Fuzzy Unification and Generalization of First-Order Terms over Similar Signatures A Constraint-Based Approach Hassan A¨ ıt-Kaci Gabriella Pasi 27th LOPSTR Namur, Belgium October 10–12, 2017
This presentation’s objective ◮ Reformulate and extend general results on (crisp & fuzzy) FOT unification and generalization (“ anti-unification ”) seen as lattice operations using (crisp & fuzzy) constraints ◮ Give declarative rulesets for operational constraint-driven deductive and inductive fuzzy inference over FOT s when some signature symbols may be similar OK. . . And why is this interesting?. . . ◮ This provides a formally clean and practically efficient way to enable approximate reasoning ( deduction and learning ) with a very popular data structure used in logic-based data and knowledge processing systems 1
Some quick but important remarks about this presentation We apologize in advance for the “ symbol soup ” in this talk ... ... but please do bear with us , as this presentation is: ◮ only meant to give you an idea. . . of what’s in the paper with more examples and all proofs available here ◮ necessary. . . since we purport to be formal ◮ not that complicated. . . at least not for this audience — we assume familiarity with Prolog’s basic data structure and Fuzzy Logic notions ◮ really always the same. . . once we get the basic gist 2
Presentation outline ◮ First-Order Terms — syntax of FOT s ◮ Subsumption — pre-order relation on FOT s ◮ Unification — glb operation on FOT s ◮ Generalization — lub operation on FOT s ◮ Weak unification — fuzzy glb of aligned FOT s ◮ Weak generalization — fuzzy lub of aligned FOT s ◮ Full fuzzy unification — fuzzy glb of misaligned FOT s ◮ Full fuzzy generalization — fuzzy lub of misaligned FOT s ◮ Conclusion — recapitulation and future work 3
The lattice of FOT s data structures that can be approximated 4
� FOT s on a signature of data constructors Σ = n ≥ 0 Σ n def T Σ , V = V def ∪ { f ( t 1 , · · · , t n ) | f ∈ Σ n , n ≥ 0 , t i ∈ T Σ , V , 1 ≤ i ≤ n } 5
FOT subsumption pre-order relation t 1 � t 2 iff ∃ σ : V → T Σ , V s.t. t 1 = t 2 σ 6
FOT subsumption lattice operations t = lub ( t 1 , t 2 ) σ 1 σ 2 t 1 = tσ 1 t 2 = tσ 2 σ σ � t 1 σ = t 2 σ � t = glb ( t 1 , t 2 ) = tσ 1 σ = tσ 2 σ 7
Declarative lattice operations on FOT s. . . using constraints 8
Unification a bit of history ◮ 1930 – Jacques Herbrand gives normalization rules for sets of term equalities in his PhD thesis ( Chap. 5, Sec. 2.4, pp. 95 – 96 ) but does not call this “ unification ” ◮ 1960 – Dag Prawitz expresses this as reduction rules as part of proof normalization procedure for Natural Deduction in F .O. Logic ( Gentzen , 1934) ◮ 1965 – J. Alan Robinson gives a procedural algorithm and uses it to lift the resolution principle from Propositional Logic to F .O. Logic — calling it “ unification ” ◮ 1967 – Jean van Heijenoort translates Chap. 5 of Herbrand’s thesis into English ◮ 1971 – Warren Goldfarb translates Herbrand’s full thesis into English 9
Unification a bit of history (ctd.) ◮ 1976 – G´ erard Huet dates the first FOT unification algorithm to initial equation normalization in Herbrand’s 1930 PhD thesis ( also in Chap. 5 in Huet’s thesis! ) ◮ 1982 – Alberto Martelli & Ugo Montanari give unification rules (with no mention of Herbrand’s thesis, although Huet’s thesis is cited) Interestingly, Martelli & Montanari use a preprocessing method that uses generalization implicitly (to compute “ common parts ” in preprocessing equations into congruence classes of equations called “ multi-equations ”) — but do not point out that it is dual to unification 10
FOT unification as a constraint ? t 1 = t 2 σ σ t 1 σ = t 2 σ 11
Declarative unification rule A unification rule rewrites a prior set of equations E into a posterior set of equations E ′ whenever an optional meta- condition holds: R ULE N AME : Prior set of equations E [ Optional meta-condition ] Posterior set of equations E ′ 12
Herbrand– Martelli-Montanari FOT unification rules T ERM D ECOMPOSITION : E ∪ { f ( s 1 , · · · , s n ) . = f ( t 1 , · · · , t n ) } [ n ≥ 0] E ∪ { s 1 . = t 1 , · · · , s n . = t n } V ARIABLE E LIMINATION : E ∪ { X . = t } � X �∈ Var ( t ) � X occurs in E E [ X ← t ] ∪ { X . = t } 13
Herbrand– Martelli-Montanari FOT unification rules (ctd.) E QUATION O RIENTATION : E ∪ { t . = X } [ t �∈ V ] E ∪ { X . = t } V ARIABLE E RASURE : E ∪ { X . = X } E 14
Moving on to. . . declarative constraint-based generalization 15
Generalization a bit of history ◮ The lattice-theoretic properties of FOT s as data structures pre-ordered by subsumption were exposed independently and simultaneously by Reynolds and Plotkin in 1970 ◮ Both gave a formal definition of FOT generalization and each proved correct a procedural specification for computing it ◮ However , . . . so far, a declarative formal specification was lacking — which we provide here ◮ Why should we care?... Well, because: – syntax-driven rules give an operational semantics as constraint solving needing no control specification (use any rule that applies in any order) – each rule’s correctness is independent of that of the others (they share no global context) – eases the formal specification of more expressive approximation over the same data structure (such as fuzzy constraints on FOT s) 16
FOT generalization judgment Statement of the form: � � � � � � σ 1 t 1 θ 1 ⊢ t σ 2 t 2 θ 2 where (for i = 1 , 2 ): • t ∈ T and t i ∈ T are FOT s • σ i : V → T and θ i : V → T are substitutions 17
FOT generalization judgment validity A generalization judgment: � � � � � � σ 1 t 1 θ 1 ⊢ t σ 2 t 2 θ 2 is deemed valid whenever: t i σ i = tθ i with θ i � σ i ( i.e. , ∃ δ i s.t. θ i = δ i σ i ) for i = 1 , 2 18
FOT generalization judgment validity as a constraint t θ 2 1 δ 1 δ 2 = θ = δ 1 σ 2 σ 1 δ 2 t 1 t 2 = tδ 1 tδ 2 = � � � � � � σ 1 t 1 θ 1 σ 1 σ 2 ⊢ t σ 2 t 2 θ 2 t 1 σ 1 t 2 σ 2 = tθ 1 tθ 2 = 19
Declarative generalization axiom Statement of the form: A XIOM N AME : [ Optional meta-condition ] Judgment J which reads: “ whenever the optional meta-condition holds, judgement J is always valid ” 20
FOT generalization axioms E QUAL V ARIABLES : � � � � � � σ 1 X σ 1 ⊢ X σ 2 X σ 2 V ARIABLE -T ERM : [ t 1 ∈ V or t 2 ∈ V ; t 1 � = t 2 ; X is new ] � � � � � � σ 1 { t 1 /X } σ 1 t 1 ⊢ X σ 2 t 2 σ 2 { t 2 /X } U NEQUAL F UNCTORS : [ m ≥ 0 , n ≥ 0; m � = n or f � = g ; X is new ] � � � � � � σ 1 f ( s 1 , · · · , s m ) σ 1 { f ( s 1 , · · · , s m ) /X } ⊢ X g ( t 1 , · · · , t n ) σ 2 { g ( t 1 , · · · , t n ) /X } σ 2 21
Declarative generalization inference rule Conditional Horn rule of generalization judgments of the form: R ULE N AME : [ Optional Meta-Condition ] Prior Judgment J 1 Prior Judgment J n · · · Posterior Judgment J (for n ≥ 0 ) — which reads: “ whenever the optional meta-condition holds, if all the n prior judgements J n are valid, then the posterior judgement J is also valid ” 22
Declarative FOT generalization rule for equal functors E QUAL F UNCTORS : [ n ≥ 0] σ n − 1 s ′ s ′ σ 0 σ 1 σ n � � � � � � � � � � � � n 1 1 1 1 1 ⊢ u 1 · · · ⊢ u n t ′ t ′ σ n − 1 σ n σ 0 σ 1 n 1 2 2 2 2 σ 0 σ n � � � � � � f ( s 1 , · · · , s n ) 1 1 ⊢ f ( u 1 , · · · , u n ) σ n σ 0 f ( t 1 , · · · , t n ) 2 2 σ i − 1 s ′ � � � � � � s i i 1 = ↑ for i = 1 , . . . , n . where def t ′ σ i − 1 t i i 2 23
“Unapplying” a pair of substitutions on a pair of FOT s Rule “ E QUAL F UNCTORS ” uses operation “ unapply ” ‘ ↑ ’ on a pair of terms t 1 , t 2 given a pair of substitutions σ 1 , σ 2 : � � X if t i = Xσ i , for i = 1 , 2 X � � � � t 1 σ 1 ↑ = def t 2 σ 2 � � t 1 otherwise t 2 24
Declarative FOT generalization rule for n = 0 NB : for n = 0 , the rule E QUAL F UNCTORS becomes an axiom; viz. , for any constant c : � � � � � � σ 1 c σ 1 ⊢ c σ 2 c σ 2 for any pair σ 1 , σ 2 25
FOT generalization example Consider the terms f ( a, a, a ) and f ( b, c, c ) to generalize; i.e. : • Find term t and substitutions σ 1 and σ 2 such that tσ 1 = f ( a, a, a ) and tσ 2 = f ( b, c, c ) : � � � � � � ∅ f ( a, a, a ) σ 1 ⊢ t ∅ f ( b, c, c ) σ 2 • By Rule E QUAL F UNCTORS , we must have t = f ( u 1 , u 2 , u 3 ) since: � � � � � � ∅ f ( a, a, a ) σ 1 ⊢ f ( u 1 , u 2 , u 3 ) ∅ f ( b, c, c ) σ 2 where: � � � � a ∅ ↑ – u 1 is the generalization of ; that is, of a and b b ∅ and by Rule U NEQUAL F UNCTORS : � � � � � � ∅ { a/X } a ⊢ X therefore u 1 = X ∅ { b/X } b 26
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