Introducing Temporal Logic

1
Introducing Temporal Logic

• Weimin Ma

2
Introduction

• Time has traditionally been represented as
  another variable in physical science.
• For example, mass conservation equation in Fluid
  Mechanics
• Which asserts that the mass of a material
  particle remains invariant.

3
Introduction

• Do we have time variables in formal logic?
• Yes, first order predicate calculus is used to
  reason about expressions containing time
  variable.
• But, there is still need for a special temporal
  logic without explicit time variable.

4
Introduction

• What is Temporal Logic?
• A formalism for describing sequences of
  transitions between states in a reactive system,
  with time not mentioned explicitly. Instead, a
  formula might specify that eventually some
  designed state is reached, or an error state is
  never entered.

5
Overview

• Precondition
• A structure of states (e.g. a sequence or tree
  of states) is the key concept that makes temporal
  logic suitable for program specification.

6
Overview

• Internal semantics of temporal logic
• -- Representing finite chunks of system
  behavior, thus leading to the chop operator.
• -- chop operator, , with the interpretation
  that pq is true on an interval so long as that
  interval can be decomposed into two continuous
  subintervals in such a way that p is true on the
  earlier of the two, and q on the later.

7
Overview

• Point semantics of temporal logic
• Requires some system behavior with respect to a
  certain reference point in time.
• -- Linear semantics
• Each moment has only one possible future
• -- Branching semantics
• Time has a tree-like nature
• -- Partial order semantics

8
Overview

• Structures to represent program executions?
• -- Maximal parallelism
• The number of instructions in concurrent
• processes that can be executed
  simultaneously is
• maximized.
• -- Interleaving executions
• Represented by interleaving their atomic
  actions.

9
Approaches to Temporal Logic

• Modal-logic approach
• -- Tense Logic
• -- Extensions to Tense Logic
• -- Semantics of tense Logic
• Predicate-logic approach
• -- Method of temporal arguments
• -- State and event-type reification

10
Tense Logic

• In addition to the usual truth-functional
  operators, Tense Logic contains above four modal
  operators with intended meanings.
• For example
• ?x(Philosopher(x) F King(x)) Someone who is
  now a philosopher will be a king at some future
  time.

11
Extensions to Tense Logic

• Binary temporal operators S and U (since and
  until)

• Next time operator ?
• This operator assumes that the time series
  consists of a discrete sequence of atomic
  times. ?p means p is true at the immediately
  succeeding time step.

12
Semantics of Tense Logic

• A temporal frame consists of a set T of entities
  called times together with an ordering relation lt
  on T.
• This defines the flow of time over which the
  meanings of the tense operators are to be
  defined.
• Example
• Fp is true at t if and only if p is true at some
  time
• such that t lt .
• Gp is true at t if and only if p is true at all
  times such that tlt .

13
Methods of Temporal Arguments

• Temporal dimension is captured by augmenting each
  predicate with an extra argument to be filled by
  an expression designating a time.
• For example, Pp means ?t(tltnow p(t)) where p(t)
  represents the result of introducing an extra
  temporal argument to the predicates occurring in
  p.
• Before the advent of Tense Logic, the method of
  temporal arguments was the natural choice for
  modeling temporal information.

14
State and event-type reification

• State and event types are denoted by terms in a
  first-order theory, their temporal incidence is
  expressed using relational predicates Holds and
  Occurs, for example
• Holds(Asleep(Mary),(1 pm,6pm)).
• The homogeneity of states and inhomogeneity of
  events is secured by axioms
• ?s,i,j(Holds(s,i) In(i,j) ?Holds(s,j))
• ?e,i,j(Occurs(e,i) In(i,j)??Occurs(e,j))

15
Application of Temporal Logic

• Applications to natural language
• Applications in artificial intelligence
• Applications in computer science

16
Applications in Computer Science

• Extensive application in specification and
  verification of programs, especially concurrent
  programs in which the computation is performed by
  two/more processors/processes in parallel.
• Among the key notations is the distinction
  between liveness properties in the form Fp,
  which ensure that desirable states will obtain in
  the course of computation, and safety
  properties in the form Gp, which ensure that
  undesirable states will never obtain.

17
Conclusions

• Temporal logic is a logic of propositions whose
  truth and falsity may depend on time, which is
  closely related to modal logics.
• Suggested references for detail explanation
• 1 Antony Galton, Temporal logics and their
  applications, Academic Press, 1987 ISBN
  0122740602
• 2 Fred Kroger, Temporal logic of programs,
  Springer-Verlag, 1987 ISBN 0387170308

18
References

• 1 Jonathan S. Ostroff, Temporal logic for
  real-time systems, Research Studies Press Ltd,
  1989
• 2 Online tutorial, Temporal logic,
  http//plato.stanford.edu/entries/logic-temporal/