This library contains an efficient representation of Abstract Dialectical Frameworks (ADf) by utilising an implementation of Ordered Binary Decision Diagrams (OBDD)
An abstract dialectical framework consists of abstract statements. Each statement has an unique label and might be related to other statements (s) in the ADF. This relation is defined by a so-called acceptance condition (ac), which intuitively is a propositional formula, where the variable symbols are the labels of the statements. An interpretation is a three valued function which maps to each statement a truth value (true, false, undecided). We call such an interpretation a model, if each acceptance condition agrees to the interpration.
They can be easily identified though:
An ordered binary decision diagram is a normalised representation of binary functions, where satisfiability- and validity checks can be done relatively cheap.
Note that one advantage of this implementation is that only one oBDD is used for all acceptance conditions. This can be done because all of them have the identical signature (i.e. the set of all statements + top and bottom concepts). Due to this uniform representation reductions on subformulae which are shared by two or more statements only need to be computed once and is already cached in the data structure for further applications.
The used algorithm to create a BDD, based on a given formula does not perform well on bigger formulae, therefore it is possible to use a state-of-the art library to instantiate the BDD (https://github.com/sybila/biodivine-lib-bdd). It is possible to either stay with the biodivine library or switch back to the variant implemented by adf-bdd. The variant implemented in this library offers reuse of already done reductions and memoisation techniques, which are not offered by biodivine. In addition some further features, like counter-model counting is not supported by biodivine.
Note that import and export only works if the naive library is chosen
Each statement is defined by an ASP-style unary predicate s, where the enclosed term represents the label of the statement. The binary predicate ac relates each statement to one propositional formula in prefix notation, with the logical operations and constants as follows:
plain
and(x,y): conjunction
or(x,y): disjunctin
iff(x,Y): if and only if
xor(x,y): exclusive or
neg(x): classical negation
c(v): constant symbol “verum” - tautology/top
c(f): constant symbol “falsum” - inconsistency/bot
```plain s(a). s(b). s(c). s(d).
ac(a,c(v)). ac(b,or(a,b)). ac(c,neg(b)). ac(d,d). ```
First parse a given ADF and sort the statements, if needed.
rust
use adf_bdd::parser::AdfParser;
use adf_bdd::adf::Adf;
// use the above example as input
let input = "s(a).s(b).s(c).s(d).ac(a,c(v)).ac(b,or(a,b)).ac(c,neg(b)).ac(d,d).";
let parser = AdfParser::default();
match parser.parse()(&input) {
Ok(_) => log::info!("[Done] parsing"),
Err(e) => {
log::error!(
"Error during parsing:\n{} \n\n cannot continue, panic!",
e
);
panic!("Parsing failed, see log for further details")
}
}
// sort lexicographic
parser.varsort_lexi();
use the naive/in-crate implementation
rust
// create Adf
let mut adf = Adf::from_parser(&parser);
// compute and print the complete models
let printer = adf.print_dictionary();
for model in adf.complete() {
print!("{}", printer.print_interpretation(&model));
}
use the biodivine implementation
rust
// create Adf
let adf = adf_bdd::adfbiodivine::Adf::from_parser(&parser);
// compute and print the complete models
let printer = adf.print_dictionary();
for model in adf.complete() {
print!("{}", printer.print_interpretation(&model));
}
use the hybrid approach implementation
rust
// create biodivine Adf
let badf = adf_bdd::adfbiodivine::Adf::from_parser(&parser);
// instantiate the internally used adf after the reduction done by biodivine
let mut adf = badf.hybrid_step();
// compute and print the complete models
let printer = adf.print_dictionary();
for model in adf.complete() {
print!("{}", printer.print_interpretation(&model));
}