6. Logical Inference

Transcription

1 Artificial Intelligence 6. Logical Inference Prof. Bojana Dalbelo Bašić Assoc. Prof. Jan Šnajder University of Zagreb Faculty of Electrical Engineering and Computing Academic Year 2016/2017 Creative Commons Attribution NonCommercial NoDerivs Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

2 Outline 1 Three inference tasks 2 Proof theory 3 Resolution in PL 4 Resolution in FOL Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

3 Outline 1 Three inference tasks 2 Proof theory 3 Resolution in PL 4 Resolution in FOL Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

4 Three inference tasks (1) Model checking: given a formula F and an interpretation I, we wish to check if I is the model of F, i.e., if F is true in interpretation I (2) Satisfiability checking or SAT problem: given a formula F, check whether there exists a model for this formula Also known as model building (3) Validity checking: given a formula F, check whether it is valid, i.e., true for every possible interpretation What about proving logical consequences? Proving logical consequences amounts to validity checking (recall the semantic deduction theorem) Q: Are these problems computationally difficult? Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

5 Complexity of inference (1) Model checking: a very simple problem for each formula F and interpretation I, we can check in finitely many steps whether F is true in I decidable in both PL and FOL (under some mild constrainds) the tool for solving this is called a model checker Satisfiability and validity: much harder problems! equally hard, because they are dual to each other: F is valid iff F is not satisfiable thus, if we have an algorithm to solve the SAT problem, we can use it for the satisfiability problem, which in turn means that we can use it to prove logical consequences Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

6 Complexity of inference (2) In PL, SAT is decidable: in finitely many steps we can check if there exists a model for a given formula Although decidable, the problem is intractable (suprapolynomial): the complexity is O(2 n ), where n is the number of propositional variables (2 n rows of a truth table must be checked) SAT problem u PL-u one of the first known NP-complete problem (Cook, 1971) In FOL, SAT is undecidable: no algorithm exists that can check the satisfiability of every formula Algorithms that solve the SAT problem are called SAT solvers. In PL, they can solve all SAT problems, whereas in FOL, they can solve only some Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

7 Problems with proving semantic consequences (1) Intractability: We need to check 2 n interpretations Imagine we wish to prove F1,..., F 100 F 1. Assuming also n = 100, we should check interpretations. This is impossible! Moreover, in this particular case it seems an overkill because it is evident that the semantic consequence relation holds (2) Undecidability: In FOL, the number of interpretations is infinite, thus we have no chance to check all interpretations Actually, FOL is semi-decidable: we can prove validity if it holds, but if it doesn t, we cannot always prove that it doesn t Is there a way out of these problems? Yes, to some extent Instead to deal with interpretations and models (i.e., semantics), we should shift to using rules of inference (i.e, proof theory) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

8 Outline 1 Three inference tasks 2 Proof theory 3 Resolution in PL 4 Resolution in FOL Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

9 Proof theory People don t really draw the conclusions by proving semantic consequences (it s not that they try to find interpretations that satisfy F, but don t satisfy G)! Instead of this, they try to show that G can be derived from the premises using a number of inference rules Each rule needs to be justified and simple Rules of inference enable us to derive new formulae from given premises, without making any explicit reference to the semantics of logic (the truth values of propositional variables) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

10 Proof theory Thus, proof theory offers two benefits over proving semantic consequences: efficiency: instead of searching exhaustively through all the interpretations, we can prove consequences faster (especially when using a smart proof strategy). Note, however, that we cannot escape undecidability of FOL interpretability: we can explain why something follows from the premises (by appealing to the rules of inference). We get a proof Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

11 Deductive consequence Deductive consequence Formula G is a deduction or deductive consequence of formulae F 1, F 2,..., F n if and only if G can be derived from the premises F 1, F 2,..., F n using rules of inference. We write F 1, F 2,..., F n G and read F 1,..., F n derives or deductively entails G. Theorem Formula G is a theorem if and only if G holds, i.e., formula G can be derived from an empty set of premises. Proving F G is equivalent to proving F G This is why we talk of theorem proving instead of deriving deductions Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

12 Proving theorems In AI, we are interested in doing theorem proving automatically This is what automated reasoning and automated theorem proving (ATP) are about Proof theory has provided us with many proof methods A tool that implements a proof method is called a theorem prover Theorem provers differ from model checkers in that they only perform syntactic symbol manipulations (they are not concerned about models and semantics) But of course, what theorem provers derive must be semantically correct, i.e., justifiable in semantic terms Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

13 Rules of inference An example of an inference rule: If two propositions are true, then their conjunction is true as well Conjunction introduction rule A B A B or A, B A B What about the following rule? A B A Intuitively, the first rule is semantically correct, whereas the second rule isn t For a rule to be correct, its deductive consequence must also be the semantic consequence of the premises Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

14 Soundness and completeness Soundness An inference rule is sound if, when applied to a set of premises, derives a formula that is a logical consequence of these premises. Formally, a rule of inference r is sound if and only if Completeness if F 1,..., F n r G then F 1,..., F n G A set of rules R is complete if and only if it can be used to derive all logical consequences: if F 1,..., F n G then F 1,..., F n R G Soundness and completeness tie together semantics and proof theory (they provide a two-way link between and ) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

15 Soundness and completeness an example Let s prove that F G, F G (modus ponens) is a sound rule. We must prove F G, F G. Direct proof: F G F G (F G) F ( (F G) F ) G Let s prove that F G, G F (abduction) is not sound. We must prove F G, G F. Direct proof: F G F G (F G) G ( (F G) G ) F Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

16 Proof methods Theorem provers utilize a particular proof method That method must be semantically sound, and preferably also complete Many methods have been developed: natural deduction systems axiomatic (Hilbert-style) systems sequent calculi tableau method resolution method We will focus on the resolution method. This method is sound and complete, and can easily be implemented on a computer. Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

17 Outline 1 Three inference tasks 2 Proof theory 3 Resolution in PL 4 Resolution in FOL Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

18 Resolution method Resolution method is used in propositional logic and first-order predicate logic First proposed by J. Robinson in 1965 The method consists of a single inference rule: Resolution rule A F A G F G which is equivalent to: or F A, A G F G A F, A G F G The advantage here is that we re working with a single rule, which greatly simplifies automated reasoning Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

19 Resolution rule soundness Let s convince ourselves that the resolution rule is sound We must prove that the deductive consequence derived with the resolution rule also is a semantic consequence. I.e., we must prove: A F, A G F G For example, using the direct method: P {}}{ A F A Q {}}{ A G P Q R {}}{ F G (P Q) R A F G Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

20 Clause Resolution rule can only be applied to premises that are disjunctions If we wish to be able to apply the resolution rule many times over, all premises must be in a form of disjunctions of literals. This form is called a clause Clause A literal is an atom or its negation. A clause is a disjunction of finitely many literals G i : G 1 G 2 G n, n 0 A clause containing a single literal is called a unit clause. Examples of literals: A, F, A, F, G, G Examples of clauses: A F, A G, A B C D, F Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

21 Resolution on clauses Resolution rule on clauses F 1 F i F n G 1 G i G m F 1 F i 1 F i+1 F n G 1 G i 1 G i+1 G n where F i and G i are complementary literals (one is the negation of the other). The premises are called the input clauses and the deduction is called the resolvent. Examples: A B C, D B E A C D E A B, A B A B, A B A A A B, A B B B A B, A B A A Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

22 Conjunctive normal form If the premises are clauses, the set of premises is a conjunction of clauses (the premises are implicitly conjoined with ) Q: Does this restrict the application of resolution? A: No! Every formula can be represented as a conjunction of clauses by conversion into the conjunctive normal form Conjunctive normal form (CNF) Formula F is in a conjunctive normal form iff F is in the form where F i is in the form F 1 F 2 F n G i1 G i2 G im where G ij are literals (atoms or their negations). Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

23 Conversion into CNF Every formula can be converted into CNF in four sequential steps: Conversion into CNF (1) Equivalence elimination: F G ( F G) ( G F ) (2) Implication elimination: F G F G (3) Moving negations onto atoms: (F G) F G (F G) F G (4) Applying distributive law: F (G H) (F G) (F H) Each step is applied repeatedly until the result no longer changes. Involution F F is applied in each step whenever possible. Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

24 Conversion into CNF an example (1) Eliminate equivalences: (2) Eliminate implications: (3) Move negations in: (C D) ( A B) (C D) ( A B) (C D) ( ( A B) ( B A) ) (C D) ( (A B) ( B A) ) (C D) ( (A B) ( B A) ) (4) Apply distributive laws: ( C D) ( (A B) ( B A) ) ( ( C D) (A B) ) ( ( C D) ( B A) ) ( ( C A B) ( D A B) ) ( ( C D) ( B A) ) ( ( C A B) ( D A B) ) ( ( C B A) ( D B A) ) ( C A B) ( D A B) ( C B A) ( D B A) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

25 Clausal form A CNF formula can be represented as a set of clauses implicitly conjoined with Each clause can be represented as set of literals implicitly conjoined with Thus, a formula can be represented as a set of sets of literals This is called the clausal form For example: ( C A B) ( D A B) ( C B) { { C, A, B}, { D, A, B}, { C, B} } We also often write the clauses one below the other: C A B D A B C B Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

26 Resolution The inference step is repeated until: (1) the goal formula is derived (2) no further formulae can be derived (3) computational resources are exhausted Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

27 Resolution an example Let us use resolution to prove: A B, B C, A C Premises in clausal form: (1) A B (2) B C (3) A Derivation: (4) A C (from 1 & 2) (5) C (from 3 & 4) Alternative derivation: (4 ) B (from 1 & 3) (5 ) C (from 2 & 4 ) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

28 Incompleteness of resolution We ve proved that that the resolution rule is sound. But is it complete? We can easily show that the resolution rule is not complete E.g., let us consider the deduction F F G We cannot derive this using the resolution rule (why not?) But F F G does hold (check this!) Since F F G holds, but we cannot derive F F G using the resolution rule, it follows that the resolution rule cannot derive all logical consequences, and hence the resolution rule is not complete Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

29 Direct vs. refutation resolution The resolution procedure we ve been using thus far is the direct resolution, which tries to derive G from F 1,..., F n There also exists refutation resolution, which instead tries to prove that F 1 F n G is inconsistent Direct resolution is incomplete, but refutation resolution is complete Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

30 Refutation resolution (1) Instead of trying to prove F 1,..., F n F 1 F n G is inconsistent As a special case of the resolution rule we have: A A NIL G, we try to prove that NIL denotes the empty clause whose semantic value is If the resolution procedure derives NIL, then this means that the premises are inconsistent (because the resolution rule is sound) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

31 Refutation resolution (2) It has been proven that, whenever the set of clauses is inconsistent, resolution will derive the NIL clause (ground resolution theorem) Thus we can always prove the inconsistency of a set of clauses This also means that we can prove all logical consequences. Q: Why? A: Because we can prove F G using the refutation method, by proving that F G is inconsistent This then means that refutation resolution is complete, because we can use it prove all logical consequences Therefore, refutation resolution is both sound and complete! Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

32 Refutation resolution example 1 Let us show that we can use refutation resolution to prove F F G: Negation of the goal formula: (F G) F G Set of clauses: (1) F (2) F (3) G From (1) and (2) we derive NIL This proves that the set of premises is inconsistent, i.e., that F G is the deductive/logical consequence of F Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

33 Refutation resolution example 2 A diplomatic problem As a Protocol Representative, you re in charge of sending out invitations for the Annual Diplomats Ball. You are to take into account the following: (1) The ambassador wants you to also invite UK, if you invite Turkey. (2) The vice-ambassador wants you to invite Turkey or Argentina. (3) Due to a recent diplomatic incident, you cannot invite both UK and Argentina. Whom to invite? Let s prove: If I invite Turkey, I will not invite Argentina Logical representation of the problem: (T U) (T A) (U A) We need to prove T A (the goal) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

34 Refutation resolution example 2 Conversion into clauses: (1) T U (2) T A (3) U A Negation of the goal: (T A) ( T A) T A Adding new clauses: (4) T (5) A Resolution procedure: (6) U (from 1 & 4) (7) A (from 3 & 6) (8) NIL (from 5 & 7) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

35 Factorization Refutation resolution is complete proviso the clauses are factorized Factorization uses equivalence G G G to eliminate multiple occurrences of a literal in a formula For example: A A, A A A A The set of clauses is inconsistent, but we ve derived a valid formula Q: Is this sound? A: Of course it is. A valid formula is a logical consequence of any formula. But this is not very useful. However, if we first factorize, we get: A, A NIL To retain completeness, we must apply factorization whenever possible Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

36 Refutation resolution algorithm Refutation resolution algorithm (for propositional logic) function plresolution(f, G) clauses cnfconvert(f G) new loop do for each (c 1, c 2 ) in selectclauses(clauses) do resolvents plresolve(c 1, c 2 ) if NIL resolvents then return true new new resolvents if new clauses then return false clauses clauses new cnfconvert converts a formula into a CNF selectclauses selects a set of clause pairs to resolve plresolve resolves two clauses and returns a set of resolvents Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

37 Refutation resolution algorithm remarks Applying the resolution rule on a pair of clauses may result in more than one resolvent, hence the algorithm uses a set of resolvents To retain completeness, factorization should be applied on each derived resolvent The number of possible distinct clauses is finite (when factorized), thus the algorithm certainly termines in a finite number of steps No pair of clauses should be resolved more than once Deriving a NIL clause from a set of clauses is a search problem: in each step we must choose which pair of clauses to resolve So we need a search strategy, which in the context of resolution is called a resolution strategy Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

38 Resolution strategies There are two types of resolution strategies: simplification strategies control strategies Simplification strategies are used to delete redundant or irrelevant clauses generated during the proof procedure, avoiding unnecessary steps later on Control strategies determine the order in which clauses are resolved The strategy needs to be complete: it must derive NIL when given a set of incomplete clauses This is not to be confused with completeness of inference rules (in general, for a proof procedure to be complete, it must combine complete inference rules with a complete search strategy) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

39 Simplification strategy Deletion strategy Removal of redundant clauses: A clause that is subsumed by another clause may be deleted Based on the absorption equivalence: F (F G) F If the set contains a pair of clauses C 1 and C 2 such that C 1 C 2, then clause C 2 may be removed from the set (clauses are represented as sets of literals) Removal of irrelevant clauses: Every valid clause (tautology) is irrelevant (why?) If the resolvent is a valid clause, it may be removed immediately Checking the validity of a clause is simple: a clause is valid iff it contains a complementary pair of literals F i and F i Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

40 Control resolution strategies Level saturation strategy Resolvents are derived level-by-level (as in breadth-first search): we resolve all pairs of first-level clauses (initial set of clauses), then we resolve second-level clauses, etc. At i-th level, the input clauses come from level 1 down to level (i 1) This strategy is complete, but very inefficient (the problem of combinatorial explosion) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

41 Control resolution strategies Set-of-support strategy (SoS) Builds on the asssumption that the set of input premises is consistent For if it were not consistent, it would logically entail any formula! Thus, to prove inconsistency in refutation resolution, we have to combine the input premises clauses with clauses from the negated goal, or with the newly derived clauses Set of support (SoS): the clauses obtained from the negated goal as well as all susequently derived clauses Set-of-support strategy: at least one parent clause always comes from the SoS SoS increases as we derive more clauses SoS strategy is complete and in principle more efficient than level saturation strategy (especially if the SoS is small) Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

42 Outline 1 Three inference tasks 2 Proof theory 3 Resolution in PL 4 Resolution in FOL Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

43 Jump to: AI-6-2-AutomatedReasoningInFOL.pdf Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44

44 Wrap-up The three inference tasks are (1) model checking, (2) satisfiability checking, and (3) validity checking The latter two are decidable but intractable in PL, whereas they are undecidable in FOL Proof theory uses rules of inference to derive deductive consequences, without explicit reference to semantics of logic Rules of inference must be sound and preferably complete Refutation resolution (with factorization and a complete strategy) is a sound and complete proof method in both PL and FOL Formulas have to be converted in clausal form before using resolution In FOL, we use unification to match two complementary literals Next topic: Logic programming Dalbelo Bašić, Šnajder (UNIGZ FER) AI Logic inference AY 2016/ / 44