# Artificial Intelligence – Temporal Logic

• Last Updated : 12 Jan, 2022

### Introduction:

The phrase temporal logic refers to any system that uses rules and symbolism for representing and reasoning about propositions that are time-limited. Tense logic is a term that is occasionally used to describe it.
More precisely, temporal logic is concerned with tense and employs modal operators in relation to temporal concepts such as sometimes, always, precedes, succeeds, and so on. Arthur Prior introduced a special modal logic-system of temporal logic in the 1960s.
In addition to the normal logical operators, temporal logic contains four modal operators with the following intended meanings:

In order to create a propositional tense logic, the operators G and F are used to refer to the future, whereas the operators H and P are used to refer to the past. The operators P & F are known as the weak tense operators, while the operators H and G are known as strong tense operators. By reason of equivalence, the two pairings are commonly viewed as inter definable. Assume Q is some closed formula in conventional Logic. The 2 axioms govern the interaction between the past and future operators :

```FQ ≅ ~G~Q
PQ ≅ ~H~Q```

NOTE: If a variable appears in quantification or is inside the scope of quantification of that variable, it is bound. Otherwise, the variable is free. A closed formula is one that has no free variables; otherwise, it is an open formula.

Arthur employed the operators to create formulas that could express the concepts of time that had been taken as axioms of a formal system based on these intended meanings.

### Distribution Axioms :

The distribution axioms are :

```G(Q -> R) -> (GQ -> GR) interpreted as:
If it will always be the case that Q implies R, then
if Q will always be the case,then R will be always be so.

H ( Q->R) -> (HQ -> HR) interpreted as:
If Q has implies R, then
if Q has always been the case,then R will be always be so.```

### General Axioms in Temporal Logic :

Depending on the assumptions we make about the structure of time, further axioms can be added to temporal logic. A set of commonly adopted axioms is :

`GQ -> FQ and HQ -> PQ`

Certain combinations of past & future tense operators may be used to express complex tense in English. Example: FPQ corresponds to a sentence Q in the future perfect tense. PPQ expresses past perfect tense.
The useful axioms in Temporal Logic :
(i) It has always been the case that Q is true is equivalent to It is not in the past that Q was false.

`HQ ≅ ~P~Q`

(ii) It will always be the case that Q will be true is equivalent to It is not in the future that Q will be false.

`GQ ≅ ~F~Q`

(iii) It will always be the case in future that Q will be true is equivalent to It will not be always that Q will be false.

`FQ ≅ ~G~Q`

(iv) It was the case that Q was true is equivalent to It has not always been the case in past that Q was false.

`PQ ≅ ~H~Q`

(v) It will not always be the case that Q will be true is equivalent to It will be the case in the future that Q will be false.

`~GQ ≅ F~Q`

(vi) It has not always been the case in the past that Q was true is equivalent to It was the case in the past that Q was false.

`~HQ ≅ P~Q`

Inference Rules in Temporal Logic :

Following inference rules in Temporal Logic can be used to infer new information from existing knowledge :

1. If it is true in future that Q will be true in the future, then we infer that Q will be true in the future. FFQ -> FQ
2. If Q is true now, then we infer that in the future it will be the case that Q was true in the past. Q -> FPQ
3. If Q has always been true in the past, then we infer that Q is true HQ -> Q
4. If Q will be always true in future then we infer that Q is true. GQ -> Q
5. Distribution Axioms :
– G(Q -> R) -> (GQ -> GR) interpreted as If it will always be the case that Q implies R, then if Q will always be the case, always be then R will always be so.
– H ( Q->R) -> (HQ -> HR) interpreted as If Q has implied R, then if Q has always been the case, then R will be always be so.
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