# Semantic Foundation of the Tagged Signal Model

### Xiaojun Liu

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2005-31

December 20, 2005

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/EECS-2005-31.pdf

The tagged signal model provides a denotational framework to study properties of various models of computation. It is a generalization of the Signals and Systems approach to system modeling and specification. Having different models of computation or aspects of them specified in the tagged signal model framework provides the following opportunities. First, one can compare certain properties of the models of computation, such as their notion of synchrony. Such comparisons highlight both the differences and the commonalities among the models of computation. Second, one can define formal relations among signals and process behaviors from different models of computation. These relations have important applications in the specification and design of heterogeneous embedded systems. Third, it facilitates the cross-fertilization of results and proof techniques among models of computation. This opportunity is exploited extensively in this dissertation. The main goal of this dissertation is to establish a semantic foundation for the tagged signal model. Both order-theoretic and metric-theoretic concepts and approaches are used. The fundamental concepts of the tagged signal model¿signals, processes, and networks of processes¿are formally defined. From few assumptions on the tag sets of signals, it is shown that the set of all signals with the same partially ordered tag set and the same value set is a complete partial order. This leads to a direct generalization of Kahn process networks to tagged process networks. Building on this result, the order-theoretic approach is further applied to study timed process networks, in which all signals share the same totally ordered tag set. The order structure of timed signals provides new characterizations of the common notion of causality and the discreteness of timed signals. Combining the causality and the discreteness conditions is proved to guarantee the non-Zenoness of timed process networks. The metric structure of tagged signals is studied from the very specific¿the Cantor metric and its properties. A generalized ultrametric on tagged signals is proposed, which provides a framework for defining more specialized metrics, such as the extension of the Cantor metric to super-dense time. The tagged signal model provides not only a framework for studying the denotational semantics of models of computation, but also useful constructs for studying implementations or simulations of tagged processes. This is demonstrated by deriving certain properties of two discrete event simulation strategies from the behavioral specifications of discrete event processes. A formulation of tagged processes as labeled transition systems provides yet another framework for comparing different implementation or simulation strategies for tagged processes. This formulation lays the foundation to future research in polymorphic implementations of tagged processes.

**Advisor:** Edward A. Lee

BibTeX citation:

@phdthesis{Liu:EECS-2005-31, Author = {Liu, Xiaojun}, Title = {Semantic Foundation of the Tagged Signal Model}, School = {EECS Department, University of California, Berkeley}, Year = {2005}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/EECS-2005-31.html}, Number = {UCB/EECS-2005-31}, Abstract = {The tagged signal model provides a denotational framework to study properties of various models of computation. It is a generalization of the Signals and Systems approach to system modeling and specification. Having different models of computation or aspects of them specified in the tagged signal model framework provides the following opportunities. First, one can compare certain properties of the models of computation, such as their notion of synchrony. Such comparisons highlight both the differences and the commonalities among the models of computation. Second, one can define formal relations among signals and process behaviors from different models of computation. These relations have important applications in the specification and design of heterogeneous embedded systems. Third, it facilitates the cross-fertilization of results and proof techniques among models of computation. This opportunity is exploited extensively in this dissertation. The main goal of this dissertation is to establish a semantic foundation for the tagged signal model. Both order-theoretic and metric-theoretic concepts and approaches are used. The fundamental concepts of the tagged signal model¿signals, processes, and networks of processes¿are formally defined. From few assumptions on the tag sets of signals, it is shown that the set of all signals with the same partially ordered tag set and the same value set is a complete partial order. This leads to a direct generalization of Kahn process networks to tagged process networks. Building on this result, the order-theoretic approach is further applied to study timed process networks, in which all signals share the same totally ordered tag set. The order structure of timed signals provides new characterizations of the common notion of causality and the discreteness of timed signals. Combining the causality and the discreteness conditions is proved to guarantee the non-Zenoness of timed process networks. The metric structure of tagged signals is studied from the very specific¿the Cantor metric and its properties. A generalized ultrametric on tagged signals is proposed, which provides a framework for defining more specialized metrics, such as the extension of the Cantor metric to super-dense time. The tagged signal model provides not only a framework for studying the denotational semantics of models of computation, but also useful constructs for studying implementations or simulations of tagged processes. This is demonstrated by deriving certain properties of two discrete event simulation strategies from the behavioral specifications of discrete event processes. A formulation of tagged processes as labeled transition systems provides yet another framework for comparing different implementation or simulation strategies for tagged processes. This formulation lays the foundation to future research in polymorphic implementations of tagged processes.} }

EndNote citation:

%0 Thesis %A Liu, Xiaojun %T Semantic Foundation of the Tagged Signal Model %I EECS Department, University of California, Berkeley %D 2005 %8 December 20 %@ UCB/EECS-2005-31 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2005/EECS-2005-31.html %F Liu:EECS-2005-31