Abstract: Many software as well digital hardware automatic synthesis methods define the set of implementations meeting the given system specifications with a boolean relation K. In such a context a fundamental step in the software (hardware) synthesis process is finding effective solutions to the functional equation defined by K. This entails finding a (set of) boolean function(s) F (typically represented using OBDDs, Ordered Binary Decision Diagrams) such that: 1) for all x for which K is satisfiable, K(x, F(x)) = 1 holds; 2) the implementation of F is efficient with respect to given implementation parameters such as code size or execution time. While this problem has been widely studied in digital hardware synthesis, little has been done in a software synthesis context. Unfortunately the approaches developed for hardware synthesis cannot be directly used in a software context. This motivates investigation of effective methods to solve the above problem when F has to be implemented with software. In this paper we present an algorithm that, from an OBDD representation for K, generates a C code implementation for F that has the same size as the OBDD for F and a WCET (Worst Case Execution Time) linear in nr, being n = |x| the number of input arguments for functions in F and r the number of functions in F.

Abstract: Model based design is particularly appealing in
software based control systems (e.g., embedded
software) design, since in such a case system
level specifications are much easier to define
than the control software behavior itself. In
turn, model based design of embedded systems
requires modeling both continuous subsystems
(typically, the plant) as well as discrete
subsystems (the controller). This is typically
done using hybrid systems. Mixed Integer Linear
Programming (MILP) based abstraction techniques
have been successfully applied to automatically
synthesize correct-by-construction control
software for discrete time linear hybrid systems,
where plant dynamics is modeled as a linear
predicate over state, input, and next state
variables. Unfortunately, MILP solvers require
such linear predicates to be conjunctions of
linear constraints, which is not a natural way of
modeling hybrid systems. In this paper we show
that, under the hypothesis that each variable
ranges over a bounded interval, any linear
predicate built upon conjunction and disjunction
of linear constraints can be automatically
translated into an equivalent conjunctive
predicate. Since variable bounds play a key role
in this translation, our algorithm includes a
procedure to compute all implicit variable bounds
of the given linear predicate. Furthermore, we
show that a particular form of linear predicates,
namely guarded predicates, are a natural and
powerful language to succinctly model discrete
time linear hybrid systems dynamics. Finally, we
experimentally show the feasibility of our
approach on an important and challenging case
study taken from the literature, namely the
multi-input Buck DC-DC Converter. As an example,
the guarded predicate that models (with 57
constraints) a 6-inputs Buck DC-DC Converter is
translated in a conjunctive predicate (with 102
linear constraints) in about 40 minutes.