Understanding PML

Paulo Pinheiro da Silva

PML

• PML is a provenance language (a language used to encode provenance knowledge) that has been proudly derived from well-established, centuries-old proof theory

Outline

• Proof Theory 101• Proofs and Provenance• Proofs vs. Justifications• PML• Conclusions

Proof Theory 101

• Proof theory is about the mechanics of one deriving new knowledge from things that one knows

• “Proof theory is syntactic in nature” (Wikipedia);

– “syntax is the way in which objects are put together; a connected or orderly system : harmonious arrangement of parts or elements”

(Merriam-Webster)

Which mechanics are you talking about?

• One knows the following:– “if it is raining then it is wet outside”– “it is raining”

• Proof theory is used to explain how one derives the following conclusion from the knowledge above:– “it is wet outside”

• The mechanics of deriving conclusions from knowledge is based on the use of inference rules

Inference Rule (1/2)

• “A rule of inference consists of (1) a set of sentence patterns called conditions, and (2) another set of sentence patterns called conclusions’ Whenever we have sentences that match the conditions of the rules, the it is acceptable to infer sentences matching the conclusions”

(Logical Foundations of Artificial Intelligence, Genesereth and Nilsson)

Inference Rule (2/2)

• Modus Ponens:

(A -> B) (A) (B). In plain English:

We know (A -> B) and (A) and we don’t know (B). If we apply the “Modus Ponens” rule to the facts that we know then we can conclude (B)

• (A-B), (A) and (B) are all sentence patterns

Inference Step (1/3)

• An inference step is a process of applying an inference rule to some antecedents to derive a conclusion

• Example:– Antecedents:• “if it is raining then it is wet outside”• “it is raining”

– Inference Rule: Modus Ponens (MP)– Conclusion:• “it is wet outside”

Inference Step (2/3)

• But how was the conclusion derived from the antecedents?

• Consider the following mappings:A = “It is raining”B = “It is wet outside”

• Then we can say that“If it is raining then it is wet outside” corresponds to (A->B)

• Thus, we can apply Modus Ponens to (A->B) and (A) and conclude B!

Inference Step (3/3)

“If it is raining then it is wet outside” “it is raining”

“it is wet outside” MP

(A -> B)(A)

(B)

(A -> B) AB

Understanding the meaning of a proof (1/2)

• What can we infer from the proof below?

• We cannot infer that it is wet outside!• Instead, we can infer the following:

If the following is true– “If it is raining then it is wet outside”– “If it is raining”

and the MP rule is sound than it is true that

– “it is wet outside”

“If it is raining then it is wet outside” “it is raining”

“it is wet outside” MP

Understanding the meaning of a proof (2/2)

• Proofs do not say what is true of false• It is up to the one using a proof to accept

(believe) the ground knowledge used in the proof to thus believe its final conclusion

• That is why every ground fact in a proof is called an axiom, which is defined as “a self-evident truth that requires no proof” (Dictionary.com)

Revisiting the Proof Terminology

“If it is raining then it is wet outside” “it is raining”

“it is wet outside” MP

• “it is raining” is a sentence• With respect to the proof, “it is raining” is an axiom (of

the proof)• With respect to the only inference step of the proof, “it

is raining” is an antecedent

A More Complex Proof

(A -> B) A

B MP

C ^I

(B^C) MP D

(B^C) -> D

Outline

• Proof Theory 101• Proofs and Provenance• Proofs vs. Justifications• PML• Conclusions

Some provenance questions may be answered by proofs:– Why is it wet outside?– Did someone say that is wet outside or was it

derived from other things one knows?– If it was derived, how was the derivation?

How proofs and provenance are related?

(“it is raining” -> “it is wet outside”) “it is raining”

“it is wet outside” MP

How proofs and provenance are related?

• Other provenance questions, however, are not answered by proofs (although proofs may help answering these questions)

(“it is raining” -> “it is wet outside”) “it is raining”

“it is wet outside” MP

Who said that it is raining?

What is MP?

Who decided to use MP?

Outline

• Proof Theory 101• Proofs and Provenance• Proofs vs. Justifications• PML• Conclusions

Proofs vs. Justifications (1/4)• Proofs

• Justifications

(“it is raining” -> “it is wet outside”) “it is raining”

“it is wet outside” MP

(KNN) k-Nearest Neighbor

algorithm

5Irregularly spaced measurements

Regularly spaced “measurements”

Proofs vs. Justifications (2/4)What are the differences?• WRT antecedents and conclusions

– “it is raining” is a object of class class Sentence

– is a “dataset” object, which is not necessarily an

instance of any formal class of objects

• WRT inference rules– MP is defined in terms of variables of strongly typed objects, i.e.,

sentences– KNN is defined in terms of parameters that are not necessarily

strongly typed, e.g., many kinds of objects can be transformed by KNN

Irregularly spaced measurements

Proofs vs. Justifications (3/4)

Basically, every proof is a justification but not every justification is a proof. Some people requires a justification to be defined in terms of the following to be considered a proof:

1. Strongly typed objects2. Inference rules that are defined in terms of patterns of

classes of objects3. Inference steps where inference rule patterns can be

verified against antecedents and conclusions

Proofs vs. Justifications (4/4)

• Many of the “proofs” that we see out there such as execution traces, information transformations traces are not formal enough to be considered proofs

• Even things that are sometimes called proofs are not formal enough to be “proper” proofs

• The term justification is used in PML to refer to both formal justifications (i.e., proofs) and informal justifications

Outline

• Proof Theory 101• Proofs and Provenance• Proofs vs. Justifications• PML• Conclusions

PML

• Four principles:1. Leverage proof theory2. Expand proof representation to support multiple

justifications3. Expand proof representation to support assertions4. Expand proof representation to support

underspecified inference rules

PML: Proof Theory

• These are PML concepts and properties derived from proof theory:– Inference Rule– Inference Step– Has Conclusion of type Information– Has Antecedent– Has Inference Engine of type Inference Engine

PML: Proofs as Graphs

• An ordinary proof can be represented by a graph composed of nodes:

(“it is raining” -> “it is wet outside”) “it is raining”

“it is wet outside” MP

(“it is raining” -> “it is wet outside”)

“it is wet outside”

“it is raining”

translated into a graph

PML: Multiple Justifications (1/2)

• PML needs to capture the diversity of justifications for any kind of conclusion:

• Multiple justifications come from the fact that a single conclusion can have more than one justification

e.g., “it is wet outside” because of the following– Paulo said so– The Weather Channel said so–

(“it is raining” -> “it is wet outside”) “it is raining”

“it is wet outside” MP

PML: Multiple Justifications (2/2)

• In an ordinary proof, a node is an inference step with the following: – A single inference rule– A single conclusion– A single set of antecedents

• PML has Node Sets, which are collection of Inference Steps with a common conclusion – A single conclusion– Zero, one or many inference steps, each step with the

following:• A single inference rule• A single set of antecedents

PML: Assertions (1/2)

• This is a part of PML that is not covered by proof theory (every ground “conclusion” in proof theory is just an axiom)

• A question: who said that “it is raining?”• An answer: Paulo said that it is raining– Paulo is a PML Source capable of asserting

Information– Source Usage is the PML concept connecting

Sources and Information

PML: Assertions (2/2)

• Why PML needs Source Usage?– the relationship between Source and Information

is 1 to many since Paulo can asserts many things– Some provenance questions are supported by

Source Usage. For example, when Paulo said that “it is raining?” According to the hasDateTime property of source usage, it is possible to say that Paulo said it yesterday

PML: Inference Rule Definition Revisited

• A rule of inference consists of (1) a set of objects called conditions, and (2) another set of objects called conclusions. Whenever we have objects that match the conditions of the rules, the it is acceptable to infer objects matching the conclusions. Rules may have no matching conditions for objects.

Inference Rule Taxonomy

Inference Rule

Declarative Rule Method Rule

Outline

• Proof Theory 101• Proofs and Provenance• Proof vs. Justification• PML• Conclusions

Conclusions

• Proof theory has been developed over centuries to support a category of provenance questions

• PML is based on proof theory• PML relaxes the proof theory notion of conclusion– Based on Information instead of (logical) sentence

• PML relaxed the proof theory notion of inference rules– Based on objects instead of (logical) sentences. Rules

do not need to provide matching conditions

Conclusions

• PML supports multiple justifications for a single conclusion– supported by Node Sets and their Inference Rules

• PML supports the encoding of assertions– Supported by Sources and SourceUsage