Benjamin Gamble

What is Time?  Can mean many different things to a computer Dynamic Equation Variable System State 2

Outline  History  Languages & Interpretations  Dimensions of the Time Modeling Problem Discrete vs. Dense Finite or Bounded Time Models Linear vs. Branching Determinism vs. Nondeterminism Time Advancement Problem Concurrency & Composition  Analysis & Verification Issues 3

History of Time in Computing  First implemented in computer hardware  Then adapted for software and computational complexity  Parallel processing mandated a further investigation in timing issues  Modern day real time systems 4

Outline  History  Languages & Interpretations  Dimensions of the Time Modeling Problem Discrete vs. Dense Finite or Bounded Time Models Linear vs. Branching Determinism vs. Nondeterminism Time Advancement Problem Concurrency & Composition  Analysis & Verification Issues 5

Choosing the Right Language 6

Two Main Classes of Languages  Operational Well suited to describe the evolution of a system starting at some initial state Based on the key concept of state and transition Modeled as evolving from a state to the next one when a certain event occurs 7

Operational Example  A safe When the correct security code is entered, the safe opens If the safe remains open for 3 minutes, it automatically closes 8

Two Main Classes of Languages  Descriptive Better suited for describing the properties that the system must satisfy Logic based formalisms Algebra based formalisms 9

Descriptive Example  A safe The safe is open if and only if the correct security code has been entered no more than 3 minutes ago 10

Outline  History  Languages & Interpretations  Dimensions of the Time Modeling Problem Discrete vs. Dense Finite or Bounded Time Models Linear vs. Branching Determinism vs. Nondeterminism Time Advancement Problem Concurrency & Composition  Analysis & Verification Issues 11

Discrete vs. Dense Time Domains  Discrete time means that there is a set of isolated points  Dense time means that for every 2 points there is always a third point in between 12

Continuous vs. Non-Continuous Time Models 13

Continuous vs. Non-Continuous Time Models 14

Finite or Bounded Time Models  Many system models assume that the behaviors may proceed indefinitely into the future and maybe even the past  This models time as an unbounded set  Some system behaviors happen within a time window  This will require a bounded time model 15

Bounded Time Model Example  Braking a car We know that it requires, at most, a few seconds Testing an anti-lock braking system may include a time window of 0-60 seconds as a temporal domain 16

Hybrid Systems  What is a hybrid system? Uses both discrete and dense time modeling  When this may be used: Square wave form Continuous sampled at certain points Discrete steps with continuous variables 17

Linear vs. Branching Time Models  Linear – interpreted over linear sequences of states 18

Linear vs. Branching Time Models  Branching – interpreted over trees of states 19

Determinism vs. Nondeterminism  Deterministic Whenever the future state of the system is uniquely determined by its current state and input values Ex. A light switch ○ Pressing the switch (input) while the light is in state ‘off’ yields the unique possible state of light ‘on’ 20

Determinism vs. Nondeterminism  Nondeterministic Systems that can evolve to different future states from the same present state and the same input by making arbitrary ‘choices’ Ex. A resource arbiter ○ Responds to 2 requests happening at the same time by ‘choosing’ arbitrarily to whom to grant the resource first 21

Implicit vs. Explicit Time Reference  Explicit Time Reference Using math such as calculus to specify system behavior and its properties  Implicit Time Reference Refers to a ‘current time’ and orders events with respect to it 22

Time Advancement Problem  This problem arises when the model of a timed system exhibits behaviors that do not progress past some instant Ex. Pushing a button ○ Actual duration to push a button can be ignored and can be represented abstractly as a zero time event ○ If these types of events are allowed then there could possibly be an infinite number of such transitions in a zero time interval 23

Solutions to the Time Advancement Problem  Priori The syntax or the semantics of the formal notation is restricted beforehand, in order to guarantee that the model of any system described with it will be exempt from time advancement problems 24

Solutions to the Time Advancement Problem  Posteriori Deals with time advancement issues after the system specification has been built It is then analyzed against a formal definition of time advancement in order to check that all of its actual behaviors do not incur into the time advancement problem 25

Concurrency & Composition  Composition also known as modularization  Problems arise when analyzing synchronous subsystems  A classification of dealing with the composition of concurrent units can be spilt in two ways Synchronous vs. Asynchronous 26

Synchronous vs. Asynchronous  Synchronous Occur at the same time or at instants of time that are related Naturally paired with a discrete time domain  Asynchronous Each activity can progress at a speed unrelated to others There is no need to know in which state each unit is at every instant 27

Outline  History  Languages & Interpretations  Dimensions of the Time Modeling Problem Discrete vs. Dense Finite or Bounded Time Models Linear vs. Branching Determinism vs. Nondeterminism Time Advancement Problem Concurrency & Composition  Analysis & Verification Issues 28

Expressiveness  The possibility of characterizing extensive classes of properties  A language is more expressive if it can be finely tuned into a set of behaviors that satisfy certain properties  For each language there exists properties that can only be expressed by them 29

Decidability & Complexity  Decidability There is a trade-off between expressiveness and decidability  Complexity is used for decidable models Used to determine the effort required by an algorithm to decide whether a property holds Measured in the amount of memory or time required as a function of the input 30

Analysis & Verification Techniques  Exhaustive Enumeration Automated Exploration of graphs  Syntactic Transformations Address the verification problem by logical deductions Specification and requirements are in descriptive form 31

Outline  History  Languages & Interpretations  Dimensions of the Time Modeling Problem Discrete vs. Dense Finite or Bounded Time Models Linear vs. Branching Determinism vs. Nondeterminism Time Advancement Problem Concurrency & Composition  Analysis & Verification Issues 32

Questions? 33