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Rule-based deduction systems The way in which a piece of knowledge is expressed by a human expert carries important information, example: if the person has fever and feels tummy- pain then she may have an infection. In logic it can be expressed as follows: x. (has_fever(x) & tummy_pain(x) has_an_infection(x)) If we convert this formula to clausal form we loose the content as then we may have equivalent formulas like: (i) has_fever(x) & ~has_an_infection(x) ~tummy_pain(x) (ii) ~has_an_infection(x) & tummy_pain(x) ~has_fever(x) We notice that (i) and (ii) despite been logically equivalent to the original sentence have lost the main information contained in its formulation.
Forward production systems The main idea behind the forward/backward production systems is to take advantage of the implicational form in which production rules are stated by the expert and use that information to help achieving the goal. In the present systems the formulas will have two forms: rules and facts, rules are the productions stated in implication form. They express specific knowledge about the problem, and facts are assertions not expressed as implications. The task of the system will be to prove a goal formula with these facts and rules. In a forward production system the rules are expressed as F-rules which operate on the global database of facts until the termination condition is achieved. This sort of proving system is a direct system rather of a refutation system. Facts Facts are expressed in AND/OR form. An expression in AND/OR form consists on sub-expressions of literals connected by & and V symbols. An expression in AND/OR form is not in clausal form.
F-rules
Rules in a forward production system will be applied to the AND/OR graph to produce new transformed graph structures. We assume that rules in a forward production system are of the form: L ==> W, where L is a literal and W is a formula in AND/OR form. Recall that a rule of the form (L1 V L2) ==> W is equivalent to the pair of rules: L1 ==> W V L2 ==> W. Steps to transform the rules into a free-quantifier form:
- Eliminate (temporarily) implication symbols.
- Reverse quantification of variables in first disjunct by moving negation symbol.
- Skolemize existential variables.
- Move all universal quantifiers to the front and drop.
- Restore implication. All variables appearing on the final expressions are assumed to be universally quantified. E.g. Original formula: x.(y. z. (p(x, y, z)) u. q(x, u)) Converted formula: p(x, y, f(x, y)) q(x, u).
A full example:
(a) Fido barks and bites, or Fido is not a dog.
(b) All terriers are dogs.
(c) Anyone who barks is noisy.
Based on these facts, prove that: “there exists someone
who is not a terrier or who is noisy.”
Logic representation: (barks(fido) & bites(fido)) v ~dog(fido) R1: terrier(x) dog(x) R2: barks(y) noisy(y)
goal: w.(~terrier(w) v noisy(w))
Backward production systems
An important property of logic is the duality between assertions and goals in theorem-proving systems. Duality between assertions and goals allows the goal expression to be treated as if it were an assertion. Conversion of the goal expression into AND/OR form:
- Elimination of implication symbols.
- Move negation symbols in.
- Skolemize universal variables.
- Drop existential quantifiers. Variables remaining in the AND/OR form are considered to be existentially quantified. Goal clauses are conjunctions of literals and the disjunction of these clauses is the clause form of the goal well-formed formula.
B-Rules We restrict B-rules to expressions of the form: W ==> L, where W is an expression in AND/OR form and L is a literal, and the scope of quantification of any variables in the implication is the entire implication. Recall that W==>(L1 & L2) is equivalent to the two rules: W==>L1 and W==>L2. An important property of logic is the duality between assertions and goals in theorem-proving systems. Duality between assertions and goals allows the goal expression to be treated as if it were an assertion. Conversion of the goal expression into AND/OR form:
- Elimination of implication symbols.
- Move negation symbols in.
- Skolemize existential variables.
- Drop existential quantifiers. Variables remaining in the AND/OR form are considered to be existentially quantified. Goal clauses are conjunctions of literals and the disjunction of these clauses is the clause form of the goal well-formed formula.
- The blocks-word situation is described by the following set of wffs: on_table(a) clear(e) on_table(c) clear(d) on(d,c) heavy(d) on(b,a) wooden(b) heavy(b) on(e,b) The following statements provide general knowledge about this blocks word: Every big, blue block is on a green block. Each heavy, wooden block is big. All blocks with clear tops are blue. All wooden blocks are blue. Represent these statements by a set of implications having single-literal consequents. Draw a consistent AND/OR solution tree (using B-rules) that solves the problem: “Which block is on a green block?”
Problem 2. Transformation of rules and goal: Facts: f1: on_table(a) f6: clear(e) f2: on_table(c) f7: clear(d) f3: on(d,c) f8: heavy(d) f4: on(b,a) f9: wooden(b) f5: heavy(b) f10: on(e,b) Rules: R1: big(y1) ^ blue(y1) green(g(y1)) Every big, blue block is on a green block. R2: big(y0) ^ blue(y0) on(y0,g(y0)) “ “ “ “ “ “ “ “ “ R3: heavy(z) ^ wooden(z) big(z) Each heavy, wooden block is big. R4: clear(x) blue(x) All blocks with clear tops are blue. R5: wooden(w) blue(w) All wooden blocks are blue. Goal: green(u) ^ on(v,u) Which block is on a green block?
Rules R1: manager(x,y) works_in(x,y) R2: works_in(x,y) & manager(x,z) boss_of(y,z) R3: works_in(x,y) & works_in(x,z) ~married(y,z) R4: married(y,z) married(z,y) R5: [married(x,y) & works_in(p-d,x) insured_by(x,eagle-corp) With these facts and rules a simple backward production system can answer a variety of questions. Build solution graphs for the following questions:
- Name someone who works in the Purchasing Department.
- Name someone who is married and works in the sales department.
- Who is Joe Smith’s boss?
- Name someone insured by Eagle Corporation.
- Is John Jones married with Sally Jones?
Planning
Planning is fundamental to “intelligent” behavior. E.g.
- assembling tasks - route finding
- planning chemical processes - planning a report Representation The planner has to represent states of the world it is operating within, and to predict consequences of carrying actions in its world. E.g. initial state: final state: a b c d on(a,b) on(b,table) on(d,c) on(c,table) clear(a) clear(d) on(a,b) on(b,c) on(c,d) on(d,table) clear(a) a b d c
Planning The Frame Problem This is the problem of how to keep track in a representation of the world of all the effects that an action may have. The action representation given is the one introduced by STRIPS (Nilson and is an attempt to a solution to the frame problem but it is only adequate for simple actions in simple worlds. If we include the problem of stability when a child plays with blocks it makes the action representation impossible. The Frame Axiom The frame axiom states that a fact is true (false) if it is not in the last delete (add) list and was true (false) in the previous state.
Planning
Control Strategies
Forward Chaining
Backward Chaining
The choice on which of these strategies to use
depends on the problem, normally backward chaining
is more effective.
Planning on(c,a) clear(c) handempty (^) upstack(c,a) putdown(c) pickup(b) stack(b,c) pickup(a) stack(a,b) holding(c) clear(a) clear(c) handempty ontable(b) clear(b) holding(b) handempty on(b,c) clear(b) ontable(a) holding(a) on(a,b) TRIANGLE TABLE {unstack(c,a), putdown(c), pickup(b), stack(b,c), pickup(a), stack(a,b)} 0 1 2 3 4 5 6
