Fuzzy Aggregators and Similarities Into a Logic Language
FASILL is a Prolog-like first order language containing variables, function symbols, predicate symbols, constants and several (arbitrary) connectives to increase language expressiveness. FASILL uses connectives to combine/propagate truth values through the rules: conjunctive operators, disjunctive operators, and aggregation operators. Additionally, FASILL contains the truth values of a complete lattice equipped with a collection of connectives.
A FASILL program is a tuple <Π, R, L> where Π is a set of rules, R is a similarity relation between the elements of the signature Σ of Π, and L is a complete lattice.
A rule has the form A ← B, where A is an atomic formula called head and B, called body, is a well-formed formula (ultimately built from atomic formulas B1, ..., Bn, truth values of the lattice and connectives).
vanguardist(hydropolis) <- 0.9. elegant(ritz) <- 0.8. close(hydropolis, taxi) <- 0.7. good_hotel(X) <- @aver(elegant(X), @very(close(X, metro))).
A complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum.
All relevant components of each lattice can be encapsulated inside a Prolog file which must contain the definitions of a minimal set of predicates defining the set of valid elements (including special mentions to the "top" and "bottom" ones), the full or partial ordering established among them, as well as the repertoire of fuzzy connectives which can be used for their subsequent manipulation.
- member/1 which is satisfied when being called with a parameter representing a valid truth degree.
- members/1 which returns in one go a list containing the whole set of truth degrees.
- bot/1 and top/1 answer with the top and bottom element of the lattice, respectively.
- leq/2 models the ordering relation among all the possible pairs of truth degrees, and it is only satisfied when it is invoked with two elements verifying that the first parameter is equal or smaller than the second one.
- Finally, given some fuzzy connectives of the form &label1 (conjunction), |label2 (disjunction) or @label3 (aggregation) with arities n1, n2 and n3 respectively, we must provide clauses defining the connective predicates and_label1/(n1+1), or_label2/(n2+1) and agr_label3/(n3+1), where the extra argument of each predicate is intended to contain the result achieved after the evaluation of the proper connective.
% Elements member(X) :- number(X), 0 =< X, X =< 1. members([0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]). % Distance distance(X,Y,Z) :- Z is abs(Y-X). % Ordering relation leq(X,Y) :- X =< Y. % Supremum and infimum bot(0.0). top(1.0). % Binary operations and_prod(X,Y,Z) :- Z is X*Y. and_godel(X,Y,Z) :- Z is min(X,Y). and_luka(X,Y,Z) :- Z is max(X+Y-1,0). or_prod(X,Y,Z) :- U1 is X*Y, U2 is X+Y, Z is U2-U1. or_godel(X,Y,Z) :- Z is max(X,Y). or_luka(X,Y,Z) :- Z is min(X+Y,1). % Aggregators agr_aver(X,Y,Z) :- Z is (X+Y)/2. agr_very(X,Y) :- Y is X*X. % Default connectives tnorm(godel). tconorm(godel).
Given a domain U and a lattice L with a fixed t-norm, a similarity relation R is a fuzzy binary relation on U fulfilling the reflexive, symmetric and transitive properties.
FASILL takes a set of similarity equations
f/n ~ g/n = r, where
g are propositional variables or constants and
r is an element of L, and generates a valid similarity relation by applying the reflexive, symmetric and transitive closures over the initial scheme.
elegant/1 ~ vanguardist/1 = 0.6. metro ~ bus = 0.5. bus ~ taxi = 0.4. ~tnorm = godel.
Install on Linux
- Download or clone the FASILL repository:
git clone https://github.com/jariazavalverde/fasill.git
- Enter the fasill folder:
- Execute the install.sh bash script:
- That's all! Now you can run FASILL by typing fasill in your terminal:
FASILL has a large set of built-in predicates for arithmetic comparison, arithmetic evaluation, atom processing, control constructs, term comparison, term unification, type testing, list manipulation, etc.
Weak Unification and Operational Semantics
As a logic language, FASILL inherits the concepts of substitution, unifier and most general unifier. Some of them are extended to cope with similarities. Concretely, the most general unifier is replaced by the concept of weak most general unifier and a weak unification algorithm is introduced to compute it.
The procedural semantics of FASILL is defined in a stepwise manner. First, an operational stage is introduced which proceeds similarly to SLD resolution in pure logic programming, returning an expression still containing values and connectives. Then, an interpretive stage evaluates these connectives and produces a final answer.
You can find all the related bibliography into the file bibliography.bib.
Source code is released under the terms of the BSD 3-Clause License.