Computing Boundary Tuples for a Fuzzy Quantale
In the process of solving a certain kind of quantitative equations over a Lawverean quantale Q (with the binary tensor operation ⊗ and partial ordering ≤), one is interested in finding finite tuples of quantale elements (q_1,...,q_n) such that their tensor product is above the given quantale element λ, called the threshold. In other words, we look for tuples such that λ ≤ q_1 ⊗ ··· ⊗ q_n holds. For a given quantale Q and threshold λ, such tuples are called good (Q,λ)-tuples. Instead of computing all good (Q,λ)-tuples, we are interested in finding ≤-minimal elements among them (so called boundary tuples or a basis), where the (partial) ordering is the componentwise extension of ≤ from quantale elements to quantale tuples.
In this bachelor's thesis, we are interested in studying this problem in a more restricted setting for a fuzzy quantale, where the domain is the unit interval [0,1], ⊗ is a left-continuous t-norm, and ≤ is the standard total ordering on numbers. Here the goal is to develop a practically useful solving algorithm and experiment with it. As a starting point, one may formulate this problem as a constraint solving problem in real arithmetic and repeatedly apply an SMT solver (e.g., Z3) for the incremental generation of boundary tuples.
Advisor: Temur Kutsia and Wolfgang Schreiner