Abductive reasoning with explicit justification Advisors Ph.D. - - PowerPoint PPT Presentation

Abductive reasoning with explicit justification

Advisors Ph.D. Francisco Hernández Quiroz (UNAM, México) Ph.D. Fernando R. Velázquez Quesada (ILLC, Amsterdam) Ph.D. Atocha Aliseda Llera (UNAM, México)

Abduction with explicit justification?

• Abduction as an input-output scheme (Estrada, 2013; Magnani, 2015, 2016).
• Justification. From justification logic (Artemov, 2008, 2010). Let Ψ be a formula and t a literal representing a

piece of evidence (i.e., a justification):

• Abductive inference with explicit justification. Let Γ be a set of justification logic formulas.

t: Ψ, “Ψ is justified by evidence t” Given “context ⇏ input”

AI input

• utput

We get “context, output ⇒ input”

AI Ψ t

Given “Γ ⇏ s: Ψ”, for every or some s We get “Γ, t ⇒ t: Ψ”

(a.k.a. AS) (a.k.a. AP)

As of this presentation…

• Show the initial steps towards our goal:

(1) Define AP and AS within justification logic framework. (2) Characterize Aliseda’s novel abductive reasoning (2006) through these definitions. (3) Expanding the taxonomy.

A taxonomy of abductive inference based on information, can be further expanded by recognizing variations on pieces justifications.

Why justification logic?

• Justification terms have a rich repertory of properties.
• A getaway from logical omniscience (Stalnaker, 1999; Dretske, 2005).
• An ever expanding set of formal tools (Baltag, Renne & Smets, 2012, 2014).

More than a non-normal modal logic, justification logic offers an alternate account of the process an agent follows in order to obtain new information: no justification means no knowledge, no matter how inferentially skilled is our agent . Its expressive power let us state a wide array of aspects regarding justifications, as “t is an accepted justification for the agent”, “t is explicit evidence of the agent”, “t is infallible evidence for the agent”… Properties regarding their semantic status (being valid, being truth), regarding their syntactic structure (being atomic, being composite). We could also define new properties from previous

• nes (being explanatory, being informative, being non-redundant), and the list goes on.

• Syntax. propositional logic + justification terms,

s, t ::= ci | xi | [s*t] | [s▹t] Where ci ∈ C, xi ∈ V, and ∀t. (t ∈ 𝓤) If t ∈ 𝓤 and φ is a formula, then, t: φ is also a formula.

• J0 system.
• A1. TAUT, such that TAUT = {φ | φ is propositionally valid}
• A2. Application Axiom: ⊢ s: (φ → ψ) → (t: φ → [s*t]: ψ)
• A3. Sum Axiom: ⊢ (s: ψ → [s▹t]: ψ), ⊢ (s: ψ → [t▹s]: ψ)
• R1. Modus Ponens: ⊢ (φ → ψ) AND ⊢ φ ⇒ ⊢ ψ
• JCS systems.

JCS = J0 + CS Where, if ci: φi ∈ CS, then ci ∈ C and φi ∈ TAUT.

Justification logic

Tracking terms

• For every t ∈ 𝓤, we get the set of atoms and sub-terms that t is made of.
• Let “◦” be a binary operation satisfying uniform substitution, such that “◦” = “*” or “◦” = “▹”.

Sub generates the set of sub-terms of any term,

• 1. Let x ∈ V. Sub(x) ≔ {x}

• 2. Let c ∈ C. Sub(c) ≔ {c}

• 3. Let r, s ∈ 𝓤. Sub([r◦s]) ≔ Sub(r) ∪ Sub(s) ∪ {[r◦s]}
• Atm generates the set of atoms of any term,
• 1. Let x ∈ V. Atm(x) ≔ {x}

• 2. Let c ∈ C. Atm(c) ≔ {c}

• 3. Let r, s ∈ 𝓤. Atm([r◦s]) ≔ Atm(r) ∪ Atm(s)

(Other tracking functions: Sub*(t), Sub▹(t), Atm*(t), Atm▹(t), etc.)

atoms sub-terms atomic terms composite terms

Sub: 𝓤 → P(𝓤) Atm: 𝓤 → P(V ∪ C)

Operability

• Operability of a term t means the difficulty to build it. We

determine its difficulty by counting over the operations

• ccurring between its sub-terms.
• For every t ∈ 𝓤, Op(t) assigns a value n ∈ ℕ,
• 1. Let x ∈ V. Op(x) ≔ 0
• 2. Let c ∈ C. Op(c) ≔ 0

^3. Let r, s ∈ 𝓤. Op([r*s]) ≔ Op(r) + Op(s) + 2

ª4. Let r, s ∈ 𝓤. Op([r▹s]) ≔ Op(r) + Op(s) + 1

^ According to A2, inferring a formula justified by a term like [r*s] requires using R1 twice.

ª According to A3, for formulas justified by terms like [r▹s], we requiere a single use of R1.

Op: 𝓤 → ℕ

Based on the method of “counting the steps” that Artemov & Kuznets (2014, p. 20-22) propose, in order to account for the complexity of inferred formulas.

!

APs and ASs in JCS

• Total AP. We have a total AP ψ in a set Γ when evidence for ψ in Γ is

non-existent. Formally,

Given that 𝓤(ψ)Γ ≔ {r ∈ 𝓤 | r: ψ ∈ Γ}, a total AP can be triggered because 𝓤(ψ)Γ = {}.

• AS. An AS for this AP is any term t, such that t: ψ ∈ ΓC.
• Partial AP. A partial AP ψ in Γ can be triggered when a specific sub-

set of terms for ψ in Γ is non-existent. E.g., ψ could be a partial AP because 𝓤(ψ)Γ = {r | r ∈ (V ∪ C)}.

• t is an AS for a partial AP iff t: ψ ∈ ΓC and, t satisfies additional

properties: e.g., t ∉ Atm(t).

• Abductive justifications. Abductive justifications for an AP ψ in Γ are those

terms that, had they been in Γ, we could build, at least, a justification for ψ in Γ. Formally, let 𝓤Γ be the set of available terms in Γ (i.e., those terms justifying formulas in Γ). Also, let AB(ψ)Γ be the set of abductive justifications for an AP ψ in Γ, 


• 1. ∀s.∀q.∀φ. (s: φ ∈ Γ AND q ∈ (((𝓤Γ)C)JCS ⇒ ¬(q: ¬φ)),
• 2. ∃t.(t ∈ ((𝓤Γ)C ∪ 𝓤Γ)JCS AND t: ψ).
• 3. If the AP is partial, every t ∈ ((𝓤Γ)C ∪ 𝓤Γ)JCS must satisfy additional properties.

What does fill the ‘gap’ between an AP and an AS?

Part of what an AS is made of are abductive justifications. r ∈ AB(ψ)Γ if and only if r ∈ (𝓤Γ)C, such that,

• We define a function whose domain is the set of formulas of

justification logic (FM), and its output a subset of terms, If φ ∈ FM is a total AP , then, every T ∈ P(𝓤) is the set of ASs for φ.

• Let ψ be a total AP in a set Γ,

Building ASs

Sol: FM → P(𝓤) Sol(ψ) ≔ {t ∈ 𝓤 | t: ψ AND t ∈ (AB(ψ)Γ ∪ 𝓤Γ)JCS}

Filtering ASs

E.g., for every t ∈ AS(ψ)Γ, it is required that for every r, s ∈ Sub(t), it is not the case that [r▹s].

• We re-define Sol function, so that its codomain excludes justification

terms combined under sum:

• For any AP ψ, Sol* generates its set of ASs, minus composite ASs

containing, at least, a pair of sub-terms combined under sum (▹):

Sol*: FM → P({s ∈ 𝓤 | Sub▹(s) = {}})

Sol*(ψ)Γ ≔ {t ∈ 𝓤 | t: ψ AND t ∈ (AB(ψ)Γ ∪ 𝓤Γ)JCS AND Sub▹(t) = {}}

Filtering ASs is also a means to build the set of ASs for partial APs.

Propositional novel abduction

• Propositional novel abductive inference. (Aliseda, 2006, pp. 46-47):

Let FMPL be the set of propositional formulas,

• Four types of novel abductive inference:

1.

• Consistent. (Δ ∪ D)PL ⇏ ⊥.

2.

• Explanatory. (Δ ⇏ ψ) AND (D ⇏ ψ).

3.

• Minimal. D is the weakest explanation not equal to (Δ︎ ⇒ ψ).
• Weak ASs. If D, E ⊆ FMPL, D is weaker than E if and only if D ⊂ E. If D, E ∈ FMPL, D is weaker

than E if and only if D is a sub-formula of E. 4.

• Preferential. D is the best explanation according to a preferential order.

(Δ ∪ D)PL ⇒ ψ

Novel AP. (Δ ⇏ ψ AND Δ ⇏ ¬ψ), such that Δ ⊆ FMPL. Novel AS. D, where (D ∈ FMPL OR D ⊆ FMPL).

Novel abductive inference in JCS

Novel abductive inference in JCS

A novel AP in JCS is a total AP.

!

((𝓤Γ ∪ AB(ψ)Γi )JCS ⇒ t: ψ), t ∈ 𝓤

Novel AP. 𝓤(ψ)Γ = {} Novel AS. t ∈ 𝓤, such that t: ψ. AB(ψ)Γi ∈ P(AB(ψ)Γ) is the set of abductive justifications in t.

Novel abductive inference in JCS

A novel AP in JCS is a total AP.

!

((𝓤Γ ∪ AB(ψ)Γi )JCS ⇒ t: ψ), t ∈ 𝓤

Novel AP. 𝓤(ψ)Γ = {} Novel AS. t ∈ 𝓤, such that t: ψ. AB(ψ)Γi ∈ P(AB(ψ)Γ) is the set of abductive justifications in t.

Propositional-based novel abduction Justification-based novel abduction

Consistent (Δ ∪ D)PL ⇏ ⊥ Explanatory (Δ ⇏ ψ) AND (D ⇏ ψ) Minimal (@product) D is the weakest explanation not equal to (Δ︎ ⇏ ψ) Minimal (@process) E.g., D’s computational complexity is the lowest (Aliseda, p. 74, 2006) ∀r. (r ∈ (AB(ψ)Γi )JCS ⇒ ∀s.∀γ. (s: γ ∈ Γ ⇒ ¬(r: ¬γ))) ¬∃t. (t: ψ ∈ Γ), ∀r. (r ∈ (AB(ψ)Γi )JCS ⇒ ¬(r: ψ)) t is exp., and ∀s. (s ∈ AS(ψ)Γ ⇒ t ∈ Sub(s)) t is exp., and ∀s. (s ∈ AS(ψ)Γ ⇒ Op(t) ≤ Op(s))

• New properties defined for justification terms lead to new kinds of AP and AS, and so, to

new kinds of abductive inference. For example:

Further kinds of abductive inference

((𝓤Γ ∪ AB(ψ)Γi )JCS ⇒ t: ψ), ∀u. (u ∈ V) ⇒ u ∉ Sub(t). 𝓤(ψ)Γ = {s ∈ 𝓤 | s ∈ V OR ∃r.(r ∈ 𝓤 AND r ∈ (Sub(s) ∩ V)} no sub-term of t is a variable term

Valid abductive inference

Bibliography

[1] Aliseda, Atocha. Abductive Reasoning: logical investigations into discovery and explanations. Springer, 2006, DOI: 10.1007/1-4020-3907-7. 
 [2] Artemov, Sergei. The logic of justification. The Review of Symbolic Logic, 1(4):477–513, december 2008. DOI: 10.1017/S1755020308090060. 
 [3] Artemov, Sergei. Why do we need justification logic? In Games, Norms and Reasons, pages 23–38. Springer Dordrecht, 2010. DOI: 10.1007/978-94-007-0714-6 2. 
 [4] Artemov, Sergei and Kuznets, Roman. Logical omniscience as infeasibility. Annals of Pure and Applied Logic, 165(1):6–25, january 2014. DOI: 10.1016/j.apal.2013.07.003. 
 [5] Baltag, Alexandru, Renne, Bryan, and Smets, Sonja. The logic of justified belief change, soft evidence and defeasible knowledge. In Logic, Language, Information and Computation Vol. 7456, pages 168–190. Springer Berlin Heidelberg, 2012. DOI: 10.1007/978-3-642-32621-9 13. 
 [6] Baltag, Alexandru, Renne, Bryan, and Smets, Sonja. The logic of justified belief, explicit knowledge, and conclusive evidence. Annals of Pure and Applied Logic, 165(1):49–81, january 2014. DOI: 10.1016/j.apal. 2013.07.005. 
 [7] Fred Dretske. Is knowledge closed under known entailment? the case against closure. In Contem- porary Debates in Epistemology, pages 13–26. Blackwell, 2005. 
 [8] Estrada-González, Luis. Remarks on some general features of abduction. Journal of Logic and Computation, 23(1):181–197, february 2013. DOI: 10.1093/logcom/exs005.

[9] Melvin Fitting. Tr-2014004: Justification logics and realization. Technical report, CUNY, 2014. Available at http:// citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.699.1614& rep=rep1&type=pdf. 
 [10] Flach, Peter A. and Kakas, Antonis C., editors. Abduction and Induction. Essays on their Relation and

• Integration. Springer Netherlands, 2000. DOI: 10.1007/978-94-017-0606-3. 


[11] Gabbay, Dov and Woods, John. A Practical Logic of Cognitive Systems Volume 2. The Reach of Abduction: Insight and Trial. Elsevier, 2005. 
 [12] Hendricks, Vincent and Symons, John. Where’s the bridge? epistemology and epistemic logic. Philosophical Studies, 128(1):137–167, march 2006. DOI: 10.1007/s11098-005-4060-0. 
 [13] Magnani, Lorenzo. Abductive Cognition. The Epistemological and Eco-Cognitive Dimensions of Hypothetical

• Reasoning. Springer Berlin Heidelberg, 2009. DOI: 10.1007/978-3-642-03631-6. 


[14] Stalnaker, Robert. The problem of logical omniscience, ii. In Context and Content. Essays on 
 Intentionality in Speech and Thought, pages 255–273. Oxford University Press, 1999. 
 [15] van Benthem, Johan. Epistemic logic and epistemology: The state of their affairs. Philosophical 
 Studies, 128(1):49–76, march 2006. DOI: 10.1007/s11098-005-4052-0. 
 [16] van Benthem, Johan, Fernández-Duque, David, and Pacuit, Eric. Evidence logic: A new look at 
 neighborhood structures. In Proceedings of Advances in Modal Logic Vol. 9, pages 97–118. King’s 
 College Press, 2012. 
 [17] van Benthem, Johan, Fernández-Duque, David, and Pacuit, Eric. Evidence and plausibility in 
 neighborhood structures. Annals of Pure and Applied Logic, 165(1):106–133, january 2014. DOI: 10.1016/j.apal. 2013.07.007.

Leftovers

Long-term goal

• Goal: the study of the role of justification on abductive

inferences, through a propositional justification-logic- based model of abduction.

• Why is abductive reasoning still worth of study? Some

if its aspects have not been profoundly analyzed. E.g., the role of justification in abductive inferences. (including F&K, 2000; Adaptives,)

IMPLICIT

propositional logic proof

1. p (hyp) 2. q (hyp) 3. p ⋀ q (I⋀)

JEC proof

1. x1: p (hyp) 2. x2: q (hyp) 3. c1: (p → (q → (p ⋀ q))) (c1 ∈ CS) 4. c1: (p → (q → (p ⋀ q))) → (x1: p → ([c1*x1]: (q → (p ⋀ q))) (A2) 5. x1: p → [c1*x1]: (q → (p ⋀ q)) (R1, 4-3) 6. [c1*x1]: (q → (p ⋀ q)) (R1, 5-1) 7. [c1*x1]: (q → (p ⋀ q)) → (x2: q → [x2*[c1*x1]]: (p ⋀ q)) (A2) 8. x2: q → [x2*[c1*x1]]: (p ⋀ q) (R1, 7-6) 9. [x2*[c1*x1]]: (p ⋀ q) (R1, 8-2)

A comparison between proofs

• An example of preferencial novel abductive inference using

justification terms. Let r, s, t be explanatory elements of SA(ψ)Γ , p any property of justification terms, and “≻” a symbol representing a preferential order (such that the minor element is preferable over the major one). If p(o) for every o ∈ Sub(r), p(q) for a single q ∈ Sub(s), and ¬p(u) for every u ∈ Sub(t), then t ≻ s ≻ r.


Expanding the taxonomy

Expanding the taxonomy

A possible account of this class of AP: atomic terms are considered less informative than composite ones.

!

𝓤(ψ)Γ = {r ∈ 𝓤 | r is atomic}

New properties for justification terms mean new classes of AP and AS, and so, new kinds

• f abductive inferences.

A taxonomy of abductive inference based on information, can be further expanded by recognizing variations on justifications.

• Atomicity.

r ∈ 𝓤 is atomic if and only if r ∈ (V ∪ C)

• There is an AP with respect to ψ in Γ, whose trigger is a set 𝓤(ψ)Γ, such that:
• Conservative terms. t is a term with n ∈ ℕ grade of conservativeness in Γ if and only if,

S = {s ∈ 𝓤 | s ∈ Sub(t) ∩ s ∈ 𝓤Γ} y |S| = n.

• t can be an AS for an AP.

Expanding the taxonomy

A possible account of this class of AP: atomic terms are considered less informative than composite ones.

!

𝓤(ψ)Γ = {r ∈ 𝓤 | r is atomic}

Expanding the taxonomy

A possible account of this class of AP: atomic terms are considered less informative than composite ones.

!

𝓤(ψ)Γ = {r ∈ 𝓤 | r is atomic}

Expanding the taxonomy

Expanding the taxonomy

• Informative and conservative abductive inference

((𝓤Γ ∪ AB(ψ)Γi )JCS ⇒ t: ψ), t ∈ ∉ Atm(t) and S = {s ∈ 𝓤 | s ∈ Sub(t) ∩ s ∈ 𝓤Γ}, so |S| = n. 𝓤(ψ)Γ = {r ∈ 𝓤 | r ∈ (V ∪ C)} t SA(ψ)Γ is composite and n-conservative.