Weighted Linear Dynamic Logic

We introduce a weighted linear dynamic logic (weighted LDL for short) and show the expressive equivalence of its formulas to weighted rational expressions. This adds a new characterization for recognizable series to the fundamental Schützenberger theorem. Surprisingly, the equivalence does not require any restriction to our weighted LDL. Our results hold over arbitrary (resp. totally complete) semirings for finite (resp. infinite) words. As a consequence, the equivalence problem for weighted LDL formulas over fields is decidable in doubly exponential time. In contrast to classical logics, we show that our weighted LDL is expressively incomparable to weighted LTL for finite words. We determine a fragment of the weighted LTL such that series over finite and infinite words definable by LTL formulas in this fragment are definable also by weighted LDL formulas

Interaction Graphs: Full Linear Logic

Interaction graphs were introduced as a general, uniform, construction of dynamic models of linear logic, encompassing all Geometry of Interaction (GoI) constructions introduced so far. This series of work was inspired from Girard's hyperfinite GoI, and develops a quantitative approach that should be understood as a dynamic version of weighted relational models. Until now, the interaction graphs framework has been shown to deal with exponentials for the constrained system ELL (Elementary Linear Logic) while keeping its quantitative aspect. Adapting older constructions by Girard, one can clearly define "full" exponentials, but at the cost of these quantitative features. We show here that allowing interpretations of proofs to use continuous (yet finite in a measure-theoretic sense) sets of states, as opposed to earlier Interaction Graphs constructions were these sets of states were discrete (and finite), provides a model for full linear logic with second order quantification

Weighted recognizability over infinite alphabets

We introduce weighted variable automata over infinite alphabets and commutative semirings. We prove that the class of their behaviors is closed under sum, and under scalar, Hadamard, Cauchy, and shuffle products, as well as star operation. Furthermore, we consider rational series over infinite alphabets and we state a Kleene-Schützenberger theorem. We introduce a weighted monadic second order logic and a weighted linear dynamic logic over infinite alphabets and investigate their relation to weighted variable automata. An application of our theory, to series over the Boolean semiring, concludes to new results for the class of languages accepted by variable automata

Digital IP Protection Using Threshold Voltage Control

This paper proposes a method to completely hide the functionality of a digital standard cell. This is accomplished by a differential threshold logic gate (TLG). A TLG with $n$ inputs implements a subset of Boolean functions of $n$ variables that are linear threshold functions. The output of such a gate is one if and only if an integer weighted linear arithmetic sum of the inputs equals or exceeds a given integer threshold. We present a novel architecture of a TLG that not only allows a single TLG to implement a large number of complex logic functions, which would require multiple levels of logic when implemented using conventional logic primitives, but also allows the selection of that subset of functions by assignment of the transistor threshold voltages to the input transistors. To obfuscate the functionality of the TLG, weights of some inputs are set to zero by setting their device threshold to be a high $V_t$. The threshold voltage of the remaining transistors is set to low $V_t$ to increase their transconductance. The function of a TLG is not determined by the cell itself but rather the signals that are connected to its inputs. This makes it possible to hide the support set of the function by essentially removing some variable from the support set of the function by selective assignment of high and low $V_t$ to the input transistors. We describe how a standard cell library of TLGs can be mixed with conventional standard cells to realize complex logic circuits, whose function can never be discovered by reverse engineering. A 32-bit Wallace tree multiplier and a 28-bit 4-tap filter were synthesized on an ST 65nm process, placed and routed, then simulated including extracted parastics with and without obfuscation. Both obfuscated designs had much lower area (25%) and much lower dynamic power (30%) than their nonobfuscated CMOS counterparts, operating at the same frequency

Bayesian Updating, Model Class Selection and Robust Stochastic Predictions of Structural Response

A fundamental issue when predicting structural response by using mathematical models is how to treat both modeling and excitation uncertainty. A general framework for this is presented which uses probability as a multi-valued conditional logic for quantitative plausible reasoning in the presence of uncertainty due to incomplete information. The fundamental probability models that represent the structure’s uncertain behavior are specified by the choice of a stochastic system model class: a set of input-output probability models for the structure and a prior probability distribution over this set that quantifies the relative plausibility of each model. A model class can be constructed from a parameterized deterministic structural model by stochastic embedding utilizing Jaynes’ Principle of Maximum Information Entropy. Robust predictive analyses use the entire model class with the probabilistic predictions of each model being weighted by its prior probability, or if structural response data is available, by its posterior probability from Bayes’ Theorem for the model class. Additional robustness to modeling uncertainty comes from combining the robust predictions of each model class in a set of competing candidates weighted by the prior or posterior probability of the model class, the latter being computed from Bayes’ Theorem. This higherlevel application of Bayes’ Theorem automatically applies a quantitative Ockham razor that penalizes the data-fit of more complex model classes that extract more information from the data. Robust predictive analyses involve integrals over highdimensional spaces that usually must be evaluated numerically. Published applications have used Laplace's method of asymptotic approximation or Markov Chain Monte Carlo algorithms