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). Given “ context ⇏ input ” input output AI (a.k.a. AP ) (a.k.a. AS ) We get “ context , output ⇒ input ” • 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): t : Ψ , “ Ψ is justified by evidence t” • Abductive inference with explicit justification . Let Γ be a set of justification logic formulas. Given “ Γ ⇏ s: Ψ ”, for every or some s Ψ t AI We get “ Γ , t ⇒ t: Ψ ”

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. 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 ones (being explanatory, being informative, being non-redundant), and the list goes on. • A getaway from logical omniscience (Stalnaker, 1999; Dretske, 2005). More than a non-normal modal logic, justification logic o ff ers 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 . • An ever expanding set of formal tools (Baltag, Renne & Smets, 2012, 2014). 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”…

Justification logic • Syntax . propositional logic + justification terms, s , t ::= c i | x i | [ s * t ] | [ s ▹ t ] Where c i ∈ C , x i ∈ V, and ∀ t. (t ∈ 𝓤 ) If t ∈ 𝓤 and φ is a formula, then, t : φ is also a formula. • J 0 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 • J CS systems . J CS = J 0 + CS Where , if c i : φ i ∈ CS , then c i ∈ C and φ i ∈ TAUT .

Tracking terms • For every t ∈ 𝓤 , we get the set of atoms and sub-terms that t is made of. atoms atomic terms sub-terms composite terms Let “ ◦ ” be a binary operation satisfying uniform substitution, such that “ ◦ ” = “*” or “ ◦ ” = “ ▹ ”. • Sub generates the set of sub-terms of any term , Sub : 𝓤 → P ( 𝓤 ) 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 , • Atm : 𝓤 → P ( V ∪ C ) 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 .)

Operability • Operability of a term t means the di ffi culty to build it. We determine its di ffi culty by counting over the operations occurring between its sub-terms. ! Based on the method of “counting the steps” that Artemov & Kuznets (2014, p. 20-22) propose, in order to For every t ∈ 𝓤 , Op ( t ) assigns a value n ∈ ℕ , • account for the complexity of inferred formulas. Op : 𝓤 → 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 ª 4. 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 .

AP s and AS s in J CS • 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 ∈ Γ C . • 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 i ff t : ψ ∈ Γ C and, t satisfies additional properties: e.g. , t ∉ Atm ( t ).


 
 What does fill the ‘gap’ between an AP and an AS ? • 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 . Part of what an AS is made of are abductive justifications . Formally, let 𝓤 Γ 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 , r ∈ AB ( ψ ) Γ if and only if r ∈ ( 𝓤 Γ ) C , such that, 1. ∀ s . ∀ q . ∀ φ . ( s : φ ∈ Γ AND q ∈ ((( 𝓤 Γ ) C ) JCS ⇒ ¬( q : ¬ φ )), 2. ∃ t .( t ∈ (( 𝓤 Γ ) C ∪ 𝓤 Γ ) JCS AND t : ψ ). 3. If the AP is partial, every t ∈ (( 𝓤 Γ ) C ∪ 𝓤 Γ ) JCS must satisfy additional properties.

Building AS s • We define a function whose domain is the set of formulas of justification logic ( FM ), and its output a subset of terms, Sol : FM → P ( 𝓤 ) , then, every T ∈ P ( 𝓤 ) is the set of If φ ∈ FM is a total AP AS s for φ . - Let ψ be a total AP in a set Γ , Sol ( ψ ) ≔ { t ∈ 𝓤 | t : ψ AND t ∈ ( AB ( ψ ) Γ ∪ 𝓤 Γ ) JCS }

Filtering AS s 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: Sol* : FM → P ({ s ∈ 𝓤 | Sub ▹ ( s ) = {}}) - For any AP ψ , Sol* generates its set of AS s, minus composite AS s containing, at least, a pair of sub-terms combined under sum ( ▹ ): Sol* ( ψ ) Γ ≔ { t ∈ 𝓤 | t : ψ AND t ∈ ( AB ( ψ ) Γ ∪ 𝓤 Γ ) JCS AND Sub ▹ ( t ) = {}} Filtering ASs is also a means to build the set of AS s for partial AP s.

Propositional novel abduction • Propositional novel abductive inference . (Aliseda, 2006, pp. 46-47): Let FM PL be the set of propositional formulas, ( Δ ∪ D ) PL ⇒ ψ Novel AP . ( Δ ⇏ ψ AND Δ ⇏ ¬ ψ ), Novel AS . D , where ( D ∈ FM PL OR D ⊆ FM PL ). such that Δ ⊆ FM PL . • 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 ( Δ ︎ ⇒ ψ ). - Weak AS s. If D , E ⊆ FM PL , D is weaker than E if and only if D ⊂ E . If D , E ∈ FM PL , 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.

Novel abductive inference in J CS

! Novel abductive inference in J CS A novel AP in J CS (( 𝓤 Γ ∪ AB ( ψ ) Γ i ) JCS ⇒ t: ψ ), t ∈ 𝓤 is a total AP . Novel AS . t ∈ 𝓤 , such that t : ψ . Novel AP . 𝓤 ( ψ ) Γ = {} AB ( ψ ) Γ i ∈ P( AB ( ψ ) Γ ) is the set of abductive justifications in t .

! Novel abductive inference in J CS A novel AP in J CS (( 𝓤 Γ ∪ AB ( ψ ) Γ i ) JCS ⇒ t: ψ ), t ∈ 𝓤 is a total AP . Novel AS . t ∈ 𝓤 , such that t : ψ . Novel AP . 𝓤 ( ψ ) Γ = {} AB ( ψ ) Γ i ∈ P( AB ( ψ ) Γ ) is the set of abductive justifications in t . Propositional -based novel Justification -based novel abduction abduction ∀ r . ( r ∈ ( AB ( ψ ) Γ i ) JCS ⇒ ∀ s . ∀ γ . ( s : γ ∈ Γ ⇒ ¬( r : ¬ γ ))) ( Δ ∪ D ) PL ⇏ ⊥ Consistent ¬ ∃ t . ( t : ψ ∈ Γ ), ∀ r . ( r ∈ ( AB ( ψ ) Γ i ) JCS ⇒ ¬( r : ψ )) ( Δ ⇏ ψ ) AND ( D ⇏ ψ ) Explanatory D is the weakest explanation Minimal t is exp., and ∀ s. ( s ∈ AS ( ψ ) Γ ⇒ t ∈ Sub ( s )) (@product) not equal to ( Δ ︎ ⇏ ψ ) E.g., D’s computational Minimal t is exp., and ∀ s. (s ∈ AS ( ψ ) Γ ⇒ Op(t) ≤ Op(s)) complexity is the lowest (@process) (Aliseda, p. 74, 2006)

Further kinds of abductive inference • New properties defined for justification terms lead to new kinds of AP and AS , and so, to new kinds of abductive inference. For example: Valid abductive inference no sub-term of t is a variable term (( 𝓤 Γ ∪ AB ( ψ ) Γ i ) 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 )}