
$$\begin{array}{l}{(20)}\\ {(21)}\end{array}$$

The definitions of satisfiability of a constraint, a fuzzy constraint, a set of **constraints,**
a set of fuzzy constraints, primitive constraint and primitive fuzzy constraint are **obvious.**
It is quite easily verified that the fuzzy entailment problem can be reduced to the unsatisfiability problem of a set of fuzzy **constraints:**

$\Sigma\cup\{\langle\alpha<n\rangle\}$ not satisfiable  $\Sigma\cup\{\langle\alpha>n\rangle\}$ not satisfiable. 
Σ|≈hα ≥ ni iff Σ ∪ {hα < ni} not satisfiable **(20)**
Σ|≈hα ≤ ni iff Σ ∪ {hα > ni} not satisfiable. **(21)**
Our calculus, determining whether a finite set S of fuzzy constraints is satisfiable or **not,** is based on a set of constraint propagation rules transforming a set S of fuzzy constraints **into**
"simpler" satisfiability preserving sets Si until either all Si contain a clash (indicating **that**
from all the Si no model of S can be build) or **some** Si is completed and clash-free, that is, no rule can be further be applied to Si and Si contains no clash (indicating that **from** Si a model of S can be **build).**
A set of fuzzy constraints S contains a clash iff it contains either one of the **constraints** in Table 1 or S contains a conjugated pair of fuzzy constraints. Each entry in Table 2 **says** us

$$\begin{array}{l}{{\langle w{:}\bot\geq n\rangle,\,\mathrm{where}\,\,n>0}}\\ {{\langle w{:}\top\leq n\rangle,\,\mathrm{where}\,\,n<1}}\\ {{\langle w{:}\bot>n\rangle,\,\langle w{:}\top<n\rangle,\,\langle w{:}C<0\rangle,\,\langle w{:}C>1\rangle}}\end{array}$$

Table 1: **Clashes**

![0_image_0.png](0_image_0.png)

under which condition the row-column pair of fuzzy constraints is a conjugated pair. **Given**
a fuzzy constraint ψ, **with** ψ c we indicate a conjugate of ψ (if there exists one). Notice **that**
a conjugate of a fuzzy constraint may be not unique, as there could be infinitely many. For instance, both ha:C < 0.6i and ha:C ≤ 0.7i are **conjugates** of ha:C ≥ 0.8i.

Concerning the rules, for each connective ⊓,⊔,¬, ∀, ∃ there is a rule for each **relation** ≥, >
, ≤, <, i.e. there are 20 rules. The rules have the **form:**

$$\Phi\rightarrow\Psi\mathrm{~if~}\Gamma$$
Φ → Ψ if Γ **(22)**
where Φ and Ψ are sequences of fuzzy constraints and Γ is a condition. A rule fires **only**
if the condition Γ holds, if the current set S of fuzzy constraints contains fuzzy **constraints**
matching the precondition Φ and the consequence Ψ is not already in S. After firing, the constraints from Ψ are added to S. The rules are the **following:**

$\left(22\right)^{2}$