Factored Edge-Valued Binary Decision Diagrams form an extension to Edge-Valued Binary Decision Diagrams. By associating both an additive and a multiplicative weight with the edges, FEVBDDs can be used to represent a wider range of functions concisely. As a result, the computational complexity for certain operations can be significantly reduced compared to EVBDDs. Additionally, the introduction of multiplicative edge weights allows us to directly represent the so-called complement edges which are used in OBDDs, thus providing a one to one mapping of all OBDDs to FEVBDDs. Applications such as integer linear programming and logic verification that have been proposed for EVBDDs also benefit from the extension. We present a complete matrix package based on FEVBDDs and apply the package to the problem of solving the Chapman-Kolmogorov equations.

Abstract. This article presents a way of implementing abstract interpretations that can be very efficient. The improvement lies in the use of a symbolic representation of boolean functions called Typed Decision Graphs (TDGs), a refinement of Binary Decision Diagrams. A general procedure for using this representation in abstract interpretation is given; we examine in particular the possibility of encoding higher order functions into TDGs. Moreover, this representation is used to design a widening operator based on the size of the objects represented, so that abstract interpretations will not fail due to insufficient memory. This approach is illustrated on strictness analysis of higher-order functions, showing a great increase in efficiency. 1

This report documents the implementation of a Boolean constraint solver and its integration with a Prolog engine. The solver comprises built-in predicates for checking consistency and entailment of a new constraint w.r.t. accumulated constraints and for generating particular solutions to a set of constraints, and extensions to the Prolog top-level for displaying answer constraints. Boolean unification was chosen as the strategy for the consistency check. Boolean unification fits well with the Prolog execution model, and allows the accumulated constraints to be associated in a natural way with the variables being eliminated. The answer constraints are computed by existentially quantifying in the accumulated set of constraints all variables not occurring in the user query. The simplified set of constraints constitutes the answer constraints. Boolean formulas are internally represented as DAGs. Details are provided on this representation and the support for it provided by the Prolog Engin...

This paper starts from the premise that the human contribution to risk must be assessed during the development of safety-critical systems. In contrast to previous approaches, discrete numerical values are rejected as means of quantifying the probability of operator `error' for many different users of many different systems. Numerical probabilities are used to rank the importance that designers attach to particular system failures. Adequate development resources must be allocated so that operators will resolve and not exacerbate high priority failures. In order to do this, human factors and systems engineers must be provided with notations that can represent risk assessments. Many techniques that are in widespread use, such as fault-tree analysis, provide inadequate support for the development of interactive systems. They do not capture the temporal properties that can determine the quality of interaction between operators and stochastic application processes. It is argued that probabil...

The strive for efficiency is ancient and universal, as time is always short for humans. Computational Complexity is a mathematical study of the what can be achieved when time (and other resources) are scarce. In this

. We propose an algorithm for the inference of decision graphs from a set of labeled instances. In particular, we propose to infer decision graphs where the variables can only be tested in accordance with a given order and no redundant nodes exist. This type of graphs, reduced ordered decision graphs, can be used as canonical representations of Boolean functions and can be manipulated using algorithms developed for that purpose. This work proposes a local optimization algorithm that generates compact decision graphs by performing local changes in an existing graph until a minimum is reached. The algorithm uses Rissanen's minimum description length principle to control the tradeoff between accuracy in the training set and complexity of the description. Techniques for the selection of the initial decision graph and for the selection of an appropriate ordering of the variables are also presented. Experimental results obtained using this algorithm in two sets of examples are presented and ...

Abstract. Extended Feature Oriented Domain Analysis (FODA) Feature Diagrams (EFD) were introduced to compensate for a purported ambiguity and lack of precision and expressiveness of the original FODA feature diagrams (OFD). However, EFD never received a formal semantics, which is the hallmark of precision and unambiguity. We propose here a semantics for both diagrams. From this we demonstrate that OFD are precise, unambiguous, and expressively complete, and thus that all extensions add no expressiveness. A finer notion is thus needed to compare these languages. Two solutions are well-established: succinctness and embeddability, that measures naturalness of a language. This tool shows that EFD indeed bring some naturalness, but are harmfully redundant and that the same naturalness can be attained with the simpler varied FD (VFD). We also show that no ambiguity is present, in fact.

In his 1938 Master’s Thesis, Shannon demonstrated that any Boolean function can be realized by a switching relay circuit, leading to the development of deterministic digital logic. Here, we replace each classical switch with a probabilistic switch (pswitch). We present algorithms for synthesizing circuits closed with a desired probability, including an algorithm that generates optimal size circuits for any binary fraction. We also introduce a new duality property for series-parallel stochastic switching circuits. Finally, we construct a universal probability generator which maps deterministic inputs to arbitrary probabilistic outputs. Potential applications exist in the analysis and design of stochastic networks in biology and engineering. I. INTRODUCTION. Claude Shannon, in his 1938 Master’s Thesis, discovered a systematic synthesis procedure to realize a given Boolean function using deterministic switches [Sha38]. This classical contribution led to the development of modern digital logic design and is at the foundation of our ability to design and manufacture digital circuits with millions of transistors.

This technical report presents a new theoretical approach to the problem of switching networks synthesis with McCulloch-Pitts feed-forward neural networks. It is shown that any n-inputs periodical symmetric Boolean function F p with the period T and the first positive transition at x = a can be implemented with a 1 + dlog n\Gammaa T e depth and size network both measured in term of neurons, when a period contains two transitions. It can be implemented with a t+dlog n\Gammaa T e depth and size network when a period contains more than two transitions, where t is the number of neural elements necessary to implement the restriction of F p to the first period, i.e. the input interval [0; T ]. An asymptotic bound of O(log n) for the network (for both size and depth) is also derived for symmetric Boolean functions that can be decomposed in l periodic symmetric Boolean sub-functions. ii iii Contents ABSTRACT ii LIST OF FIGURES vi 1 Periodic Symmetric Functions with Feed-Forward Neural...

BDDs (binary decision diagrams) are a very succesful tool for handling boolean functions, but one which has not yet attracted the attention of many automated deduction specialists. We give an overview of BDDs from an automated deduction perspective, showing what can be done with them in propositional and first-order logic, and discuss the parallels to well-known methods like tableaux and resolution.