Lollipops taste of Lollipops taste of Lollipops taste of Vanilla too Vanilla too Vanilla too Post-ICLP'94 Workshop on Proof-Theoretical Extensions of Logic Programming Iliano Cervesato Iliano Cervesato Dipartimento di Informatica Università di Torino Corso Svizzera, 185 10149 Torino - Italy iliano@di.unito.it S. Margherita Ligure, Italy June 18 th , 1994 Iliano Cervesato - Lollipops taste of Vanilla too 1
Overview Overview Overview • Meta-programming in logic programming – Meta-programs – The current trend – Granularity of a meta-program – The Vanilla meta-interpreter for Prolog • The linear logic programming language Lolli – Linear logic – Uniform proofs – The Τ , &, −ο , ⇒ , ∀ fragment – Lolli – Operational semantics • A Vanilla meta-interpreter for Lolli – Syntactic restrictions – The representation function – A vanilla meta-interpreter for the core of Lolli – Soundness of the meta-interpreter – Extensions and examples of use • Conclusions and future work Iliano Cervesato - Lollipops taste of Vanilla too 2
Meta-programs Meta-programs Meta-programs ... are programs that treat other programs as data . Advantages: – powerful extensions of the base language are easily encoded – programming tools can be developed as needed – is a valid support to rapid prototyping Drawbacks: – low efficiency (in part recovered through partial evaluation) Applications: – software development (integrated programming environments) – software analysis – artificial intelligence (knowledge based systems) – theorem proving Iliano Cervesato - Lollipops taste of Vanilla too 3
The current trend The current trend The current trend There have been many proposal to improve the meta-programming capabilities of Prolog: – 1982: Bowen & Kowalski's paper – 1985: MetaProlog (Bowen et al.) – 1989: Reflective Prolog (Costantini, Lanzarone) – 1990: Gödel (Hill, Lloyd) – 1991: 'Log (Cervesato, Rossi) No or little work has been done to investigate the meta-programming attitude of the new generation logic programming languages ( λ -Prolog, Lolli, Forum, Elf, ...). The underlying theory makes new techniques available to explore the properties of the meta- programs. Iliano Cervesato - Lollipops taste of Vanilla too 4
The granularity of a The granularity of a The granularity of a meta-program meta-program meta-program ... is the ratio between what is simulated and what is passed over to the underlying interpreter Direct Simulation execution Efficiency Flexibility Examples: • solve(G) :- G. Vanilla • solve(true). solve((A,B)) :- solve(A), solve(B). solve(A) :- clause(A,B), solve(B). [Bowen-Kowalski] • solve(P,Gs) :- empty(G). solve(P,Gs) :- select(Gs,G,Gs'), member(C,P), rename(C,G,C'), parts(C',H,B), unify(H,G,S), append(G',B,G"), apply(S,G",G'"), solve(P,G'"). Iliano Cervesato - Lollipops taste of Vanilla too 5
The Vanilla meta- The Vanilla meta- The Vanilla meta- interpreter for Prolog interpreter for Prolog interpreter for Prolog solve(true). • Vanilla is a medium solve((A,B)) :- granularity meta-interpreter solve(A), solve(B). • it achieves the function- solve(A) :- alities of the Horn core of clause(A,B), Prolog solve(B). Extra-logical functionalities can be added: solve((A;B)) :- solve(A); Disjunction solve(B). solve(not A) :- Negation as failure not solve(A). solve(bagof(X,G,Xs) :- Grouping bagof(X,solve(G),Xs). solve(A) :- functor(A,F,N), System calls system(F,N), A. Vanilla can be enhanced to deal correctly with control directives such as cut ( ! ). Iliano Cervesato - Lollipops taste of Vanilla too 6
Linear logic Linear logic Linear logic ... refines traditional logic by constraining the number of times an assumption is used in a proof. Weakening and contraction are ruled out. This calls for a finer set of connectives: Connectives Context ¬ ∧ ∨ → Traditional T F Unbound 1 ⊥ ⊗ ℘ −ο ⊥ Multiplicative Split Τ 0 ⊕ Additive & Copy ! ? Exponential Unbound Example of sequent rules: ∆ → ll ∆ → ll ∆ → ll ∆ → ll B C B C 1 2 ⊗ & L L ∆ ∆ → ll ⊗ ∆ → ll , & B C B C 1 2 Controlled forms of weakening and contraction are made availabe through the use of ! and ? ∆ → ll ∆ → ll ∆ → ll ,! ,! , E B B E B E ! ! ! W C D ∆ → ll ∆ → ll ∆ → ll ,! ,! ,! B E B E B E Iliano Cervesato - Lollipops taste of Vanilla too 7
Uniform proofs Uniform proofs Uniform proofs The logical connectives in a logic programming language should be used by the interpreter as search directives for finding proofs of goals Solving a goal G in a program ∆ = building a ∆ → bottom up deduction for the sequent G Such a proof must be: • cut-free • goal directed A cut-free proof Ξ of the sequent is ∆ → G uniform iff every left rules is applied to sequents with an atomic rhs only. • Horn intuitionistic logic YES • Full intuitionistic logic NO! • Full linear logic NO! We can identify maximal fragments of logic having the uniform proof property • hereditary Harrop formulas (the language freely generated from Τ , ∧ , → and ∀ ) Iliano Cervesato - Lollipops taste of Vanilla too 8
The Τ , &, −ο , ⇒ , ∀ The Τ , &, −ο , ⇒ , ∀ The Τ , &, −ο , ⇒ , ∀ fragment fragment fragment Intuitionistic implication : A ⇒ B = !A −ο B Then, the language freely generated from Τ , &, −ο , ⇒ , ∀ HAS the uniform proof property The uniform proof property is preserved if we allow positive occurrences of 1 , ⊕ , ⊗ , ! and ∃ : R ::= Τ | A | R 1 & R 2 | G −ο R | G ⇒ ⇒ R | ∀ ∀ x . R G ::= Τ | A | G 1 & G 2 | R −ο G | R ⇒ ⇒ G | ∀ ∀ x . G | 1 | G 1 ⊗ G 2 | G 1 ⊕ G 2 | ! G | ∃ x . G Specialized sequents: − Γ ∆ → ll ; G Unbound context: Bound Goal Γ = ! A 1 , ..., ! A n context formula set of R-Formulas multiset of G-Formula R-Formulas Iliano Cervesato - Lollipops taste of Vanilla too 9
Sequent calculus rules Sequent calculus rules Sequent calculus rules Γ ∆ → , ; , B B C id absorb Γ → Γ ∆ → ; , ; A A B C 1 T R R 1 Γ ∆ → Γ ∅ → ; T ; Γ ∆ → Γ ∆ → Γ ∆ → Γ ∆ → ; ; ; ; B C B C 1 2 ⊗ & R R Γ ∆ → Γ ∆ ∆ → ⊗ ; & ; , B C B C 1 2 Γ ∆ → Γ ∅ → ; , ; B C B − o ! R R Γ ∆ → − Γ ∅ → ; o ; ! B C B Γ ∆ → Γ ∆ → , ; ; B C B i ⇒ ⊕ R Ri Γ ∆ → ⇒ Γ ∆ → ⊕ ; ; B C B B 1 2 Γ ∆ → Γ ∆ → ; [ / ] ; [ / ] B c x B t x ∀ ∃ R R Γ ∆ → ∀ Γ ∆ → ∃ ; . ; . x B x B ∃ ∀ where is a term in , and is not present in the lower sequent of t R c R Γ ∅ → Γ ∅ → Γ ∆ → Γ ∆ → ; ... ; ; ... ; B B C C 1 n 1 1 m m BC Γ ∆ ∆ → ; ,..., , B A 1 m ≥ σ provided that , n m 0, is atomic and there exist a substitution A ≡ σ such that & ... & B D D 1 p = ο − ⊗ ⊗ ⊗ ⊗ ⊗ with ! ... ! ... D A B B C C m 1 1 n 1 Iliano Cervesato - Lollipops taste of Vanilla too 10
Lolli Lolli Lolli Concrete syntax Τ −>> erase 1 −>> true −>> { A } ! A −>> A & B A & B A ⊗ B −>> A , B A ⊕ B −>> A ; B A −ο B −>> A -o B and B :− A ⇒ B −>> A ⇒ A => B and B <= A ∀ x . A −>> ∀ forall x \ A ∃ x . A −>> exists x \ A Example: toggling a switch s toggle s G :- on s, Without resource (off s -o G). consumption, turning toggle s G :- s on would keep it off s, also off (on s -o G). LINEAR off s. Iliano Cervesato - Lollipops taste of Vanilla too 11
Operational semantics Operational semantics Operational semantics • D ; ∅ �� LOLLI true • D ; R �� LOLLI erase • D ; R �� LOLLI { G } iff D ; ∅ �� LOLLI G • D ; R 1 + R 2 �� LOLLI G 1 , G 2 iff D ; R 1 �� LOLLI G 1 and D ; R 2 �� LOLLI G 2 • D ; R �� LOLLI G 1 & G 2 iff D ; R �� LOLLI G 1 and D ; R �� LOLLI G 2 • D ; R �� LOLLI G 1 ; G 2 iff D ; R �� LOLLI G 1 or D ; R �� LOLLI G 2 • D ; R �� LOLLI r -o G iff D ; R + r �� LOLLI G • D ; R �� LOLLI d => G iff D ∪ d ; R �� LOLLI G • D ; R �� LOLLI forall x \ G iff D ; R �� LOLLI [ c / x ] G where c is a new constant • D ; R �� LOLLI exists x \ G iff D ; R �� LOLLI [ t / x ] G for some term t • D ∪ h :- b ; R �� LOLLI A σ = match( A , h ) and if D ∪ h :- b ; R �� LOLLI b σ • D ; R + h :- b �� LOLLI A σ = match( A , h ) and if D ; R �� LOLLI b σ • D ; R + c 1 & ... & c n �� LOLLI A D ; R + c i �� LOLLI A if for some i =1... n where σ = match( A , A 1 & ... & A n ) iff A σ = A i σ for some i =1... n Iliano Cervesato - Lollipops taste of Vanilla too 12
Recommend
More recommend
