In case you have ever been in control of managing advanced enterprise logic, you know the way nested if-else statements is usually a jungle: painful to navigate and straightforward to get misplaced. In terms of mission-critical duties, for instance formal verification or satisfiability, many builders attain for classy instruments resembling 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 an easy, clear guidelines engine?
The important thing concept for constructing such a light-weight engine depends on an idea that we have been taught to be insightful however impractical: fact tables. Exponential development, their deadly flaw, makes them unfit for real-world issues. So we have been advised.
A easy statement modifications every part: In virtually all sensible circumstances, the “impossibly massive” fact desk is definitely not dense with info; it’s in truth a sparse matrix in disguise.
This reframing makes the reality tables each conceptually clear and computationally tractable.
This text exhibits you easy methods to 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 should utilize 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 underneath 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 deal with 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 are 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 will use this framework to construct a complete brute-force map of each potential 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} ]
All the things we have to carry out logical inference is encoded within the fact desk. Let’s see it in motion.
Logical Inference
Think about a basic instance of the transitivity of implication. Suppose we all know that… By the way in which, every part we are saying “we all know” is named a premise. Suppose now we have two premises:
- If (x_1) is true, then (x_2) have to be true ((x_1 to x_2))
- If (x_2) is true, then (x_3) have to be true ((x_2 to x_3))
It’s straightforward to guess the conclusion: “If (x_1) is true, then (x_3) have to 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 overlaying 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 incorporates eight rows, one for every project 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 massive the desk is, even for this easy case!
The subsequent 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. Subsequently, any row the place both (f_1) or (f_2) is fake represents an not possible situation which needs to 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 confirmed that (f_3) logically follows from (or is entailed by) (f_1) and (f_2).
A sublime and intuitive methodology 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 1,000,000. 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 will repair that.
The Sparse Illustration
The important thing concept for addressing exponentially rising fact tables lies in a compact illustration enabling lossless compression.
Past simply compressing the reality tables, we’ll want an environment friendly solution to carry out logical inference. We are going to obtain this by introducing “state vectors” — which signify units of states (fact desk rows) — and adopting set principle operations like union and intersection to govern them.
The Compressed Fact Desk
First, we return to system (f = (x_1 to x_2)). Let’s determine the rows that make the system true. We use the image (sim) to signify 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*}]
Word 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 rely upon? The basic 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 may very well be compressed, and, for this function, 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 are able to signify (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 will generalise this by postulating that any row containing dashes is equal to the set of a number of rows obtained by all potential substitutions of 0s and 1s within the locations of dashes. For instance (strive 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, by 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 project 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-about 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) potential 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 signify one or many states.
Conceptually, shifting the main target from tables to units is an important step. Bear in mind how we carried out inference utilizing the reality desk methodology: we used premises (f_1) and (f_2) as a filter, conserving solely the rows the place each premises have been true. This operation, by way of the language of set principle, 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 at the moment are represented sparsely, and we reinterpret them as units (state vectors). We additionally established that logical inference will be completed by multiplying the state vectors.
We’re almost there. However earlier than we are able to apply this principle to develop an environment friendly inference algorithm, we’d like 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 at the moment are able to go to the subsequent section — creating an algebraic engine to govern state vectors effectively. The basic constructing block of our building is the t-object — our compact, wildcard-powered illustration of a single row in a state vector.
Word 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. As an illustration:
[
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 incorporates all potential 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). As an illustration (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 an identical, apart from one variable, which is 0 in a single and 1 within the different. As an illustration:
[
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 are an identical, apart 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 are able to 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 an important operation for our guidelines engine: intersection (by way of set principle), represented as multiplication (by way of algebra). How do we discover the states widespread to the 2 t-objects?
The rule governing this operation is easy: to multiply two t-objects, one types the union of their superscripts, in addition to the union of their subscripts (we go away 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 haven’t any states in widespread; 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’d like. 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 straightforward: to seek out the set of legitimate states, we have to multiply all state vectors akin to premises and evidences.
If now we 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 by 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 easy:
- 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 potential.
- Multiply: Multiply the state vectors of all premises collectively. The result’s a single state vector representing all states constant together with your premises.
- Scale back Once more: The ultimate product might have reducible phrases, so simplify it one final time.
Instance: Proving Transitivity, The Algebraic Method
Let’s revisit our basic 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 are able to 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 by 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 part 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 essentially 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}^{{coloration{purple}1}}_{{coloration{purple}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 is no such thing as a potential state the place (f_1) and (f_2) are true, however (f_3) is fake. Subsequently, (f_3) should observe from the premises.
Proof by contradiction just isn’t the one solution to clear up this downside. You can see 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 excessiveYour entire data base will be compiled right into a single massive 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 info which are true and 0s to info which are false, leaving all unobserved info 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 (resembling danger) given the proof. No messy chain of “if-then-else” statements was wanted.
Scaling Up: Optimisation Methods
As with mechanical engines, there are various methods to make our engine extra environment friendly.
Let’s face the fact: logical inference issues are computationally onerous, which means that the worst-case runtime is non-polynomial. Put merely, regardless of how compact the illustration is, or how sensible the algorithm is, within the worst-case situation, the runtime will probably be extraordinarily lengthy. So lengthy that more than likely, you’ll have to cease the computation earlier than the result’s calculated.
The rationale SAT solvers are doing a fantastic job just isn’t as a result of they alter actuality. It’s as a result of nearly all of real-life issues should not worst-case situations. The runtime on an “common” downside will probably be extraordinarily delicate to the heuristic optimisations that your algorithm makes use of for computation.
Thus, optimisation heuristics may very well be one of the vital vital parts of the engine to attain significant scalability. Right here, we simply trace at potential locations the place optimisation will be thought-about.
Word that when multiplying many state vectors, the variety of intermediate t-objects can develop considerably earlier than ultimately shrinking, however we are able to 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 will trigger extra t-objects to fade early, pruning the calculation.
Conclusion
We’ve taken you on a journey to find how propositional logic will be forged into the formalism of state vectors, such that we are able to use primary algebra to carry out logical inference. The class of this strategy lies in its simplicity and effectivity.
At no level does inference require the calculation of large 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 carried out in a couple of simple steps.
Whereas this algorithm doesn’t purpose to compete with cutting-edge SAT solvers and formal verification algorithms, it gives a good looking, intuitive means of representing logic in a extremely compact type. It’s a robust software for constructing light-weight guidelines engines, and a fantastic psychological mannequin for eager about logical inference.
Attempt It Your self
One of the simplest ways to grasp this method is to make use of it. We’ve packaged all the algorithm into an open-source Python library referred to as vector-logic. It may be put in immediately from PyPI:
pip set up vector-logicThe total supply code, together with extra examples and documentation, is offered on
We encourage you to discover the repository, strive it by yourself logic issues, and contribute.
In case you’re keen on delving deeper into mathematical principle, take a look at the unique paper [1]. The paper covers some matters which we couldn’t embrace on this sensible information, resembling canonical discount, orthogonalisation and plenty 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. Out there at: https://arxiv.org/abs/2509.10326
