Class first logic order formmulas

First-Order Logic (Chapters 8 - 9)

First-Order Logic (FOL or FOPC) Syntax

• User defines these primitives:
  • Constant symbols (i.e., the "individuals" in the world) E.g., Mary, 3
  • Function symbols (mapping individuals to individuals) E.g., father-of(Mary) = John, color-of(Sky) = Blue
  • Predicate symbols (mapping from individuals to truth values) E.g., greater(5,3), green(Grass), color(Grass, Green)
  • Variable symbols. E.g., x, y
  • Connectives. Same as in PL: not (~), and (^), or (v), implies (=>), if and only if ( )
  • Quantifiers: Universal (A) and Existential (E)
    • Universal quantification corresponds to conjunction ("and") in that (Ax)P(x) means that P holds for all values of x in the domain associated with that variable. E.g., (Ax) dolphin(x) => mammal(x)
    • Existential quantification corresponds to disjunction ("or") in that (Ex)P(x) means that P holds for some value of x in the domain associated with that variable. E.g., (Ex) mammal(x) ^ lays-eggs(x)
    • Universal quantifiers usually used with "implies" to form "if-then rules." E.g., (Ax) cs540-student(x) => smart(x) means "All cs540 students are smart." You rarely use universal quantification to make blanket statements about every individual in the world: (Ax)cs540-student(x) ^ smart(x) meaning that everyone in the world is a cs540 student and is smart.
    • Existential quantifiers usually used with "and" to specify a list of properties or facts about an individual. E.g., (Ex) cs540-student(x) ^ smart(x) means "there is a cs540 student who is smart." A common mistake is to represent this English sentence as the FOL sentence: (Ex) cs540-student(x) => smart(x) But consider what happens when there is a person who is NOT a cs540-student.
    • Switching the order of universal quantifiers does not change the meaning: (Ax)(Ay)P(x,y) is logically equivalent to (Ay)(Ax)P(x,y). Similarly, you can switch the order of existential quantifiers.
    • Switching the order of universals and existentials does change meaning:
      • Everyone likes someone: (Ax)(Ey)likes(x,y)
      • Someone is liked by everyone: (Ey)(Ax)likes(x,y)
      • A term (denoting a real-world individual) is a constant symbol, a variable symbol, or an n-place function of n terms. For example, x and f(x1, . xn) are terms, where each xi is a term.
      • An atom (which has value true or false) is either an n-place predicate of n terms, or, if P and Q are atoms, then ~P, P V Q, P ^ Q, P => Q, P Q are atoms
      • A sentence is an atom, or, if P is a sentence and x is a variable, then (Ax)P and (Ex)P are sentences
      • A well-formed formula (wff) is a sentence containing no "free" variables. I.e., all variables are "bound" by universal or existential quantifiers. E.g., (Ax)P(x,y) has x bound as a universally quantified variable, but y is free.

      Translating English to FOL

      • Every gardener likes the sun.
        (Ax) gardener(x) => likes(x,Sun)
      • You can fool some of the people all of the time.
        (Ex) (person(x) ^ (At)(time(t) => can-fool(x,t)))
      • You can fool all of the people some of the time.
        (Ax) (person(x) => (Et) (time(t) ^ can-fool(x,t)))
      • All purple mushrooms are poisonous.
        (Ax) (mushroom(x) ^ purple(x)) => poisonous(x)
      • No purple mushroom is poisonous.
        ~(Ex) purple(x) ^ mushroom(x) ^ poisonous(x)
        or, equivalently,
        (Ax) (mushroom(x) ^ purple(x)) => ~poisonous(x)
      • There are exactly two purple mushrooms.
        (Ex)(Ey) mushroom(x) ^ purple(x) ^ mushroom(y) ^ purple(y) ^ ~(x=y) ^ (Az) (mushroom(z) ^ purple(z)) => ((x=z) v (y=z))
      • Deb is not tall.
        ~tall(Deb)
      • X is above Y if X is on directly on top of Y or else there is a pile of one or more other objects directly on top of one another starting with X and ending with Y.
        (Ax)(Ay) above(x,y) (on(x,y) v (Ez) (on(x,z) ^ above(z,y)))

      Inference Rules for FOL

      • Inference rules for PL apply to FOL as well. For example, Modus Ponens, And-Introduction, And-Elimination, etc.
      • New (sound) inference rules for use with quantifiers:
        • Universal Elimination
          If (Ax)P(x) is true, then P(c) is true, where c is a constant in the domain of x. For example, from (Ax)eats(Ziggy, x) we can infer eats(Ziggy, IceCream). The variable symbol can be replaced by any ground term, i.e., any constant symbol or function symbol applied to ground terms only.
        • Existential Introduction
          If P(c) is true, then (Ex)P(x) is inferred. For example, from eats(Ziggy, IceCream) we can infer (Ex)eats(Ziggy, x). All instances of the given constant symbol are replaced by the new variable symbol. Note that the variable symbol cannot already exist anywhere in the expression.
        • Existential Elimination
          From (Ex)P(x) infer P(c). For example, from (Ex)eats(Ziggy, x) infer eats(Ziggy, Cheese). Note that the variable is replaced by a brand new constant that does not occur in this or any other sentence in the Knowledge Base. In other words, we don't want to accidentally draw other inferences about it by introducing the constant. All we know is there must be some constant that makes this true, so we can introduce a brand new one to stand in for that (unknown) constant.
        • Given two sentences (P1 v . v PN) and (t=s v Q1 v . v QM) where each Pi and Qi is a literal (see definition below) and Pj contains a term t, derive new sentence (P1 v . v Pj-1 v Pj[s] v Pj+1 v . v PN v Q1 v . v QM) where Pj[s] means a single occurrence of the term t is replaced by the term s in Pj
        • Example: From P(a) and a=b derive P(b)
        • Combines And-Introduction, Universal-Elimination, and Modus Ponens
        • Example: from P(c), Q(c), and (Ax)(P(x) ^ Q(x)) => R(x), derive R(c)
        • In general, given atomic sentences P1, P2, . PN, and implication sentence (Q1 ^ Q2 ^ . ^ QN) => R, where Q1, . QN and R are atomic sentences, and subst(Theta, Pi) = subst(Theta, Qi) for i=1. N, derive new sentence: subst(Theta, R)
        • subst(Theta, alpha) denotes the result of applying a set of substitutions defined by Theta to the sentence alpha
        • A substitution list Theta = means to replace all occurrences of variable symbol vi by term ti. Substitutions are made in left-to-right order in the list. Example: subst(, eats(y,x)) = eats(Ziggy, IceCream)

        Automated Inference for FOL

        • Automated inference using FOL is harder than using PL because variables can take on potentially an infinite number of possible values from their domain. Hence there are potentially an infinite number of ways to apply Universal-Elimination rule of inference
        • Godel's Completeness Theorem says that FOL entailment is only semidecidable. That is, if a sentence is true given a set of axioms, there is a procedure that will determine this. However, if the sentence is false, then there is no guarantee that a procedure will ever determine this. In other words, the procedure may never halt in this case.
        • The Truth Table method of inference is not complete for FOL because the truth table size may be infinite
        • Natural Deduction is complete for FOL but is not practical for automated inference because the "branching factor" in a search is too large, caused by the fact that we would have to potentially try every inference rule in every possible way using the set of known sentences
        • Generalized Modus Ponens is not complete for FOL
        • Generalized Modus Ponens is complete for KBs containing only Horn clauses
          • A Horn clause is a sentence of the form:
            (Ax) (P1(x) ^ P2(x) ^ . ^ Pn(x)) => Q(x)
            where there are 0 or more Pi's, and the Pi's and Q are positive (i.e., un-negated) literals
          • Horn clauses represent a subset of the set of sentences representable in FOL. For example, P(a) v Q(a) is a sentence in FOL but is not a Horn clause.
          • Natural deduction using GMP is complete for KBs containing only Horn clauses. Proofs start with the given axioms/premises in KB, deriving new sentences using GMP until the goal/query sentence is derived. This defines a forward chaining inference procedure because it moves "forward" from the KB to the goal.
            • Example: KB = All cats like fish, cats eat everything they like, and Ziggy is a cat. In FOL, KB =
              
              1. (Ax) cat(x) => likes(x, Fish)
              2. (Ax)(Ay) (cat(x) ^ likes(x,y)) => eats(x,y)
              3. cat(Ziggy)

            Goal query: Does Ziggy eat fish?

            1. Use GMP with (1) and (3) to derive: 4. likes(Ziggy, Fish)
            2. Use GMP with (3), (4) and (2) to derive eats(Ziggy, Fish)
            3. So, Yes, Ziggy eats fish.

            To prove eats(Ziggy, Fish), first see if this is known from one of the axioms directly. Here it is not known, so see if there is a Horn clause that has the consequent (i.e., right-hand side) of the implication matching the goal. Here,

            1. Goal matches RHS of Horn clause (2), so try and prove new sub-goals cat(Ziggy) and likes(Ziggy, Fish) that correspond to the LHS of (2)
            2. cat(Ziggy) matches axiom (3), so we've "solved" that sub-goal
            3. likes(Ziggy, Fish) matches the RHS of (1), so try and prove cat(Ziggy)
            4. cat(Ziggy) matches (as it did earlier) axiom (3), so we've solved this sub-goal
            5. There are no unsolved sub-goals, so we're done. Yes, Ziggy eats fish.

            Resolution Refutation Procedure (aka Resolution Procedure)

            • Resolution procedure is a sound and complete inference procedure for FOL
            • Resolution procedure uses a single rule of inference: the Resolution Rule (RR), which is a generalization of the same rule used in PL.
            • Resolution Rule for PL:
              From sentence P1 v P2 v . v Pn and sentence ~P1 v Q2 v . v Qm derive resolvent sentence: P2 v . v Pn v Q2 v . v Qm
              • Examples
                
                • From P and ~P v Q, derive Q (Modus Ponens)
                  
                • From (~P v Q) and (~Q v R), derive ~P v R
                  
                • From P and ~P, derive False
                  
                • From (P v Q) and (~P v ~Q), derive True
                • Example
                  
                  • From clause P(x, f(a)) v P(x, f(y)) v Q(y) and clause ~P(z, f(a)) v ~Q(z), derive resolvent clause P(z, f(y)) v Q(y) v ~Q(z) using Theta =

                  Unification

                  • Unification is a "pattern matching" procedure that takes two atomic sentences, called literals, as input, and returns "failure" if they do not match and a substitution list, Theta, if they do match. That is, unify(p,q) = Theta means subst(Theta, p) = subst(Theta, q) for two atomic sentences p and q.
                  • Theta is called the most general unifier (mgu)
                  • All variables in the given two literals are implicitly universally quantified
                  • To make literals match, replace (universally-quantified) variables by terms
                  • Unification algorithm

                  procedure unify(p, q, theta) Scan p and q left-to-right and find the first corresponding terms where p and q "disagree" ; where p and q not equal If there is no disagreement, return theta ; success Let r and s be the terms in p and q, respectively, where disagreement first occurs If variable(r) then theta = union(theta, ) unify(subst(theta, p), subst(theta, q), theta) else if variable(s) then theta = union(theta, ) unify(subst(theta, p), subst(theta, q), theta) else return "failure" end 

                  Literal 1	Literal 2	Result of Unify
                  parents(x, father(x), mother(Bill))	parents(Bill, father(Bill), y)
                  parents(x, father(x), mother(Bill))	parents(Bill, father(y), z)
                  parents(x, father(x), mother(Jane))	parents(Bill, father(y), mother(y))	Failure

                  Resolution Refutation Procedure (aka Resolution Procedure)

                  • Proof by contradiction method
                  • Given a consistent set of axioms KB and goal sentence Q, we want to show that KB |= Q. This means that every interpretation I that satisfies KB, satisfies Q. But we know that any interpretation I satisfies either Q or ~Q, but not both. Therefore if in fact KB |= Q, an interpretation that satisfies KB, satisfies Q and does not satisfy ~Q. Hence KB union ~Q> is unsatisfiable, i.e., that it's false under all interpretations.
                  • In other words, (KB |- Q) (KB ^ ~Q |- False)
                  • What's the gain? If KB union ~Q is unsatisfiable, then some finite subset is unsatisfiable
                  • Resolution procedure can be used to establish that a given sentence Q is entailed by KB; however, it cannot, in general, be used to generate all logical consequences of a set sentences. Also, the resolution procedure cannot be used to prove that Q is not entailed by KB.
                  • Resolution procedure won't always give an answer since entailment is only semidecidable. And you can't just run two proofs in parallel, one trying to prove Q and the other trying to prove ~Q, since KB might not entail either one.
                  • Algorithm
                    

                  procedure resolution-refutation(KB, Q) ;; KB is a set of consistent, true FOL sentences ;; Q is a goal sentence that we want to derive ;; return success if KB |- Q, and failure otherwise KB = union(KB, ~Q) while false not in KB do pick 2 sentences, S1 and S2, in KB that contain literals that unify if none, return "failure" resolvent = resolution-rule(S1, S2) KB = union(KB, resolvent) return "success" end

                  Example using PL Sentences

                  • From the sentence "Heads I win, tails you lose," prove that "I win."
                  1. "Heads I win, tails you lose."
                     (Heads => IWin) or, equivalently, (~Heads v IWin)
                     (Tails => YouLose) or, equivalently, (~Tails v YouLose)
                  2. Add some general knowledge axioms about coins, winning, and losing:
                     (Heads v Tails)
                     (YouLose => IWin) or, equivalently, (~YouLose v IWin)
                     (IWin => YouLose) or, equivalently, (~IWin v YouLose)

                  Sentence 1	Sentence 2	Resolvent
                  ~IWin	~Heads v IWin	~Heads
                  ~Heads	Heads v Tails	Tails
                  Tails	~Tails v YouLose	YouLose
                  YouLose	~YouLose v IWin	IWin
                  IWin	~IWin	False

                  Problems Yet to Be Addressed

                  • Resolution rule of inference is only applicable with sentences that are in the form P1 v P2 v . v Pn, where each Pi is a negated or un-negated predicate and contains functions, constants, and universally quantified variables, so can we convert every FOL sentence into this form?
                  • How to pick which pair of sentences to resolve?
                  • How to pick which pair of literals, one from each sentence, to unify?

                  Converting FOL Sentences to Clause Form

                  • Every FOL sentence can be converted to a logically equivalent sentence that is in a "normal form" called clause form
                  • Steps to convert a sentence to clause form:
                  • Eliminate all connectives by replacing each instance of the form (P Q) by the equivalent expression ((P => Q) ^ (Q => P))
                  • Eliminate all => connectives by replacing each instance of the form (P => Q) by (~P v Q)
                  • Reduce the scope of each negation symbol to a single predicate by applying equivalences such as converting ~~P to P; ~(P v Q) to ~P ^ ~Q; ~(P ^ Q) to ~P v ~Q; ~(Ax)P to (Ex)~P, and ~(Ex)P to (Ax)~P
                  • Standardize variables: rename all variables so that each quantifier has its own unique variable name. For example, convert (Ax)P(x) to (Ay)P(y) if there is another place where variable x is already used.
                  • Eliminate existential quantification by introducing Skolem functions. For example, convert (Ex)P(x) to P(c) where c is a brand new constant symbol that is not used in any other sentence. c is called a Skolem constant. More generally, if the existential quantifier is within the scope of a universal quantified variable, then introduce a Skolem function that depend on the universally quantified variable. For example, (Ax)(Ey)P(x,y) is converted to (Ax)P(x, f(x)). f is called a Skolem function, and must be a brand new function name that does not occur in any other sentence in the entire KB.
                    • Example: (Ax)(Ey)loves(x,y) is converted to (Ax)loves(x,f(x)) where in this case f(x) specifies the person that x loves. (If we knew that everyone loved their mother, then f could stand for the mother-of function.

                    Convert the sentence (Ax)(P(x) => ((Ay)(P(y) => P(f(x,y))) ^ ~(Ay)(Q(x,y) => P(y))))

                    1. Eliminate
                       Nothing to do here.
                    2. Eliminate =>
                       (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ ~(Ay)(~Q(x,y) v P(y))))
                    3. Reduce scope of negation
                       (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Ey)(Q(x,y) ^ ~P(y))))
                    4. Standardize variables
                       (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Ez)(Q(x,z) ^ ~P(z))))
                    5. Eliminate existential quantification
                       (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Q(x,g(x)) ^ ~P(g(x)))))
                    6. Drop universal quantification symbols
                       (~P(x) v ((~P(y) v P(f(x,y))) ^ (Q(x,g(x)) ^ ~P(g(x)))))
                    7. Convert to conjunction of disjunctions
                       (~P(x) v ~P(y) v P(f(x,y))) ^ (~P(x) v Q(x,g(x))) ^ (~P(x) v ~P(g(x)))
                    8. Create separate clauses
                       
                       • ~P(x) v ~P(y) v P(f(x,y))
                       • ~P(x) v Q(x,g(x))
                       • ~P(x) v ~P(g(x))
                    9. Standardize variables
                       
                       • ~P(x) v ~P(y) v P(f(x,y))
                       • ~P(z) v Q(z,g(z))
                       • ~P(w) v ~P(g(w))

                    Revised Resolution Refutation Procedure

                    procedure resolution-refutation(KB, Q) ;; KB is a set of consistent, true FOL sentences ;; Q is a goal sentence that we want to derive ;; return success if KB |- Q, and failure otherwise KB = union(KB, ~Q) KB = clause-form(KB) ; convert sentences to clause form while false not in KB do pick 2 clauses, C1 and C2, that contain literals that unify if none, return "failure" resolvent = resolution-rule(C1, C2) KB = union(KB, resolvent) return "success" end

                    Example: Hoofers Club

                    • Problem Statement

                    Tony, Shi-Kuo and Ellen belong to the Hoofers Club. Every member of the Hoofers Club is either a skier or a mountain climber or both. No mountain climber likes rain, and all skiers like snow. Ellen dislikes whatever Tony likes and likes whatever Tony dislikes. Tony likes rain and snow.

                    Is there a member of the Hoofers Club who is a mountain climber but not a skier?

                    Let S(x) mean x is a skier, M(x) mean x is a mountain climber, and L(x,y) mean x likes y, where the domain of the first variable is Hoofers Club members, and the domain of the second variable is snow and rain. We can now translate the above English sentences into the following FOL wffs:

                    1. (Ax) S(x) v M(x)
                    2. ~(Ex) M(x) ^ L(x, Rain)
                    3. (Ax) S(x) => L(x, Snow)
                    4. (Ay) L(Ellen, y) ~L(Tony, y)
                    5. L(Tony, Rain)
                    6. L(Tony, Snow)

                    1. S(x1) v M(x1)
                    2. ~M(x2) v ~L(x2, Rain)
                    3. ~S(x3) v L(x3, Snow)
                    4. ~L(Tony, x4) v ~L(Ellen, x4)
                    5. L(Tony, x5) v L(Ellen, x5)
                    6. L(Tony, Rain)
                    7. L(Tony, Snow)
                    8. Negation of the Query: ~M(x7) v S(x7)

                    Clause 1	Clause 2	Resolvent	MGU (i.e., Theta)
                    8	1	9. S(x1)
                    9	3	10. L(x1, Snow)
                    10	4	11. ~L(Tony, Snow)
                    11	7	12. False	<>

                    Clause 1	Clause 2	Resolvent	MGU (i.e., Theta)
                    ~M(x7) v S(x7) v (M(x7) ^ ~S(x7))	1	9. S(x1) v (M(x1) ^ ~S(x1))
                    9	3	10. L(x1, Snow) v (M(x1) ^ ~S(x1))
                    10	4	11. ~L(Tony, Snow) v (M(Ellen) ^ ~S(Ellen))
                    11	7	12. M(Ellen) ^ ~S(Ellen)	<>

                    Resolution Procedure as Search through a Search Space

                    • Resolution procedure can be thought of as the bottom-up construction of a search tree, where the leaves are the clauses produced by KB and the negation of the goal. When a pair of clauses generates a new resolvent clause, add a new node to the tree with arcs directed from the resolvent to the two parent clauses. The resolution procedure succeeds when a node containing the False clause is produced, becoming the root node of the tree.
                    • A strategy is complete if its use guarantees that the empty clause (i.e., false) can be derived whenever it is entailed
                    • Some Strategies for Controlling Resolution's Search
                      • Breadth-First
                        • Level 0 clauses are those from the original axioms and the negation of the goal. Level k clauses are the resolvents computed from two clauses, one of which must be from level k-1 and the other from any earlier level.
                        • Compute all level 1 clauses possible, then all possible level 2 clauses, etc.
                        • Complete, but very inefficient.
                        • At least one parent clause must be from the negation of the goal or one of the "descendents" of such a goal clause (i.e., derived from a goal clause)
                        • Complete (assuming all possible set-of-support clauses are derived)
                        • At least one parent clause must be a "unit clause," i.e., a clause containing a single literal
                        • Not complete in general, but complete for Horn clause KBs
                        • At least one parent from the set of original clauses (from the axioms and the negation of the goal)
                        • Not complete in general, but complete for Horn clause KBs

                        Last modified October 14, 1998
                        Copyright © 1996 by Charles R. Dyer. All rights reserved.