arXiv subjects

Udi Boker

Publications and source records attributed to Udi Boker.

21 records · Page 2Linked to original sources

arXiv2014

Discounting in LTL

In recent years, there is growing need and interest in formalizing and reasoning about the quality of software and hardware systems. As opposed to traditional verification, where one handles the question of whether a system satisfies, or not, a given specification, reasoning about quality addresses the question of \emph{how well} the system satisfies the specification. One direction in this effort is to refine the "eventually" operators of temporal logic to {\em discounting operators}: the satisfaction value of a specification is a value in $[0,1]$, where the longer it takes to fulfill eventuality requirements, the smaller the satisfaction value is. In this paper we introduce an augmentation by discounting of Linear Temporal Logic (LTL), and study it, as well as its combination with propositional quality operators. We show that one can augment LTL with an arbitrary set of discounting functions, while preserving the decidability of the model-checking problem. Further augmenting the logic with unary propositional quality operators preserves decidability, whereas adding an average-operator makes some problems undecidable. We also discuss the complexity of the problem, as well as various extensions.

cs.LO↗

arXiv2014

Exact and Approximate Determinization of Discounted-Sum Automata

A discounted-sum automaton (NDA) is a nondeterministic finite automaton with edge weights, valuing a run by the discounted sum of visited edge weights. More precisely, the weight in the i-th position of the run is divided by $λ^i$, where the discount factor $λ$ is a fixed rational number greater than 1. The value of a word is the minimal value of the automaton runs on it. Discounted summation is a common and useful measuring scheme, especially for infinite sequences, reflecting the assumption that earlier weights are more important than later weights. Unfortunately, determinization of NDAs, which is often essential in formal verification, is, in general, not possible. We provide positive news, showing that every NDA with an integral discount factor is determinizable. We complete the picture by proving that the integers characterize exactly the discount factors that guarantee determinizability: for every nonintegral rational discount factor $λ$, there is a nondeterminizable $λ$-NDA. We also prove that the class of NDAs with integral discount factors enjoys closure under the algebraic operations min, max, addition, and subtraction, which is not the case for general NDAs nor for deterministic NDAs. For general NDAs, we look into approximate determinization, which is always possible as the influence of a word's suffix decays. We show that the naive approach, of unfolding the automaton computations up to a sufficient level, is doubly exponential in the discount factor. We provide an alternative construction for approximate determinization, which is singly exponential in the discount factor, in the precision, and in the number of states. We also prove matching lower bounds, showing that the exponential dependency on each of these three parameters cannot be avoided. All our results hold equally for automata over finite words and for automata over infinite words.

cs.FL↗

arXiv2005

Comparing Computational Power

It is common practice to compare the computational power of different models of computation. For example, the recursive functions are strictly more powerful than the primitive recursive functions, because the latter are a proper subset of the former (which includes Ackermann's function). Side-by-side with this "containment" method of measuring power, it is standard to use an approach based on "simulation". For example, one says that the (untyped) lambda calculus is as powerful--computationally speaking--as the partial recursive functions, because the lambda calculus can simulate all partial recursive functions by encoding the natural numbers as Church numerals. The problem is that unbridled use of these two ways of comparing power allows one to show that some computational models are strictly stronger than themselves! We argue that a better definition is that model A is strictly stronger than B if A can simulate B via some encoding, whereas B cannot simulate A under any encoding. We then show that the recursive functions are strictly stronger in this sense than the primitive recursive. We also prove that the recursive functions, partial recursive functions, and Turing machines are "complete", in the sense that no injective encoding can make them equivalent to any "hypercomputational" model.

cs.LO↗