Logic, Resolution and Prolog Michael Zakharyaschev Department of - - PowerPoint PPT Presentation
Logic, Resolution and Prolog Michael Zakharyaschev Department of Computer Science and Information Systems Birkbeck, University of London email: zmishaz@gmail.com homepage: http://www.dcs.bbk.ac.uk/~michael course site:
zmishaz@gmail.com
http://www.dcs.bbk.ac.uk/~michael
http://www.dcs.bbk.ac.uk/~michael/res/res.html
A Horn program, P, consists of sentences of the form – ∀X1 . . . ∀Xk (A1 ∧ A2 ∧ · · · ∧ An → A0) where the Ai are atoms usually written as A0 ← A1, . . . , An and called a rule or clause
A0 the head and A1, . . . , An the body
– ∀X1 . . . ∀Xk A where A is an atom usually written as A0 ← and called a fact – A goal is a conjunction of atoms G = C1 ∧ · · · ∧ Cl usually written as ?C1, . . . , Cl The variables Y1, . . . , Ym in G are goal variables (or answer variables) An answer to G over P is any substitution θ = {Y1/t1, . . . , Ym/tm} An answer θ is a correct answer if | = P → ∀Z1 . . . ∀ZN Gθ where Z1, . . . , ZN are all the variables in t1, . . . , tm This gives the declarative semantics of Horn programs
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G : ? elem(X, w.o.r.d.nil) P : elem(X, X.L) elem(X, Y.L) ← elem(X, L) concat(nil, Y, Y ) concat(U.X, Y, U.Z) ← concat(X, Y, Z) common(X, Y ) ← elem(U, X), elem(U, Y ) There are four correct answers to G over P θ1 = {X/w}, θ2 = {X/o}, θ3 = {X/r}, θ4 = {X/d} because | =
∀X∀L elem(X, X.L)
How to compute the correct answers? that is, what is the procedural (or operational) semantics?
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Given a goal G(Y1, . . . Ym) = ? C1, C2, . . . , Cm over a Horn program P = {D1, . . . , DN} we consider the validity problem | = D1 ∧ · · · ∧ DN → ∃Y1 . . . ∃Yk (C1 ∧ C2 ∧ · · · ∧ Cm), which is equivalent to the unsatisfiability problem for the sentence ¬(D1 ∧ · · · ∧ DN → ∃Y1 . . . ∃Yk(C1 ∧ C2 ∧ · · · ∧ Cm), which is equivalent to unsatisfiability of D1 ∧ · · · ∧ DN ∧ ∀Y1 . . . ∀Yk(¬C1 ∨ ¬C2 ∨ · · · ∨ ¬Cm), which is represented as the set of clauses SP,G = {D1, . . . , DN, ¬C1 ∨ ¬C2 ∨ · · · ∨ ¬Cm}, which is treated by the resolution method using a special strategy
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❄ ✏ ✏ ✏ ✏ ✏ ✏ ✮
❄ ✏ ✏ ✏ ✏ ✏ ✏ ✮
q q q
❄ ✏ ✏ ✏ ✏ ✏ ✏ ✮
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Suppose – G = ? C1, . . . , Ci, . . . , Cm is a goal with a selected sub-goal Ci – D′ = A′
0 ← A′ 1, A′ 2, . . . , A′ n is a variant of a rule in P
which does not have common variables with G – θ ∈ MGU(Ci, A′
0) is an MGU of the sub-goal Ci and the head A′ 0 of D′
Then the goal – G′ = ?(C1, . . . , Ci−1, A′
1, A′ 2, . . . , A′ n, Ci+1, . . . , Cm)θ
is called an SLD-resolvent of D′ and G for the selected sub-goal Ci and the unifier θ In other words, we apply the resolution rule: G = ¬C1 ∨ · · · ∨ ¬Ci ∨ · · · ∨ ¬Cm D′ = A′
0 ∨ ¬A′ 1 ∨ ¬A′ 2 ∨ · · · ∨ A′ n
θ = MGU(Ci, A′
0)
G′ = (¬C1 ∨ · · · ∨ ¬Ci−1 ∨ ¬A′
1 ∨ ¬A′ 2 ∨ · · · ∨ ¬A′ n ∨ ¬Ci+1 ∨ · · · ∨ ¬Cm)θ Logic, Resolution and Prolog (5) 5
Given a goal G1(Y1, . . . Ym) = ? C1, C2, . . . , Cm over a Horn program P = {D1, . . . , DN}, an SLD-derivation from G1 and P is a (finite or infinite) sequence of triples (Dj1, θ1, G1), (Dj2, θ2, G2), . . . , (Djn, θn, Gn), . . . such that, for every i, i ≥ 1, – Dji ∈ P, θi is a substitution, Gi is a goal – Gi is an SLD-resolvent of Dji and Gi−1 with unifier θi A finite SLD-derivation (Dj1, θ1, G1), (Dj2, θ2, G2), . . . , (Djk, θn, Gn) is successful (or an SLD-refutation) if Gn = The restriction of θ = θ1θ2 . . . θn to the goal variables Y1, . . . , Yk is called a computed answer for P and G1 If Gn has no SLD-resolvent, then the derivation is called failed
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Program P: elem(X, X.L) elem(X, Y.L) ← elem(X, L) Goal G1: ? elem(X, a.b.c.nil)
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Program P: elem(X, X.L) elem(X, Y.L) ← elem(X, L) Goal G1: ? elem(X, a.b.c.nil) Goal G2: ? elem(c, a.b.nil)
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Program P: elem(X, X.L) elem(X, Y.L) ← elem(X, L) Goal G1: ? elem(X, a.b.c.nil) Goal G2: ? elem(c, a.b.nil) Goal G3: ? elem(a, X)
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Program P: elem(X, X.L) elem(X, Y.L) ← elem(X, L) Goal G1: ? elem(X, a.b.c.nil) Goal G2: ? elem(c, a.b.nil) Goal G3: ? elem(a, X) Are the computed answers correct? Can all the correct answers be computed?
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Theorem: Every computed answer is correct. For every correct answer θ, there is a computed answer η such that θ = ηρ, for some substitution ρ
(this holds for any selection function for subgoals
Example: Program P: P (f(X), X) ← R(X) R(Y ) ← Q(c) ← Goal: ?P (U, V ) The substitution θ = {U/f(c), V/c} is a correct answer, η = {U/f(Y ), V/Y } is the only computed answer θ = η{Y/c}
Standard selection function: always take the leftmost sub-goal
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Goal: ?P (U, V ), R(U) Program P: P (X, Y ) ← R(X), Q(Y ) P (X, X) ← Q(X) R(b) ← Q(c) ← By choosing the first rule in P, we obtain a successful derivation with the answer {U/b, V/c} The second rule leads to failure
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An SLD-derivation tree for a goal G0 and a program P is a labelled tree TG0,P such that – G0 the root of the tree – the immediate successors of a vertex G are all the SLD-resolvents of G (under the standard selection function for sub-goals) – The leaves of the tree are and goals without resolvents (failures) SLD-derivation trees can be finite or infinite, with a finite or infinite sets of branches, etc.
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P: P (X.L) ← P (L), R(X) P (nil) ← R(a) ← R(c) ←
?P(X.Y) s ❄ ?P(Y), R(X)
(1), {X1/X, L1/Y }
s
❅ ❅ ❅ ❘ s s ?P(L2), R(X2), R(X) ?R(X)
(1), {Y/X2.L2} (2), {Y/nil}
❄ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ ❄ ❅ ❅ ❅ ❘ s s s s ?R(X2), R(X) ?P(L3), R(X3), R(X2), R(X) ? ?
(2), {L2/nil} (1), {L2/X3.L3} (3), {X/a} (4), {X/c}
✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ ❄
❅ ❅ ❅ ❘ s s s ?R(X3), R(X2), R(X) ∞ ?R(X) ?R(X)
(3), {X1/a} (3), {X1/a} (2), {L3/nil}
❄ ❅ ❅ ❅ ❘ s s ? ?
(3), {X/a} (4), {X/c}
❄ ❅ ❅ ❅ ❘ s s ? ?
(3), {X/a} (4), {X/c}
How to search derivation trees: breadth-first? depth-first? some other way? Standard (compromise) strategy:
it is effective, but incomplete! (because termination is not guaranteed) The order of clauses in programs matters!
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What can be computed by Horn programs? Horn programs are algorithmically complete in the sense that they can compute any function computable by Turing machines (or by programs in any programming language) Exercise: given a Turing machine M, construct a Horn program PM simulating the computations of M (Hint: represent words on the machine tape as lists) Theorem: There is no algorithm that can decide, given a goal G and a Horn program P,
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