logic is everywhere Logik ist überall Hikmat har Jaga Hai Mantık her yerde la logica è dappertutto lógica está em toda parte

Transcription

1 Steffen Hölldobler logika je všude logic is everywhere la lógica está por todas partes Logika ada di mana-mana lógica está em toda parte la logique est partout Hikmat har Jaga Hai Human Reasoning, Logic Programs and the CORE Method International Center for Computational Logic Technische Universität Dresden Germany Human Reasoning Three-Valued Logic Programs A CORE Method for Human Reasoning Contractions Positive vs Negative Information Affirmation of the Consequent Open Problems logika je svuda Mantık her yerde la logica è dappertutto Logik ist überall Logica este peste tot Human Reasoning, Logic Programs and the CORE Method 1

2 Human Reasoning Two Examples Instructions on boarding card distributed at Amsterdam Schiphol Airport: If it s thirty minutes before your flight departure, make your way to the gate. As soon as the gate number is confirmed, make your way to the gate. Notice in London Underground: If there is an emergency then you press the alarm signal bottom. The driver will stop if any part of the train is in a station. Observations Intended meaning differs from literal meaning. Rigid adherence to classical logic is no help in modeling the examples. There seems to be a reasoning process towards more plausible meanings. The driver will stop the train in a station if the driver is alerted to an emergency and any part of the train is in the station. (Kowalski: Computational Logic and Human Life: How to be Artificially Intelligent. To appear 011) Human Reasoning, Logic Programs and the CORE Method

3 Human Reasoning Forward Reasoning Byrne: Suppressing Valid Inferences with Conditionals. Cognition 31, 61-83: 1989 If she has an essay to write then she will study late in the library. She has an essay to write. Modus Ponens. Byrne % of subjects conlude that she will study late in the library. If she has an essay to write then she will study late in the library. She has an essay to write. If she has a textbook to read she will study late in the library. Alternative Arguments. Byrne % of subjects conlude that she will study late in the library. If she has an essay to write then she will study late in the library. She has an essay to write. If the library stays open she will study late in the library Byrne % of subjects conlude that she will study late in the library. Additional arguments lead to suppression of earlier conclusions. Human Reasoning, Logic Programs and the CORE Method 3

4 Reasoning Towards an Appropriate Logical Form Context independent rules If she has an essay to write and the library is open then she will study late in the library. If the library is open and she has a reason for studying in the library then she will study late in the library. Context dependent rule plus exception If she has an essay to write then she will study late in the library. However, if the library is not open, then she will not study late in the library. The last sentence is the contrapositive of the converse of the original sentence! Human Reasoning, Logic Programs and the CORE Method 4

5 Human Reasoning Denial of Antecedent If she has an essay to write then she will study late in the library. She does not have an essay to write. Byrne % of subjects conlude that she will not study late in the library. If she has an essay to write then she will study late in the library. She does not have an essay to write. If she has a textbook to read she will study late in the library. Byrne % of subjects conlude that she will not study late in the library. If she has an essay to write then she will study late in the library. She does not have an essay to write. If the library stays open she will study late in the library. Byrne % of subjects conlude that she will not study late in the library. Human Reasoning, Logic Programs and the CORE Method 5

6 Human Reasoning The Search for Models Goal find a logic which adequately models human reasoning. How about classical two-valued propositional logic? Let s consider a direct encoding: {l e, e} {l e, e, l t} {l e, e, l o} {l e, e} {l e, e, l t} {l e, e, l o} Human Reasoning, Logic Programs and the CORE Method 6

7 Two-Valued Interpretations Let L be a language of propositional logic. A (two-valued) interpretation is a mapping L {, } represented by I, where I is a set containing all atoms which are mapped to. All atoms which do not occur in I are mapped to. Let I denote the set of all interpretations. (I, ) is a lattice. An interpretation I is a model for a program P, in symbols I = P, iff I(P) =. = {l e, e} {e} = {l e, e} {l} = {l e, e} {e, l} = {l e, e} {e} {e, l} {l} Human Reasoning, Logic Programs and the CORE Method 7

8 Two-Valued Interpretations Let L be a language of propositional logic. A (two-valued) interpretation is a mapping L {, } represented by I, where I is a set containing all atoms which are mapped to. All atoms which do not occur in I are mapped to. Let I denote the set of all interpretations. (I, ) is a lattice. An interpretation I is a model for a program P, in symbols I = P, iff I(P) =. = {l e, e} {l} = {l e, e} {e} = {l e, e} {e, l} = {l e, e} {e} {e, l} {l} Human Reasoning, Logic Programs and the CORE Method 8

9 Logical Consequence (1) A formula G is a logical consequence of a set of formulas F, in symbols F = G, iff all models for F are also models for G. {l e, e} = l {l e, e} = e {e, l} {e} {l} Human Reasoning, Logic Programs and the CORE Method 9

10 Logical Consequence () A formula G is a logical consequence of a set of formulas F, in symbols F = G, iff all models for F are also models for G. {l e, e} = e {l e, e} = l {l e, e} = l {e, l} {e} {l} Human Reasoning, Logic Programs and the CORE Method 10

11 Human Reasoning A Classical Logic Approach Recall the examples: {l e, e} = l modus ponens {l e, e, l t} = l classical logic is monoton {l e, e, l o} = l upps, humans don t do this {l e, e} = l denial of antecendent {l e, e, l t} = l {l e, e, l o} = l Conclusion classical logic is inadequate. Often mistakenly generalized to logic is inadequate. Human Reasoning, Logic Programs and the CORE Method 11

12 Human Reasoning A Computational Logic Approach Goal find a logic which adequately models human reasoning. Solution I propose the following: Logic programs under (weak) completion semantics Non-monotonicity Reasoning towards an appropriate logical form Logic programs Three-valued Lukasiewicz logic Least models An appropriate semantic operator Least fixed points are least models Least fixed points can be computed by iterating the operator Reasoning with respect to the least models A connectionist realization Human Reasoning, Logic Programs and the CORE Method 1

13 Logic Programs A (logic) program is a finite set of clauses. A (program) clause is an expression of the form A B 1 B n, where n 1, A is an atom, and each B i, 1 i n, is either a literal, or. A is called head and B 1 B n body of the clause. A clause of the form A is called a positive fact. A clause of the form A is called a negative fact. {l e, e } {l e, e, l t} {l e, e } Here I consider only propositional programs, but the approach extends to first-order programs. The language L underlying a program P shall contain precisely the relation symbols occurring in P, and no others. Human Reasoning, Logic Programs and the CORE Method 13

14 Program Completion Let P be a program. Consider the following transformation: 1 All clauses with the same head A Body 1, A Body,... are replaced by A Body 1 Body.... If an atom A is not the head of any clause in P then add A. 3 All occurrences of are replaced by. The resulting set is called completion of P or c P. If step is omitted then the resulting set is called weak completion of P or wc P. Human Reasoning, Logic Programs and the CORE Method 14

15 Program Completion Example 1 P 1 = {l e, e } c P 1 = {l e, e } wc P 1 = {l e, e } The models of c P 1 and wc P 1 {e, l} {e} {l} Hence, c P 1 = l and wc P 1 = l Human Reasoning, Logic Programs and the CORE Method 15

16 Program Completion Example P = {l e, e } c (P ) = {l e, e } wc (P ) = {l e, e } The models of c P and wc P {e, l} {e} {l} Hence, c P = l and wc P = l Remember, P = l Human Reasoning, Logic Programs and the CORE Method 16

17 Program Completion Example 3 P 3 = {l e, e, l t} c P 3 = {l e t, e, t } wc P 3 = {l e t, e } The models of c P 3 {e, l} The models of wc P 3 {e, l} {e, l, t} Hence, c P 3 = t whereas wc P 3 = t and wc P 3 = t. Human Reasoning, Logic Programs and the CORE Method 17

18 Monotonicity Let F and F be sets of formulas and G a formula. A logic is monotonic if the following holds: If F = G then F F = G. Classical logic is monotonic. A logic based on completion semantics is non-monotonic. Consider Then P 3 = {l e, e, l t} P 3 = P {t } c P 3 = t c P 3 = t Human Reasoning, Logic Programs and the CORE Method 18

19 Reasoning Towards an Appropriate Logical Form Stenning, van Lambalgen: Human Reasoning and Cognititve Science. MIT Press: 008 Represent conditionals as licences for conditionals. If she has an essay to write then she will study late in the library. She has an essay to write. P 4 = {l e ab, ab, e } If she has an essay to write then she will study late in the library. She has an essay to write. If she has a textbook to read she will study late in the library. P 5 = {l e ab 1, ab 1, l t ab, ab, e } Reason about additional premises. If she has an essay to write then she will study late in the library. She has an essay to write. If the library stays open she will study late in the library. P 6 = {l e ab 1, ab 1 o, l o ab, ab e, e } Human Reasoning, Logic Programs and the CORE Method 19

20 Reasoning Towards an Appropriate Logical Form Denial of Antecedent (DA) If she has an essay to write then she will study late in the library. She does not have an essay to write. P 7 = {l e ab, ab, e } If she has an essay to write then she will study late in the library. She does not have an essay to write. If she has a textbook to read she will study late in the library. P 8 = {l e ab 1, ab 1, l t ab, ab, e } If she has an essay to write then she will study late in the library. She does not have an essay to write. If the library stays open she will study late in the library. P 9 = {l e ab 1, ab 1 o, l o ab, ab e, e } Human Reasoning, Logic Programs and the CORE Method 0

21 Three-Valued Logics L L K K C U U U U U U U U U U U U U U U U U U U U U U U U U U U U U Lukasiewicz (L) semantics 190 L L Kleene (K) semantics 195 K K Fitting (F) semantics 1985 K C (F G) { 3L 3F } (F G) (G F ) Human Reasoning, Logic Programs and the CORE Method 1

22 Principles of Lukasiewicz Lukasiewicz used 1,, and 0 instead of,u, and, respectively. Principles of identity and non-identity ( ) ( ) (U U). ( ) ( ). ( U) (U ) ( U) (U ) U. Principles of implication ( ) ( ) (U U). ( ) ( U) (U ). ( ). ( U) (U ) U. Definitions of negation, disjunction, and conjunction F ( F ) (F G) (G (G F )). (F B) ( F G). Human Reasoning, Logic Programs and the CORE Method

23 Some Common Laws and Literature Laws L K F Equivalence F G (F G) (G F ) yes yes no Implication F G F G no yes yes Syllogism (F G) (G H) F H no yes yes Excluded Middle F F no no no Contradiction F F no no no Literature Fitting: A Kripke-Kleene Semantics for Logic Programs. Journal of Logic Programming, 95-31: Kleene: Introduction to Metamathematics. North-Holland: 195. Lukasiewicz: O logice trójwartościowej. Ruch Filozoficzny 5, : 190. English translation: On Three-Valued Logic. In: Jan Lukasiewicz Selected Works. (L. Borkowski, ed.), North Holland, 87-88, Human Reasoning, Logic Programs and the CORE Method 3

24 Three-Valued Interpretations A (three-valued) interpretation is a mapping L {,, U} represented by I, I, where I contains all atoms which are mapped to, I contains all atoms which are mapped to, I I =. All atoms which occur neither in I nor I are mapped to U. Human Reasoning, Logic Programs and the CORE Method 4

25 Interpretations and Models Let I denote the set of all interpretations. Fitting 1985 (I, ) is a complete semi-lattice. {p, q}, {p}, {q} {q}, {p}, {p, q} {p}, {q},, {q}, {p}, An interpretation I is a model for a program P, in symbols I = 3 P, iff I(P) =., { =3L = 3F } {p q} Human Reasoning, Logic Programs and the CORE Method 5

26 Logic Programs under Three-Valued L-Semantics We consider Lukasiewicz semantics. Let P be a logic program and I = I, I be an interpretation. Proposition 1 If I = I, I = 3L P then I = I, = 3L P. Proof Suppose I = I, I = 3L P Let A Body P. To show I = 3L A Body. We distinguish three cases A I In this case, I = 3L A Body. A I A I I Human Reasoning, Logic Programs and the CORE Method 6

27 Proof of Proposition 1 Case A I In this case I(A) = and I (A) = U. Because I = 3L A Body we conclude I(Body) =. Hence, we find L Body such that I(L) =. L = B In this case I(B) = and, hence, I (B) = I (L) = U. L = B In this case I(B) = and, hence, I (B) = and I (L) =. Consequently, I (Body) {U, }. Because I (A) = U we conclude I = 3L A Body. Human Reasoning, Logic Programs and the CORE Method 7

28 Proof of Proposition 1 Case 3 A I I In this case I(A) = I (A) = U. I(Body) = As in the previous case we find I (Body) {, U} Consequently, I = 3L A Body. I(Body) = U In this case we find L Body with I(L) = U. Then, I (L) = U. Consequently, I (Body) = U. Hence, I = 3L A Body. qed Example Consider P = {p q r}. {p, q}, {r} is a model for P, and so is {p, q},. {p, r}, {q} is a model for P, and so is {p, r},. {r}, {q} is a model for P, and so is {r},. Human Reasoning, Logic Programs and the CORE Method 8

29 Intersection of Two Models We consider Lukasiewicz semantics; let P be a logic program. Proposition Let I 1 = I 1, and I = I, be two models of P. Then, I 3 = I 1 I, is also a model for P. Proof Suppose I 3 = 3L P. Then, we find A Body P such that I 3 (A Body). We distinguish the following cases: I 3 (A) = and I 3 (Body) = Impossible, because I 3 =. I 3 (A) = and I 3 (Body) = U Impossible, because I 3 =. I 3 (A) = U and I 3 (Body) = We find j {1, } with I j (A) = U. Because I j = 3L A Body we find I j (Body) {U, }. Because I 3 (Body) = and I 3 = we find for all L Body that L is an atom and L I 3. Hence, for all L Body we find L I j, j {1, }. Consequently, I j (Body) =, j {1, } contradicting ( ) ( ) qed Human Reasoning, Logic Programs and the CORE Method 9

30 Intersection of Two Models Example Proposition does not hold for arbitrary models. Consider P = {p q 1 r 1, p q q }. Let I 1 =, {p, q 1, r } and I =, {p, q, r 1 }. Both, I 1 and I, are models for P. However, I 1 I =, {p} is not a models for P. Human Reasoning, Logic Programs and the CORE Method 30

31 Model Intersection We consider Lukasiewicz semantics; let P be a logic program. Theorem 3 The model intersection property holds for P, i.e., {I I = 3L P} = 3L P. Proof Follows immediately from Propositions 1 and Let lm 3L P denote the least model of P under Lukasiewicz semantics. qed lm 3L {p q} =, Observation Theorem 3 does not hold under Fitting semantics: {p, q}, = 3F {p q}, {p, q} = 3F {p q} but, = 3F {p q} Human Reasoning, Logic Programs and the CORE Method 31

32 Weakly Completed Logic Programs under L-Semantics We consider Lukasiewicz semantics. Let P be a logic program. Theorem 4 The model intersection property holds for wc P as well. Theorem 5 If I = 3L wc P then I = 3L P. Observation Theorem 5 does not hold under F-semantics., = 3F wc {p q} = {p q}, but, = 3F {p q} Human Reasoning, Logic Programs and the CORE Method 3

33 Reasoning with Respect to the Least Model of wc P Recall our examples wc P 4 = {l e ab, ab, e } lm 3L wc P 4 = {e, l}, {ab} lm 3L wc P 4 (l) = wc P 5 = {l (e ab 1 ) (t ab ), ab 1, ab, e } lm 3L wc P 5 = {e, l}, {ab 1, ab } lm 3L wc P 5 (l) = wc P 6 = {l (e ab 1 ) (o ab ), ab 1 o, ab e, e } lm 3L wc P 6 = {e}, {ab } lm 3L wc P 6 (l) = U wc P 7 = {l e ab, ab, e } lm 3L wc P 7 =, {e, l, ab} lm 3L wc P 7 (l) = wc P 8 = {l (e ab 1 ) (t ab ), ab 1, ab, e } lm 3L wc P 8 =, {e, ab 1, ab } lm 3L wc P 8 (l) = U wc P 9 = {l (e ab 1 ) (o ab ), ab 1 o, ab e, e } lm 3L wc P 9 = {ab }, {e, l} lm 3L wc P 9 (l) = This logic appears to be adequate! Human Reasoning, Logic Programs and the CORE Method 33

34 Completion versus Weak Completion Recall P 8 wc P 8 = {l (e ab 1 ) (t ab ), ab 1, ab, e } lm 3L wc P 8 =, {e, ab 1, ab } lm 3L wc P 8 (l) = U c P 8 = {l (e ab 1 ) (t ab ), ab 1, ab, e, t } lm 3L c P 8 =, {e, t, l, ab 1, ab } lm 3L c P 8 (l) = A logic using completion instead of weak completion is not adequate. Human Reasoning, Logic Programs and the CORE Method 34

35 Computing the Least Models of Weakly Completed Programs How can we compute the least models of weakly completed programs? A first candidate: Fitting s immediate consequence operator Φ F,P (I) = J, J, where J = {A there exists A Body P with I(Body) = } and J = {A for all A Body P we find I(Body) = }. Uses F-semantics. Some well-known results mostly due to Fitting 1985: (1) Φ F,P is monotone on (I, ). () Φ F,P is continuous and, hence, admits a least fixed point denoted by lfp Φ F,P. (3) lfp Φ F,P can be computed by iterating Φ F,P on,. (3) The least F-model of c P is the least fixed point of Φ F,P. Inadequate for human reasoning. Human Reasoning, Logic Programs and the CORE Method 35

36 The Stenning and van Lambalgen Operator Stenning and van Lambalgen s operator Φ SvL,P (I) = J, J, where J = {A there exists A Body P with I(Body) = } and J = {A there exists A Body P and for all A Body P we find I(Body) = }. Theorem 4 (1) Φ SvL,P is monotone on (I, ). () Φ SvL,P is continuous and, hence, admits a least fixed point denoted by lfp Φ SvL,P. (3) lfp Φ SvL,P can be computed by iterating Φ SvL,P on,. (3) lm 3L wc P = lfp Φ SvL,P. Human Reasoning, Logic Programs and the CORE Method 36

37 Computing Least Fixed Points Recall some of our examples P 4 = {l e ab, ab, e } wc P 4 = {l e ab, ab, e } Φ SvL,P4 (, ) = {e}, {ab} Φ SvL,P4 ( {e}, {ab} ) = {e, l}, {ab} = lfp Φ SvL,P4 = lm 3L wc P 4 P 9 = {l e ab 1, ab 1 o, l o ab, ab e, e } wc P 9 = {l (e ab 1 ) (o ab ), ab 1 o, ab e, e } Φ SvL,P9 (, ) =, {e} Φ SvL,P9 (, {e} ) = {ab }, {e} Φ SvL,P9 ( {ab }, {e} ) = {ab }, {e, l} = lfp Φ SvL,P9 = lm 3L wc P 9 Human Reasoning, Logic Programs and the CORE Method 37

38 Cores for Three-Valued Logic Programs Consider {p q}. A translation algorithm translates programs into a core. Recurrent connections connect the output to the input layer. Kalinke 1995, Seda, Lane 004 Φ F new Φ SvL p p p p p p p p q q q q q - q q q - - Human Reasoning, Logic Programs and the CORE Method 38

39 A CORE Method for Human Reasoning Consider three-layer feed forward networks of binary threshold units. The input as well as the output layer shall represent interpretations. Theorem 5 For each program P there exists a feed-forward core computing Φ SvL,P. Add recurrent connections between corresponding units in the output and the input layer. Corollary 6 The recurrent network reaches a stable state representing lfp Φ SvL,P if initialized with,. Human Reasoning, Logic Programs and the CORE Method 39

40 Human Reasoning Modus Ponens If she has an essay to write, she will study late in the library. She has an essay to write. Byrne % of subjects conclude that she will study late in the library. Stenning, van Lambalgen 008 P 4 = {l e ab, e, ab }. e e l l ab ab e e l l ab ab lfp Φ SvL,P4 = lm 3L wc P 4 = {l, e}, {ab}. From {l, e}, {ab} follows that she will study late in the library. Human Reasoning, Logic Programs and the CORE Method 40

41 Human Reasoning Alternative Arguments If she has an essay to write, she will study late in the library. She has an essay to write. If she has some textbooks to read, she will study late in the library. Byrne % of subjects conclude that she will study late in the library. Stenning, van Lambalgen 008 P 5 = {l e ab 1, e, ab 1, l t ab, ab }. e e l 3 l ab 1 ab 1 t t ab ab e e l l ab 1 ab 1 t t - ab ab lfp Φ SvL,P5 = lm 3L wc P 5 = {e, l}, {ab 1, ab }. From {e, l}, {ab 1, ab } follows that she will study late in the library. Human Reasoning, Logic Programs and the CORE Method 41

42 Human Reasoning Additional Argument If she has an essay to write, she will study late in the library. She has an essay to write. If the library stays open, she will study late in the library. Byrne % of subjects conclude that she will study late in the library. Stenning, van Lambalgen 008 P 6 = {l e ab 1, e, l o ab, ab 1 o, ab e, }. e e l 3 l ab 1 ab 1 o o ab ab e e l l ab 1 ab 1 o o - ab ab lfp Φ SvL,P6 = lm 3L wc P 6 = {e}, {ab }. From {e}, {ab } follows that it is unknown whether she will study late in the library. Human Reasoning, Logic Programs and the CORE Method 4

43 Human Reasoning Denial of Antecendent (DA) If she has an essay to write, she will study late in the library. She does not have an essay to write. Byrne % of subjects conclude that she will not study late in the library. Stenning, van Lambalgen 008 P 7 = {l e ab, e, ab }. e e l l ab ab e e l l ab ab lfp Φ SvL,P7 = lm 3L wc P 7 =, {ab, e, l}. From, {ab, e, l} follows that she will not study late in the library. Human Reasoning, Logic Programs and the CORE Method 43

44 Human Reasoning Alternative Argument and DA If she has an essay to write, she will study late in the library. She does not have an essay to write. If she has textbooks to read, she will study late in the library. Byrne % of subjects conclude that Marian will not study late in the library. Stenning, van Lambalgen 008 P 8 = {l e ab 1, e, ab 1, l t ab, ab }. e e l 3 l ab 1 ab 1 t t ab ab e e l l ab 1 ab 1 t t - ab ab lfp Φ SvL,P8 = lm 3L wc P 8 =, {ab 1, ab, e}. From, {ab 1, ab, e} follows that it is unknown whether she will study late in the library. Human Reasoning, Logic Programs and the CORE Method 44

45 Human Reasoning Additional Argument and DA If she has an essay to write, she will study late in the library. She does not have an essay to write. If the library is open, she will study late in the library. Byrne % of subjects conclude that she will not study late in the library. Stenning, van Lambalgen 008 P 9 = {l e ab 1, e, l o ab, ab 1 o, ab e}. e e l 3 l ab 1 ab 1 o o ab ab e e l l ab 1 ab 1 o o - ab ab lfp Φ SvL,P9 = lm 3L wc P 9 = {ab }, {e, l}. From {ab }, {e, l} follows that she will not study late in the library. Human Reasoning, Logic Programs and the CORE Method 45

46 Summary Under Lukasiewicz semantics we obtain Byrne 1989 Program lm 3L wc P i (l) Modus Ponens l (96%) P 4 Alternative Arguments l (96%) P 5 Additional Arguments l (38%) P 6 U Modus Ponens and DA l (46%) P 7 Alternative Arguments and DA l (4%) P 8 U Additional Arguments and DA l (63%) P 9 The approach appears to be adequate. Fitting semantics or completion is inadequate. Human Reasoning, Logic Programs and the CORE Method 46

47 Contraction Mappings Do we have to initialize the networks with the empty interpretation? Banach s Contraction Mapping Theorem A contraction mapping f on a complete metric space has a unique fixed point; the sequence x, f(x), f(f(x)),... converges to this fixed point, where x is an arbitrary element from the metric space. Let P be a program. A level mapping is a mapping l from the set of atoms to the set of natural numbers. It is extended to negative atoms by defining l( A) = l(a) for each atom A. Let I be the set of all interpretations and I, J I. d l (I, J) = 1 n if I J and I(A) = J(A) U for all A with l(a) < n and I(A) J(A) or I(A) = J(A) = U for some A with l(a) = n, 0 otherwise. Proposition 7 (Kencana Ramli 009) (I, d l ) is a complete metric space. Human Reasoning, Logic Programs and the CORE Method 47

48 Contraction Properties of the Semantic Operators Observation If P is an acceptable program then Φ SvL,P is not necessarily a contraction. Consider P = {r q p, r p q}. P is acceptable. Both,, and, {p, q}, are fixed points of Φ SvL,P. By Banach s contraction mapping theorem Φ SvL,P is not a contraction. Theorem 8 If P is an acyclic program then Φ SvL,P is a contraction. If P is acyclic then we find a level mapping l such that for each L 1... L n A P we have l(l i ) < l(a). d l (Φ SvL,P (I), Φ SvL,P (J)) 1 d l(i, J). Corollary 9 If P is an acyclic program then Φ SvL,P has a unique fixed point which can be reached by iterating Φ SvL,P starting from some interpretation. Human Reasoning, Logic Programs and the CORE Method 48

49 Positive versus Negative Information Consider P = {p, p }. lm 3L wc P = {p},. 1 p 1 p p 1 3 p 1 Positive information dominates negative information. However, we may take integrity constraints into account, i.e. expressions of the form L 1... L n. Human Reasoning, Logic Programs and the CORE Method 49

50 Long Term versus Short Term Memory Consider P = {p }. 1 1 ext 1 ext p 1 p p 1 p p 1 p p 1 p p 1 p p 1 p If unit p in the output layer is clamped, then the input and output layers do not represent interpretations any more. Adding a connection between the units p and p in the output layer with weight yields the desired effect. Human Reasoning, Logic Programs and the CORE Method 50

51 Human Reasoning Affirmation of the Consequent (AC) If she has an essay to write then she will study late in the library. She will study late in the library. Byrne % of subjects conlude that she has an essay to write. If she has an essay to write then she will study late in the library. She will study late in the library. If she has a textbook to read she will study late in the library. Byrne % of subjects conlude that she has an essay to write. If she has an essay to write then she will study late in the library. She will study late in the library. If the library stays open she will study late in the library. Byrne % of subjects conlude that she has an essay to write. Human Reasoning, Logic Programs and the CORE Method 51

52 Human Reasoning Modus Tollens If she has an essay to write then she will study late in the library. She will not study late in the library. Byrne % of subjects conlude that she does not have an essay to write. If she has an essay to write then she will study late in the library. She will not study late in the library. If she has a textbook to read she will study late in the library. Byrne % of subjects conlude that she does not have an essay to write. If she has an essay to write then she will study late in the library. She will not study late in the library. If the library stays open she will study late in the library. Byrne % of subjects conlude that she does not have an essay to write. Human Reasoning, Logic Programs and the CORE Method 5

53 Abductive Frameworks and Observations Remember Let K L be a set of formulas called the knowledge base, A L be a set of formulas called abducibles, and = L L a logical consequence relation. K, A, = is called abductive framework. An observation O is a subset of the language L. Here K is a logic program P (with negative facts). L is the language underlying P. R D P = {A R P A Body P} is the set of defined predicates in P. R U P = R P \ R D P is the set of undefined predicates in P. A is the set {A A R U P } {A A RU P }. = is = lm wc 3L lm wc, where P = 3L F iff lm 3L wc P(F ) =. Observations are usually sets containing a single literal, in which case we simply write O = L instead of O = {L}. Human Reasoning, Logic Programs and the CORE Method 53

54 Explanations Let L be a language. Let K, A, = be an abductive framework and O an observation. O is explained by E (or E is an explanation for O) iff E A, K E is satisfiable, K E = L for each L O. An explanation E for O is said to be a minimal iff there is no explanation E E for O. Human Reasoning, Logic Programs and the CORE Method 54

55 Reasoning Let P, A, = lm wc 3L P is a logic program and be an abductive framework, where A = {A A R U P } {A A RU P } the set of abducibles Let O be an observation and F a formula in the language underlying P. F follows sceptically by abduction from P and O (in symbols P, O = s A F ) lm wc iff O can be explained and for all minimal explanations E we find P E = 3L F. F follows credulously by abduction from P and O (in symbols P, O = c A F ) lm wc iff there exists a minimal explanation E such that P E = 3L F. Human Reasoning, Logic Programs and the CORE Method 55

56 Human Reasoning Modus Ponens and AC If she has an essay to write then she will study late in the library. She will study late in the library. Byrne % of subjects conlude that she has an essay to write. We obtain Thus P 10 = {l e ab, ab } A = {e, e } O = l lm 3L wc P 10 =, {ab} lm 3L wc (P 10 {e }) = {e, l}, {ab} lm 3L wc (P 10 {e }) =, {e, l, ab} Hence, {e } is the only minimal explanation and P 10, O = s A e. Human Reasoning, Logic Programs and the CORE Method 56

57 Human Reasoning Alternative Arguments and AC If she has an essay to write then she will study late in the library. She will study late in the library. If she has a textbook to read she will study late in the library. Byrne % of subjects conlude that she has an essay to write. We obtain P 11 = {l e ab 1, ab 1, l t ab, ab } A = {e, e, t, e } O = l Thus lm 3L wc P 11 =, {ab 1, ab } lm 3L wc (P 11 {e }) = {e, l}, {ab 1, ab } lm 3L wc (P 11 {e }) =, {e, ab 1, ab } lm 3L wc (P 11 {t }) = {t, l}, {ab 1, ab } lm 3L wc (P 11 {t }) =, {t, ab 1, ab } Hence, {e } and {t } are minimal explanations and P 11, O = s A e. Human Reasoning, Logic Programs and the CORE Method 57

58 Human Reasoning Additional Arguments and AC If she has an essay to write then she will study late in the library. She will study late in the library. If the library stays open she will study late in the library. Byrne % of subjects conlude that she has an essay to write. We obtain P 1 = {l e ab 1, ab 1 o, l o ab, ab e} A = {e, e, o, o } O = l Thus lm 3L wc P 1 =, lm 3L wc (P 1 {e }) = {e}, {ab } lm 3L wc (P 1 {e }) = {ab }, {e, l} lm 3L wc (P 1 {o }) = {t}, {ab 1 } lm 3L wc (P 1 {o }) = {ab 1 }, {o, l} lm 3L wc (P 1 {e, o }) = {t, e, l}, {ab 1, ab } {e, o } is the only minimal explanation and P 1, O = s A e. Human Reasoning, Logic Programs and the CORE Method 58

59 Human Reasoning Modus Tollens (MT) If she has an essay to write then she will study late in the library. She will not study late in the library. Byrne % of subjects conlude that she does not have an essay to write. We obtain Thus P 10 = {l e ab, ab } A = {e, e } O = l lm 3L wc P 10 =, {ab} lm 3L wc (P 10 {e }) = {e, l}, {ab} lm 3L wc (P 10 {e }) =, {e, l, ab} Hence, {e } is the only minimal explanation and P 10, O = s A e. Human Reasoning, Logic Programs and the CORE Method 59

60 Human Reasoning Alternative Argument and MT If she has an essay to write then she will study late in the library. She will not study late in the library. If she has a textbook to read she will study late in the library. Byrne % of subjects conlude that she does not have an essay to write. We obtain P 11 = {l e ab 1, ab 1, l t ab, ab } A = {e, e, t, e } O = l Thus lm 3L wc P 11 =, {ab 1, ab } lm 3L wc (P 11 {e }) = {e, l}, {ab 1, ab } lm 3L wc (P 11 {e }) =, {e, ab 1, ab } lm 3L wc (P 11 {t }) = {t, l}, {ab 1, ab } lm 3L wc (P 11 {t }) =, {t, ab 1, ab } lm 3L wc (P 11 {e, t }) =, {e, l, t, ab 1, ab } {e, t } is the only minimal explanation and P 11, O = s A e. Human Reasoning, Logic Programs and the CORE Method 60

61 Human Reasoning Additional Arguments and MT If she has an essay to write then she will study late in the library. She will not study late in the library. If the library stays open she will study late in the library. Byrne % of subjects conlude that she does not have an essay to write. We obtain P 1 = {l e ab 1, ab 1 o, l o ab, ab e} A = {e, e, o, o } O = l Thus lm 3L wc P 1 =, lm 3L wc (P 1 {e }) = {e}, {ab } lm 3L wc (P 1 {e }) = {ab }, {e, l} lm 3L wc (P 1 {o }) = {t}, {ab 1 } lm 3L wc (P 1 {o }) = {ab 1 }, {o, l} Hence, {e } and {o } are minimal explanations and P 1, O = s A e. Human Reasoning, Logic Programs and the CORE Method 61

62 Weak Completion is Needed Reconsider the case modus ponens with positive observation, i.e. P 10 = {l e ab, ab }, A = {e, e }, and O = l. lm wc Now consider P 10, A, = 3L instead of P 10, A, = 3L P 10 = 3L l P 10 {e } = 3L l (because ab can be mapped to ). P 10 {e } = 3L l P 10 A = 3L l Hence, the observation can not be explained at all (in contrast to Byrne 1989). Human Reasoning, Logic Programs and the CORE Method 6

63 Completion is Insufficient Reconsider the case modus ponens with positive observation, i.e. P 10 = {l e ab, ab }, A = {e, e }, and O = l. Now consider P 10, A, = c 3L instead of P lm wc 10, A, = where P = c 3L F iff F holds in all models for c P. 3L, c P 10 = {l e ab, ab, e } = 3L l c P 10 = 3L e The observation is inconsistent with the knowledge base and, thus, cannot be explained at all (in contrast to Byrne 1989). Human Reasoning, Logic Programs and the CORE Method 63

64 Explanations must be Completed as well Reconsider the case modus ponens with positive observation, i.e. P 10 = {l e ab, ab }, A = {e, e }, and O = l. Now (weakly) complete only the program, but not the explanations. wc P 10 = {l e ab, ab }. wc P 10 = 3L l wc (P 10 ) {e } = 3L l wc (P 10 ) {e } = 3L l (because e can be mapped to ). wc (P 10 ) A = 3L l Hence, the observation can not be explained (in contrast with Byrne 1989). Human Reasoning, Logic Programs and the CORE Method 64

65 Sceptical versus Creduluos Reasoning Reconsider the case alternative arguments with positive observation, i.e., P 11 = {l e ab 1, ab 1, l t ab, ab }, A = {e, e }, and O = l. Now consider P 11, A, = lm wc 3L and reason credulously: There are two minimal explanations, viz. {e } and {e }. Hence, P 11, l = s A e, but P 11, l = c A e. Creduluos reasoning is inconsistent with Byrne Human Reasoning, Logic Programs and the CORE Method 65

66 Summary Let P 10 = {l e ab, ab } P 11 = {l e ab 1, ab 1, l t ab, ab } P 1 = {l e ab 1, ab 1 o, l o ab, ab e} We obtain P 10, l = s A e P 11, l = s A e P 1, l = s A e P 10, l = s A e P 11, l = s A e P 1, l = s A e Byrne 1989 e(53%) e(16%) e(55%) e(69%) e(69%) e(44%) Human Reasoning, Logic Programs and the CORE Method 66

67 Summary Logic appears to be adequate for human reasoning if weak completion, Lukasiewicz semantics, the Stenning and van Lambalgen semantic operator, and abduction are used Human Reasoning is modeled by reasoning towards an appropriate logic program P and, thereafter, reasoning with respect to the least L-model of the weak completion of P. This approach matches data from studies in human reasoning. There is a connectionist encoding. Human Reasoning, Logic Programs and the CORE Method 67

68 Discussion Stenning, van Lambalgen 008 propose spreading-activation networks like KBANN (Towell, Shavlik 1993) with two units for each propositional letter and an inhibitory link between them. Logical threshold units can be replaced by bipolar sigmoidal ones following d Avila Garcez, Zaverucha, Carbalho Networks can be trained by backpropagation, but backpropagation is not neurally plausible. Human Reasoning, Logic Programs and the CORE Method 68

69 Some Open Problems (1) Negation How is negation treated in human reasoning? Errors How can frequently made errors be explained in the proposed approach? Lukasiezicz logic In a Lukasiezicz logic the semantic deduction theorem does not hold. Is this adequate with respect to human reasoning? Completion Under which conditions is human reasoning adequately modeled by completion and/or weak completion? Human Reasoning, Logic Programs and the CORE Method 69

70 Some Open Problems () Contractions Do humans exhibit a behavior which can be adequately modeled by contractional semantic operators? Can we generate appropriate level mappings by studying the behavior of humans? Explanations Do humans consider minimal explanations? In which order are (minimal) explanations generated by humans if there are several? Does attention play a role in the selection of (minimal) explanations? Stable coalitions Do stable coalitions occur in human reasoning? How are they deactivated? Human Reasoning, Logic Programs and the CORE Method 70

71 Some Open Problems (3) Reasoning Do humans reason sceptically or credulously? How does a connectionist realization of sceptical reasoning looks like? Theory revision How is theory revision modeled in human reasoning? Relation to other semantics What is the relation between the proposed approach and well-founded and/or stable and/or projection semantics? Human Reasoning, Logic Programs and the CORE Method 71