivial state vector that accommodates all attainable states.
From Formulation to T-Objects
A state vector is the union of its rows or, in our new notation, the sum of its t-objects. We name this row decomposition. For instance, the system (f=(x_1to x_2)) will be represented as:
[begin{align*}
fquadsimquad
begin{Bmatrix}
x_1 & x_2 & ldots & x_n hline
0 & 0 & ldots & –
0 & 1 & ldots & –
1 & 1 & ldots & –
end{Bmatrix} = mathbf{t}_{12} + mathbf{t}_1^2 + mathbf{t}^{12}
end{align*}]
Discover that this decomposition doesn’t change if we add extra variables ((x_3, x_4, dots)) to the system, which exhibits that our strategy is inherently scalable.
The Rule of Contradiction
The identical index can not seem in each the higher and decrease positions of a t-object. If this happens, the t-object is null (an empty set). For example (we highlighted the conflicting index):
[
mathbf{t}^{1{color{red}3}}_{2{color{red}3}} = 0
]
That is the algebraic equal of a logical contradiction. A variable ((x_3) on this case) can’t be each true (superscript) and false (subscript) on the identical time. Any such t-object represents an not possible state and vanishes.
Simplifying Expressions: Atomic Discount
Atomic discount will be expressed cleanly utilizing the newly launched t-object notation. Two rows will be diminished if they’re similar, aside from one variable, which is 0 in a single and 1 within the different. For example:
[
begin{align*}
begin{Bmatrix}
x_1 & x_2 & x_3 & x_4 & x_5 hline
1 & – & 0 & 0 & –
1 & – & 0 & 1 & –
end{Bmatrix} =
begin{Bmatrix}
x_1 & x_2 & x_3 & x_4 & x_5 hline
1 & – & 0 & – & –
end{Bmatrix}
end{align*}
]
In algebraic phrases, that is:
[
mathbf{t}^1_{34} + mathbf{t}^{14}_3 = mathbf{t}^1_3
]
The rule for this operation follows immediately from the definition of the t-objects: If two t-objects have index units which might be similar, aside from one index that may be a superscript in a single and a subscript within the different, they mix. The clashing index (4 on this instance) is annihilated, and the 2 t-objects merge.
By making use of atomic discount, we will simplify the decomposition of the system (f = (x_1 to x_2)). Noticing that (mathbf{t}_{12} + mathbf{t}_1^2 = mathbf{t}_1), we get:
[
f quad simquad mathbf{t}_{12} + mathbf{t}_1^2 + mathbf{t}^{12} = mathbf{t}_1 + mathbf{t}^{12}
]
The Core Operation: Multiplication
Lastly, allow us to talk about a very powerful operation for our guidelines engine: intersection (when it comes to set concept), represented as multiplication (when it comes to algebra). How do we discover the states frequent to the 2 t-objects?
The rule governing this operation is simple: to multiply two t-objects, one varieties the union of their superscripts, in addition to the union of their subscripts (we depart the proof as a easy train for a curious reader):
[
mathbf{t}^{alpha_1}_{beta_1},mathbf{t}^{alpha_2}_{beta_2} = mathbf{t}^{alpha_1 cup alpha_2}_{beta_1cupbeta_2}
]
The rule of contradiction nonetheless applies. If the ensuing superscript and subscript units overlap, the product vanishes:
[
mathbf{t}^{alpha_1 cup alpha_2}_{beta_1cupbeta_2} = 0 quad iff quad
(alpha_1 cup alpha_2) cap (beta_1cupbeta_2) not = emptyset
]
For instance:
[
begin{align*}
& mathbf{t}^{12}_{34},mathbf{t}^5_6 = mathbf{t}^{125}_{346} && text{Simple combination}
& mathbf{t}^{12}_{34} ,mathbf{t}^{4} = mathbf{t}^{12{color{red}4}}_{3{color{red}4}} = 0 && text{Vanishes, because 4 is in both sets}
end{align*}
]
A vanishing product signifies that the 2 t-objects don’t have any states in frequent; subsequently, their intersection is empty.
These guidelines full our building. We outlined a sparse illustration of logic and algebra for manipulating the objects. These are all of the theoretical instruments that we want. We’re able to assemble them right into a sensible algorithm.
Placing It All Collectively: Inference With State Algebra
The engine is prepared, it’s time to show it on! In its core, the thought is easy: to search out the set of legitimate states, we have to multiply all state vectors akin to premises and evidences.
If we now have two premises, represented by the state vectors ((mathbf{t}_{(1)} + mathbf{t}_{(2)})) and ((mathbf{t}_{(3)} + mathbf{t}_{(4)})), the set of states during which each are true is their product:
[
left(mathbf{t}_{(1)} + mathbf{t}_{(2)}right),left(mathbf{t}_{(3)} + mathbf{t}_{(4)}right) =
mathbf{t}_{(1)},mathbf{t}_{(3)} +
mathbf{t}_{(1)},mathbf{t}_{(4)} +
mathbf{t}_{(2)},mathbf{t}_{(3)} +
mathbf{t}_{(2)},mathbf{t}_{(4)}
]
This instance will be simply generalised to any arbitrary variety of premises and t-objects.
The inference algorithm is simple:
- Decompose: Convert every premise into its state vector illustration (a sum of t-objects).
- Simplify: Use atomic discount on every state vector to make it as compact as attainable.
- Multiply: Multiply the state vectors of all premises collectively. The result’s a single state vector representing all states constant together with your premises.
- Cut back Once more: The ultimate product could have reducible phrases, so simplify it one final time.
Instance: Proving Transitivity, The Algebraic Method
Let’s revisit our traditional instance of implication transitivity: if (f_1 = (x_1to x_2)) and (f_2 = (x_2to x_3)) are true, show that (f_3=(x_1to x_3)) should even be true. First, we write the simplified state vectors for our premises as follows:
[
begin{align*}
& f_1 quad sim quad mathbf{t}_1 + mathbf{t}^{12}
& f_2 quad sim quad mathbf{t}_2 + mathbf{t}^{23}
end{align*}
]
To show the conclusion, we will use a proof by contradiction. Let’s ask: does a situation exist the place our premises are true, however our conclusion (f_3) is fake?
The states that invalidate (f_3 = (x_1 to x_3)) are these during which (x_1) is true (1) and (x_3) is fake (0). This corresponds to a single t-object: (mathbf{t}^1_3).
Now, let’s see if this “invalidating” state vector can coexist with our premises by multiplying every little thing collectively:
[
begin{gather*}
(text{Premise 1}) times (text{Premise 2}) times (text{Invalidating State Vector})
(mathbf{t}_1 + mathbf{t}^{12}),(mathbf{t}_2 + mathbf{t}^{23}), mathbf{t}^1_3
end{gather*}
]
First, we multiply by the invalidating t-object, because it’s probably the most restrictive time period:
[
(mathbf{t}_1 mathbf{t}^1_3 + mathbf{t}^{12} mathbf{t}^1_3),(mathbf{t}_2 + mathbf{t}^{23}) = (mathbf{t}^{{color{red}1}}_{{color{red}1}3} + mathbf{t}^{12}_3),(mathbf{t}_2 + mathbf{t}^{23})
]
The primary time period, (mathbf{t}^{{colour{crimson}1}}_{{colour{crimson}1}3}), vanishes as a consequence of contradiction. So we’re left with:
[
mathbf{t}^{12}_3,(mathbf{t}_2 + mathbf{t}^{23}) =
mathbf{t}^{12}_3 mathbf{t}_2 + mathbf{t}^{12}_3 mathbf{t}^{23} =
mathbf{t}^{1{color{red}2}}_{{color{red}2}3} + mathbf{t}^{12{color{red}3}}_{{color{red}3}} =
0 + 0 = 0
]
The intersection is empty. This proves that there isn’t any attainable state the place (f_1) and (f_2) are true, however (f_3) is fake. Due to this fact, (f_3) should observe from the premises.
Proof by contradiction isn’t the one approach to resolve this downside. You will see that a extra elaborate evaluation within the “State Algebra” paper [1].
From Logic Puzzles to Fraud Detection
This isn’t nearly logic puzzles. A lot of our world is ruled by guidelines and logic! For instance, take into account a rule-based fraud-detection system.
Your data base is a algorithm like
IF card_location is abroad AND transaction_amount > $1000, THEN danger is excessiveThe complete data base will be compiled right into a single giant state vector.
Now, a transaction happens. That is your proof:
card_location = abroad, transaction_amount > $1000, user_logged_in = falseThis proof is a single t-object, assigning 1s to noticed information which might be true and 0s to information which might be false, leaving all unobserved information as wildcards.
To decide, you merely multiply:
[
text{Knowledge Base Vector}times text{Evidence T-object}
]
The ensuing state vector immediately tells you the worth of the goal variable (comparable to danger) given the proof. No messy chain of “if-then-else” statements was wanted.
Scaling Up: Optimisation Methods
As with mechanical engines, there are a lot of methods to make our engine extra environment friendly.
Let’s face the truth: logical inference issues are computationally exhausting, that means that the worst-case runtime is non-polynomial. Put merely, regardless of how compact the illustration is, or how good the algorithm is, within the worst-case situation, the runtime can be extraordinarily lengthy. So lengthy that most definitely, you’ll have to cease the computation earlier than the result’s calculated.
The rationale SAT solvers are doing a terrific job isn’t as a result of they modify actuality. It’s as a result of the vast majority of real-life issues should not worst-case situations. The runtime on an “common” downside can be extraordinarily delicate to the heuristic optimisations that your algorithm makes use of for computation.
Thus, optimisation heuristics could possibly be one of the crucial necessary parts of the engine to realize significant scalability. Right here, we simply trace at attainable locations the place optimisation will be thought of.
Observe that when multiplying many state vectors, the variety of intermediate t-objects can develop considerably earlier than ultimately shrinking, however we will do the next to maintain the engine operating easily:
- Fixed Discount: After every multiplication, run the discount algorithm on the ensuing state vector. This retains intermediate outcomes compact.
- Heuristic Ordering: The order of multiplication issues. It’s typically higher to multiply smaller or extra restrictive state vectors first, as this may trigger extra t-objects to fade early, pruning the calculation.
Conclusion
We’ve got taken you on a journey to find how propositional logic will be solid into the formalism of state vectors, such that we will use primary algebra to carry out logical inference. The magnificence of this strategy lies in its simplicity and effectivity.
At no level does inference require the calculation of big fact tables. The data base is represented as a set of sparse matrices (state vector), and the logical inference is diminished to a set of algebraic manipulations that may be applied in a couple of easy steps.
Whereas this algorithm doesn’t purpose to compete with cutting-edge SAT solvers and formal verification algorithms, it presents a wonderful, intuitive method of representing logic in a extremely compact kind. It’s a strong instrument for constructing light-weight guidelines engines, and a terrific psychological mannequin for desirous about logical inference.
Strive It Your self
One of the best ways to grasp this method is to make use of it. We’ve packaged your entire algorithm into an open-source Python library known as vector-logic. It may be put in immediately from PyPI:
pip set up vector-logicThe complete supply code, together with extra examples and documentation, is offered on
We encourage you to discover the repository, attempt it by yourself logic issues, and contribute.
In case you’re all in favour of delving deeper into mathematical concept, try the unique paper [1]. The paper covers some matters which we couldn’t embrace on this sensible information, comparable to canonical discount, orthogonalisation and lots of others. It additionally establishes an summary algebraic illustration of propositional logic based mostly on t-objects formalism.
We welcome any feedback or questions.
Who We Are
References
[1] Dmitry Lesnik and Tobias Schäfer, “State Algebra for Propositional Logic,” arXiv preprint arXiv:2509.10326, 2025. Obtainable at: https://arxiv.org/abs/2509.10326
