Horn-formulas as Types for Structural Resolution

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1 Horn-formulas as Types for Structural Resolution Peng Fu, Ekaterina Komendantskaya University of Dundee School of Computing

2 2 / 17 Introduction: Background Logic Programming(LP) is based on first-order Horn formula The execution of LP is by SLD-resolution SLD-resolution uses Robinson s unification

3 3 / 17 Introduction: Example of SLD-resolution Connectivity of graph with 3 nodes: κ 1 : Connect(x, y), Connect(y, z) Connect(x, z) κ 2 : Connect(node 1, node 2 ) κ 3 : Connect(node 2, node 3 )

4 3 / 17 Introduction: Example of SLD-resolution Connectivity of graph with 3 nodes: κ 1 : Connect(x, y), Connect(y, z) Connect(x, z) κ 2 : Connect(node 1, node 2 ) κ 3 : Connect(node 2, node 3 ) query: Connect(node 1, node 3 )?

5 3 / 17 Introduction: Example of SLD-resolution Connectivity of graph with 3 nodes: κ 1 : Connect(x, y), Connect(y, z) Connect(x, z) κ 2 : Connect(node 1, node 2 ) κ 3 : Connect(node 2, node 3 ) query: Connect(node 1, node 3 )? Execution trace: {Connect(node 1, node 3 )} κ1,[node 1 /x,node 3 /z] {Connect(node 1, y), Connect(y, node 3 )} κ2,[node 1 /x,node 2 /y,node 3 /z] {Connect(node 2, node 3 )} κ3

6 3 / 17 Introduction: Example of SLD-resolution Connectivity of graph with 3 nodes: κ 1 : Connect(x, y), Connect(y, z) Connect(x, z) κ 2 : Connect(node 1, node 2 ) κ 3 : Connect(node 2, node 3 ) query: Connect(node 1, node 3 )? Execution trace: {Connect(node 1, node 3 )} κ1,[node 1 /x,node 3 /z] {Connect(node 1, y), Connect(y, node 3 )} κ2,[node 1 /x,node 2 /y,node 3 /z] {Connect(node 2, node 3 )} κ3 So the answer for Connect(node 1, node 3 ) is yes.

7 4 / 17 Introduction: Motivation Assumptions of LP Provide answers only when a query yields terminating execution Answering a query as proving a formula The notion of proof seems to be of little use in LP Difficulties Hard to model infinite data, where the execution may not terminate How to understand the meaning a query when the query is not terminating

8 5 / 17 Introduction: Resolution by Term-Matching Let s call LP by SLD-resolution LP-Unif How about resolution by term-matching(lp-tm)? Unifiable t 1 γ t 2, i.e. γt 1 γt 2. Matchable t 1 σ t 2, i.e. σt 1 t 2. A use case for LP-TM: Theorem proving

9 5 / 17 Introduction: Resolution by Term-Matching Let s call LP by SLD-resolution LP-Unif How about resolution by term-matching(lp-tm)? Unifiable t 1 γ t 2, i.e. γt 1 γt 2. Matchable t 1 σ t 2, i.e. σt 1 t 2. A use case for LP-TM: Theorem proving Given axioms: Q(x) Q(x) P(x) Is P(x) provable? P(x) Q(x)

10 5 / 17 Introduction: Resolution by Term-Matching Let s call LP by SLD-resolution LP-Unif How about resolution by term-matching(lp-tm)? Unifiable t 1 γ t 2, i.e. γt 1 γt 2. Matchable t 1 σ t 2, i.e. σt 1 t 2. A use case for LP-TM: Theorem proving Given axioms: Q(x) Q(x) P(x) Is P(x) provable? P(x) Q(x) Given axioms: Q(c) Q(x) P(x) Is P(x) provable? P(x) Q(x)

11 6 / 17 Execution behavior of LP-TM Consider following Stream predicate: κ : Stream(y) Stream(cons(x, y))

12 6 / 17 Execution behavior of LP-TM Consider following Stream predicate: κ : Stream(y) Stream(cons(x, y)) For query Stream(cons(x, y)), in LP-Unif: {Stream(cons(x, y))} κ,[x/x1,y/y 1 ] {Stream(y)} κ,[cons(x2,y 2 )/y,x/x 1,cons(x 2,y 2 )/y 1,] {Stream(y 2 )}...

13 6 / 17 Execution behavior of LP-TM Consider following Stream predicate: κ : Stream(y) Stream(cons(x, y)) For query Stream(cons(x, y)), in LP-Unif: {Stream(cons(x, y))} κ,[x/x1,y/y 1 ] {Stream(y)} κ,[cons(x2,y 2 )/y,x/x 1,cons(x 2,y 2 )/y 1,] {Stream(y 2 )}... In LP-TM: {Stream(cons(x, y))} κ,[x/x1,y/y 1 ] {Stream(y)}

14 7 / 17 Limitations of LP-TM LP-TM not quite suitable for problem solving The following logic program can describe finite bit list κ 1 : Bit(0) κ 2 : Bit(1) κ 3 : BList(nil) κ 4 : Bit(x), BList(y) BList(cons(x, y))

15 7 / 17 Limitations of LP-TM LP-TM not quite suitable for problem solving The following logic program can describe finite bit list κ 1 : Bit(0) κ 2 : Bit(1) κ 3 : BList(nil) κ 4 : Bit(x), BList(y) BList(cons(x, y)) Consider query BList(cons(x, y)): {BList(cons(x, y))} κ4,[x/x 1,y/y 1] {Bit(x), BList(y)}

16 7 / 17 Limitations of LP-TM LP-TM not quite suitable for problem solving The following logic program can describe finite bit list κ 1 : Bit(0) κ 2 : Bit(1) κ 3 : BList(nil) κ 4 : Bit(x), BList(y) BList(cons(x, y)) Consider query BList(cons(x, y)): {BList(cons(x, y))} κ4,[x/x 1,y/y 1] {Bit(x), BList(y)} But what is the answer for x, y?

17 7 / 17 Limitations of LP-TM LP-TM not quite suitable for problem solving The following logic program can describe finite bit list κ 1 : Bit(0) κ 2 : Bit(1) κ 3 : BList(nil) κ 4 : Bit(x), BList(y) BList(cons(x, y)) Consider query BList(cons(x, y)): {BList(cons(x, y))} κ4,[x/x 1,y/y 1] {Bit(x), BList(y)} But what is the answer for x, y? We need unification to compute substitution: x 0, y nil

18 7 / 17 Limitations of LP-TM LP-TM not quite suitable for problem solving The following logic program can describe finite bit list κ 1 : Bit(0) κ 2 : Bit(1) κ 3 : BList(nil) κ 4 : Bit(x), BList(y) BList(cons(x, y)) Consider query BList(cons(x, y)): {BList(cons(x, y))} κ4,[x/x 1,y/y 1] {Bit(x), BList(y)} But what is the answer for x, y? We need unification to compute substitution: x 0, y nil The combination of LP-TM with substitution computed by unification leads to Structural Resolution

19 8 / 17 Formalism: LP-Unif, LP-TM and LP-Struct Term-matching reduction: Φ {A 1,..., A i,..., A n } κ,σ {A 1,..., σb 1,..., σb m,..., A n }, if there exists κ : x.b 1,..., B n C Φ such that C σ A i.

20 8 / 17 Formalism: LP-Unif, LP-TM and LP-Struct Term-matching reduction: Φ {A 1,..., A i,..., A n } κ,σ {A 1,..., σb 1,..., σb m,..., A n }, if there exists κ : x.b 1,..., B n C Φ such that C σ A i. Unification reduction: Φ {A 1,..., A i,..., A n } κ,γ γ {γa 1,..., γb 1,..., γb m,..., γa n }, if there exists κ : x.b 1,..., B n C Φ such that C γ A i.

21 8 / 17 Formalism: LP-Unif, LP-TM and LP-Struct Term-matching reduction: Φ {A 1,..., A i,..., A n } κ,σ {A 1,..., σb 1,..., σb m,..., A n }, if there exists κ : x.b 1,..., B n C Φ such that C σ A i. Unification reduction: Φ {A 1,..., A i,..., A n } κ,γ γ {γa 1,..., γb 1,..., γb m,..., γa n }, if there exists κ : x.b 1,..., B n C Φ such that C γ A i. Substitutional reduction: Φ {A 1,..., A i,..., A n } κ,γ γ {γa 1,..., γa i,..., γa n }, if there exists κ : x.b 1,..., B n C Φ such that C γ A i.

22 8 / 17 Formalism: LP-Unif, LP-TM and LP-Struct Term-matching reduction: Φ {A 1,..., A i,..., A n } κ,σ {A 1,..., σb 1,..., σb m,..., A n }, if there exists κ : x.b 1,..., B n C Φ such that C σ A i. Unification reduction: Φ {A 1,..., A i,..., A n } κ,γ γ {γa 1,..., γb 1,..., γb m,..., γa n }, if there exists κ : x.b 1,..., B n C Φ such that C γ A i. Substitutional reduction: Φ {A 1,..., A i,..., A n } κ,γ γ {γa 1,..., γa i,..., γa n }, if there exists κ : x.b 1,..., B n C Φ such that C γ A i. LP-TM: (Φ, ) LP-Unif: (Φ, ) LP-Struct: (Φ, µ 1 )

23 9 / 17 LP-Struct: Stream κ : Stream(y) Stream(cons(x, y)) For query Stream(cons(x, y)), in LP-Struct: {Stream(cons(x, y))} {Stream(y)}

24 9 / 17 LP-Struct: Stream κ : Stream(y) Stream(cons(x, y)) For query Stream(cons(x, y)), in LP-Struct: {Stream(cons(x, y))} {Stream(y)} [cons(x1,y 1 )/y] {Stream(cons(x 1, y 1 ))} {Stream(y 1 )}

25 9 / 17 LP-Struct: Stream κ : Stream(y) Stream(cons(x, y)) For query Stream(cons(x, y)), in LP-Struct: {Stream(cons(x, y))} {Stream(y)} [cons(x1,y 1 )/y] {Stream(cons(x 1, y 1 ))} {Stream(y 1 )} [cons(x2,y 2 )/y 1,cons(x 1,cons(x 2,y 2 ))/y] {Stream(cons(x 2, y 2 ))} {Stream(y 2 )}

26 9 / 17 LP-Struct: Stream κ : Stream(y) Stream(cons(x, y)) For query Stream(cons(x, y)), in LP-Struct: {Stream(cons(x, y))} {Stream(y)} [cons(x1,y 1 )/y] {Stream(cons(x 1, y 1 ))} {Stream(y 1 )} [cons(x2,y 2 )/y 1,cons(x 1,cons(x 2,y 2 ))/y] {Stream(cons(x 2, y 2 ))} {Stream(y 2 )} [cons(x3,y 3 )/y 2,cons(x 2,cons(x 3,y 3 ))/y 1,cons(x 1,cons(x 2,cons(x 3,y 3 )))/y] {Stream(cons(x 3, y 3 ))} {Stream(y 3 )}

27 9 / 17 LP-Struct: Stream κ : Stream(y) Stream(cons(x, y)) For query Stream(cons(x, y)), in LP-Struct: {Stream(cons(x, y))} {Stream(y)} [cons(x1,y 1 )/y] {Stream(cons(x 1, y 1 ))} {Stream(y 1 )} [cons(x2,y 2 )/y 1,cons(x 1,cons(x 2,y 2 ))/y] {Stream(cons(x 2, y 2 ))} {Stream(y 2 )} [cons(x3,y 3 )/y 2,cons(x 2,cons(x 3,y 3 ))/y 1,cons(x 1,cons(x 2,cons(x 3,y 3 )))/y] {Stream(cons(x 3, y 3 ))} {Stream(y 3 )}... Partial answer: cons(x 1, cons(x 2, cons(x 3, y 3 )))/y

28 10 / 17 Question: Relation between LP-Unif and LP-Struct? Both LP-Unif and LP-Struct are sound w.r.t. Herbrand Model

29 10 / 17 Question: Relation between LP-Unif and LP-Struct? Both LP-Unif and LP-Struct are sound w.r.t. Herbrand Model Operationally, They seem similar but a little different

30 10 / 17 Question: Relation between LP-Unif and LP-Struct? Both LP-Unif and LP-Struct are sound w.r.t. Herbrand Model Operationally, They seem similar but a little different Again, the graph example κ 1 : Connect(x, y), Connect(y, z) Connect(x, z) κ 2 : Connect(node 1, node 2 ) κ 3 : Connect(node 2, node 3 )

31 10 / 17 Question: Relation between LP-Unif and LP-Struct? Both LP-Unif and LP-Struct are sound w.r.t. Herbrand Model Operationally, They seem similar but a little different Again, the graph example κ 1 : Connect(x, y), Connect(y, z) Connect(x, z) κ 2 : Connect(node 1, node 2 ) κ 3 : Connect(node 2, node 3 ) Connect(node 1, node 3 ) in LP-Unif terminates.

32 10 / 17 Question: Relation between LP-Unif and LP-Struct? Both LP-Unif and LP-Struct are sound w.r.t. Herbrand Model Operationally, They seem similar but a little different Again, the graph example κ 1 : Connect(x, y), Connect(y, z) Connect(x, z) κ 2 : Connect(node 1, node 2 ) κ 3 : Connect(node 2, node 3 ) Connect(node 1, node 3 ) in LP-Unif terminates. For LP-Struct: Φ {Connect(node 1, node 3 )} κ1,[node 1 /x,node 3 /z] {Connect(node 1, y), Connect(y, node 3 )} κ1,[node 1 /x,y/z] {Connect(node 1, y 1 ), Connect(y 1, y), Connect(y, node 3 )} κ1...

33 11 / 17 Formalization of a Type System Term t ::= x f (t 1,..., t n ) Atomic Formula A, B, C, D ::= P(t 1,..., t n ) (Horn) Formula F ::= A 1,..., A n A Proof Term p, e ::= κ a λa.e e e

34 Formalization of a Type System Term t ::= x f (t 1,..., t n ) Atomic Formula A, B, C, D ::= P(t 1,..., t n ) (Horn) Formula F ::= A 1,..., A n A Proof Term p, e ::= κ a λa.e e e Girard s observation on intuitionistic sequent calculus with atomic formulas B A axiom B C σb σc subst A D B, D C A, B C cut 11 / 17

35 11 / 17 Formalization of a Type System Term t ::= x f (t 1,..., t n ) Atomic Formula A, B, C, D ::= P(t 1,..., t n ) (Horn) Formula F ::= A 1,..., A n A Proof Term p, e ::= κ a λa.e e e Girard s observation on intuitionistic sequent calculus with atomic formulas B A axiom B C σb σc subst A D B, D C A, B C Is Q provable? cut

36 Formalization of a Type System Term t ::= x f (t 1,..., t n ) Atomic Formula A, B, C, D ::= P(t 1,..., t n ) (Horn) Formula F ::= A 1,..., A n A Proof Term p, e ::= κ a λa.e e e Girard s observation on intuitionistic sequent calculus with atomic formulas B A axiom B C σb σc subst A D B, D C cut A, B C Is Q provable? We internalized as and add proof term annotations κ : x.f axiom e : F e : x.f gen e : x.f e : [t/x]f inst e 1 : A D e 2 : B, D C λa.λb.(e 2 b) (e 1 a) : A, B C cut 11 / 17

37 12 / 17 Soundness of LP-TM and LP-Unif Soundness of LP-Unif If Φ {A} γ, then there exists a proof e : x. γa given axioms Φ. Soundness of LP-TM If Φ {A}, then there exists a proof e : x. A given axioms Φ. For example, the LP-Unif reductions: {Connect(node 1, node 3 )} κ1,[node 1 /x,node 3 /z] {Connect(node 1, y), Connect(y, node 3 )} κ2,[node 1 /x,node 2 /y,node 3 /z] {Connect(node 2, node 3 )} κ3 The reduction yields a proof (λb.(κ 1 b) κ 3 ) κ 2 for the formula Connect(node 1, node 3 ).

38 13 / 17 Useful Properties about the Type System Strong Normalization If e : F, then e is strongly normalizable w.r.t. beta-reduction on proof terms. First Orderness If e : [ x.]a B given axioms Φ, then either e is a proof term constant or it is normalizable to the form λa.n, where n is first order normal proof term. If e : [ x.] B, then e is normalizable to a first order proof term.

39 14 / 17 Realizability Transformation Inspired from Kleene s realizability: ϕ realize A B iff for any number a realizes A and ϕ(a) realizes B.

40 14 / 17 Realizability Transformation Inspired from Kleene s realizability: ϕ realize A B iff for any number a realizes A and ϕ(a) realizes B. Representing First Order Proof Term Let φ be a mapping from proof term variables to first order terms. a φ := φ(a) κ p 1...p n φ := f κ ( p 1 φ,..., p n φ )

41 14 / 17 Realizability Transformation Inspired from Kleene s realizability: ϕ realize A B iff for any number a realizes A and ϕ(a) realizes B. Representing First Order Proof Term Let φ be a mapping from proof term variables to first order terms. a φ := φ(a) κ p 1...p n φ := f κ ( p 1 φ,..., p n φ ) For A P(x), we write A[y] P(x, y). Similarly, A[t] P(x, t)

42 14 / 17 Realizability Transformation Inspired from Kleene s realizability: ϕ realize A B iff for any number a realizes A and ϕ(a) realizes B. Representing First Order Proof Term Let φ be a mapping from proof term variables to first order terms. a φ := φ(a) κ p 1...p n φ := f κ ( p 1 φ,..., p n φ ) For A P(x), we write A[y] P(x, y). Similarly, A[t] P(x, t) Realizability transformation F on normal proofs

43 14 / 17 Realizability Transformation Inspired from Kleene s realizability: ϕ realize A B iff for any number a realizes A and ϕ(a) realizes B. Representing First Order Proof Term Let φ be a mapping from proof term variables to first order terms. a φ := φ(a) κ p 1...p n φ := f κ ( p 1 φ,..., p n φ ) For A P(x), we write A[y] P(x, y). Similarly, A[t] P(x, t) Realizability transformation F on normal proofs F(κ : x.a1,..., A m B) := κ : x. y.a 1 [y 1 ],..., A m [y m ] B[f κ (y 1,..., y m )]

44 14 / 17 Realizability Transformation Inspired from Kleene s realizability: ϕ realize A B iff for any number a realizes A and ϕ(a) realizes B. Representing First Order Proof Term Let φ be a mapping from proof term variables to first order terms. a φ := φ(a) κ p 1...p n φ := f κ ( p 1 φ,..., p n φ ) For A P(x), we write A[y] P(x, y). Similarly, A[t] P(x, t) Realizability transformation F on normal proofs F(κ : x.a1,..., A m B) := κ : x. y.a 1 [y 1 ],..., A m [y m ] B[f κ (y 1,..., y m )] F(λa.n : [ x].a 1,..., A m B) := λa.n : [ x. y].a 1 [y 1 ],..., A m [y m ] B[ n [y/a] ]

45 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 )

46 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 ) LP-Struct reduction for Connect(node 1, node 3, u).

47 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 ) LP-Struct reduction for Connect(node 1, node 3, u). {Connect(node1, node 3, u)}

48 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 ) LP-Struct reduction for Connect(node 1, node 3, u). {Connect(node1, node 3, u)} κ1,[node 1/x,node 3/z,f κ1 (u 1,u 2)/u] {Connect(node 1, node 3, f κ1 (u 1, u 2 ))} κ1 {Connect(node 1, y, u 1 ), Connect(y, node 3, u 2 )}

49 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 ) LP-Struct reduction for Connect(node 1, node 3, u). {Connect(node1, node 3, u)} κ1,[node 1/x,node 3/z,f κ1 (u 1,u 2)/u] {Connect(node 1, node 3, f κ1 (u 1, u 2 ))} κ1 {Connect(node 1, y, u 1 ), Connect(y, node 3, u 2 )} κ2,[c κ2 /u 1,node 1/x,node 2/y,,node 3/z,f κ1 (c κ2,u 2)/u]

50 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 ) LP-Struct reduction for Connect(node 1, node 3, u). {Connect(node1, node 3, u)} κ1,[node 1/x,node 3/z,f κ1 (u 1,u 2)/u] {Connect(node 1, node 3, f κ1 (u 1, u 2 ))} κ1 {Connect(node 1, y, u 1 ), Connect(y, node 3, u 2 )} κ2,[c κ2 /u 1,node 1/x,node 2/y,,node 3/z,f κ1 (c κ2,u 2)/u] {Connect(node1, node 2, c κ2 ), Connect(node 2, node 3, u 2 )} κ2 {Connect(node 2, node 3, u 2 )}

51 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 ) LP-Struct reduction for Connect(node 1, node 3, u). {Connect(node1, node 3, u)} κ1,[node 1/x,node 3/z,f κ1 (u 1,u 2)/u] {Connect(node 1, node 3, f κ1 (u 1, u 2 ))} κ1 {Connect(node 1, y, u 1 ), Connect(y, node 3, u 2 )} κ2,[c κ2 /u 1,node 1/x,node 2/y,,node 3/z,f κ1 (c κ2,u 2)/u] {Connect(node1, node 2, c κ2 ), Connect(node 2, node 3, u 2 )} κ2 {Connect(node 2, node 3, u 2 )} κ3,[c κ3 /u 2,c κ2 /u 1,node 3/z,node 1/x,node 2/y,f κ1 (c κ2,c κ3 )/u] {Connect(node 2, node 3, c κ3 )} κ3

52 15 / 17 Realizability Transformation: Example Connectivity after realizability transformation: κ 1 : Connect(x, y, u 1), Connect(y, z, u 2) Connect(x, z, f κ1 (u 1, u 2)) κ 2 : Connect(node 1, node 2, c κ2 ) κ 3 : Connect(node 2, node 3, c κ3 ) LP-Struct reduction for Connect(node 1, node 3, u). {Connect(node1, node 3, u)} κ1,[node 1/x,node 3/z,f κ1 (u 1,u 2)/u] {Connect(node 1, node 3, f κ1 (u 1, u 2 ))} κ1 {Connect(node 1, y, u 1 ), Connect(y, node 3, u 2 )} κ2,[c κ2 /u 1,node 1/x,node 2/y,,node 3/z,f κ1 (c κ2,u 2)/u] {Connect(node1, node 2, c κ2 ), Connect(node 2, node 3, u 2 )} κ2 {Connect(node 2, node 3, u 2 )} κ3,[c κ3 /u 2,c κ2 /u 1,node 3/z,node 1/x,node 2/y,f κ1 (c κ2,c κ3 )/u] {Connect(node 2, node 3, c κ3 )} κ3 Answer: f κ1 (c κ2, c κ3 )/u

53 16 / 17 Results about Realizability Transformation Termination of term-matching reduction For any (Φ, µ 1 ), we have (F(Φ), ν 1 ) Preserve Provability Given axioms Φ, if e : [ x].a B holds with e in normal form, then F(e : [ x].a B) holds for axioms F(Φ) Recording Proof Suppose F(Φ) {A[y]} γ. We have p : x. γa[γy] for F(Φ), where p is in normal form and p = γy Preserve Unification Φ {A} iff F(Φ) {A[y]} Operational Equivalent of LP-Unif and LP-Struct F(Φ) {A[y]} iff F(Φ) {A[y]}( ν 1 ).

54 17 / 17 Summary and Future Work We define a type system to model LP-TM, LP-Unif and LP-Struct We define a transformation called realizability transformation Realizability transformation preserves proof content We show LP-Unif and LP-Struct are operationally equivalent after the tranformation Future works: Apply LP-TM to analyze type class inference in functional langauges

55 18 / 17 Future Work In type class inference, proof has computational meaning: class Eq A where eq :: Eq A => A -> A -> Bool instance => Eq Int where.. instance Eq A => Eq (List A) where.. test = eq [] [1] test function will generate a query Eq (List Int) Eq (List Int) ==> Eq Int ==> empty The proof of the query Eq (List Int) will be passed as an input for eq

Structural Resolution

Structural Resolution Katya Komendantskaya School of Computing, University of Dundee, UK 12 May 2015 Outline Motivation Coalgebraic Semantics for Structural Resolution The Three Tier Tree calculus for