Elements of AI Luxembourg - session 3

1. Logic and Automated Reasoning Or: Machine languages for everything? Alexander Steen University of Luxembourg Elements of AI, Webinar III, 2021

2. C. B. Francke, Herzog Anton Ulrich-Museum ”If we had it [a characteristica unversalis], we should be able to reason in metaphysics and morals in much the same way as in geometry and analysis.” — G.W. Leibniz, 1677 (translated by Russell, 1900)

3. C. B. Francke, Herzog Anton Ulrich-Museum ”If we had it [a characteristica universalis], we should be able to reason in metaphysics and morals in much the same way as in geometry and analysis.” — G.W. Leibniz, 1677 (translated by Russell, 1900)

4. Leibniz’ Vision ”[...] quando orientur controversiae, non magis disputatione opus erit inter duos philosophus, quam inter duos computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo [...] dicere: calcule- mus” — G.W. Leibniz, 1684 ”[...] if controversies were to arise, there would be no more need of disputation between two philosophers than between two calculators. For it would suffice for them to take their pencils in their hands and to sit down at the abacus, and to say to each other [...]: Let us calculate.” , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 3

7. Leibniz’ Vision (2) The ultimate goal ”[...] it would suffice [...] to say [...]: Let us calculate” , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 4

8. Leibniz’ Vision (2) The ultimate goal ”[...] it would suffice [...] to say [...]: Let us calculate” Dispute Â National Gallery of Victoria, Melbourne/Felton Bequest, via NGV Formalization Â Calculation Â Result Â , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 4

9. Leibniz’ Vision (2) The ultimate goal ”[...] it would suffice [...] to say [...]: Let us calculate” Dispute Â National Gallery of Victoria, Melbourne/Felton Bequest, via NGV Formalization Â Calculation Â Result Â ↑ Characteristica universalis ↑ Calculus ratiocinator , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 4

10. Leibniz’ Vision (2) The ultimate goal ”[...] it would suffice [...] to say [...]: Let us calculate” Dispute Â National Gallery of Victoria, Melbourne/Felton Bequest, via NGV Formalization Â Calculation Â Result Â ↑ Characteristica universalis ↑ Calculus ratiocinator ≈ Logic ≈ (Automated) Reasoning , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 4

11. Leibniz’ Vision (2) The ultimate goal ”[...] it would suffice [...] to say [...]: Let us calculate” Dispute Â National Gallery of Victoria, Melbourne/Felton Bequest, via NGV Formalization Â Calculation Â Result Â ↑ Characteristica universalis ↑ Calculus ratiocinator ≈ Logic (?) ≈ (Automated) Reasoning (?) , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 4

12. AI and Logic What does this has to do with AI? Automated Reasoning is a core subfield of Artificial Intelligence also: ”good old-fashioned AI (GOFAI)” , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 5

13. AI and Logic What does this has to do with AI? A ”dispute” can be anything: É Internal decision processes of banks for loans Is this person eligible for a loan? É Emergency actions of autonomous vehicles Which action should be taken? É Facial recognition Does this picture show person X? É Planning Which route is the fastest/shortest? É ... and much more ... , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 5

14. AI and Logic What does this has to do with AI? A ”dispute” can be anything: É Internal decision processes of banks for loans Is this person eligible for a loan? É Emergency actions of autonomous vehicles Which action should be taken? É Facial recognition Does this picture show person X? É Planning Which route is the fastest/shortest? É ... and much more ... , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 5

15. Formal Logic and Reasoning What is a Logic? (1) A language Composed by number of building blocks: É not A (¬A) É A and B (A ∧ B) É A or B (A ∨ B) É if A then B (A → B) É ... many more possible (2) Collection of reasoning patterns Recipes for processing information (rules for argumentation): É If A → B holds and A holds then B follows (modus ponens) É If A ∨ B holds and ¬A holds then B follows (modus tollens) É ... Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 6

18. Formal Logic and Reasoning What is a Logic? (1) A language Composed by number of building blocks: É not A (¬A) É A and B (A ∧ B) É A or B (A ∨ B) É if A then B (A → B) É ... many more possible (2) Collection of reasoning patterns Recipes for processing information (rules for argumentation): É If A → B holds and A holds then B follows (modus ponens) É If A ∨ B holds and ¬A holds then B follows (modus tollens) É ... Important property: Unambiguous Consider the sentence ”Time flies like an arrow” What do we mean? Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 6

19. Formal Logic and Reasoning What is a Logic? (1) A language Composed by number of building blocks: É not A (¬A) É A and B (A ∧ B) É A or B (A ∨ B) É if A then B (A → B) É ... many more possible (2) Collection of reasoning patterns Recipes for processing information (rules for argumentation): É If A → B holds and A holds then B follows (modus ponens) É If A ∨ B holds and ¬A holds then B follows (modus tollens) É ... Important property: Unambiguous Consider the sentence ”Time flies like an arrow” What do we mean? (1) There is species of flies, called time flies, that like a specific arrow Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 6

20. Formal Logic and Reasoning What is a Logic? (1) A language Composed by number of building blocks: É not A (¬A) É A and B (A ∧ B) É A or B (A ∨ B) É if A then B (A → B) É ... many more possible (2) Collection of reasoning patterns Recipes for processing information (rules for argumentation): É If A → B holds and A holds then B follows (modus ponens) É If A ∨ B holds and ¬A holds then B follows (modus tollens) É ... Important property: Unambiguous Consider the sentence ”Time flies like an arrow” What do we mean? (2) Time (considered as an ”object”) flies through space like an arrow Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 6

21. Formal Logic and Reasoning What is a Logic? (1) A language Composed by number of building blocks: É not A (¬A) É A and B (A ∧ B) É A or B (A ∨ B) É if A then B (A → B) É ... many more possible (2) Collection of reasoning patterns Recipes for processing information (rules for argumentation): É If A → B holds and A holds then B follows (modus ponens) É If A ∨ B holds and ¬A holds then B follows (modus tollens) É ... Important property: Unambiguous Consider the sentence ”Time flies like an arrow” What do we mean? (3) Time passes quite quickly Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 6

23. Formal Logic and Reasoning What is a Logic? (1) A language Composed by number of building blocks: É not A (¬A) É A and B (A ∧ B) É A or B (A ∨ B) É if A then B (A → B) É ... many more possible (2) Collection of reasoning patterns Recipes for processing information (rules for argumentation): É If A → B holds and A holds then B follows (modus ponens) É If A ∨ B holds and ¬A holds then B follows (modus tollens) É ... Example: 1. It’s daytime or it’s nighttime (day ∨ night) 2. If the sun is shining, it is not nighttime (sun → ¬night) 3. The sun is shining (sun) Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 6

24. Formal Logic and Reasoning What is a Logic? (1) A language Composed by number of building blocks: É not A (¬A) É A and B (A ∧ B) É A or B (A ∨ B) É if A then B (A → B) É ... many more possible (2) Collection of reasoning patterns Recipes for processing information (rules for argumentation): É If A → B holds and A holds then B follows (modus ponens) É If A ∨ B holds and ¬A holds then B follows (modus tollens) É ... Example: 1. It’s daytime or it’s nighttime (day ∨ night) 2. If the sun is shining, it is not nighttime (sun → ¬night) 3. The sun is shining (sun)            It follows: It’s daytime Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 6

25. Logical Reasoning in AI Reasoning: How is this useful in AI? Deduction From A → B and A to B Given: Rules, Situation Search: Conclusions ⇒ Decide on actions Induction From A and B to A → B Given: Situation, Conclusions Search: Rules ⇒ Learn principles Abduction From A → B and B to A Given: Rules, Conclusions Search: Situation ⇒ Explain actions Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 7

26. Logical Reasoning in AI Reasoning: How is this useful in AI? Deduction From A → B and A to B Given: Rules, Situation Search: Conclusions ⇒ Decide on actions Induction From A and B to A → B Given: Situation, Conclusions Search: Rules ⇒ Learn principles Abduction From A → B and B to A Given: Rules, Conclusions Search: Situation ⇒ Explain actions , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 7

27. Logical Reasoning in AI Reasoning: How is this useful in AI? Deduction From A → B and A to B Given: Rules, Situation Search: Conclusions ⇒ Decide on actions Induction From A and B to A → B Given: Situation, Conclusions Search: Rules ⇒ Learn principles Abduction From A → B and B to A Given: Rules, Conclusions Search: Situation ⇒ Explain actions , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 7

30. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

31. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

32. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality É Example 1: A bag of coins, pick some: É First pick: A penny É Second pick: ... penny É Third pick: ... penny É , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

36. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality É Example 1: A bag of coins, pick some: É First pick: A penny É Second pick: ... penny É Third pick: ... penny É Therefore, every coin is a penny (prediction) , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

37. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality É Example 1: A bag of coins, pick some: É First pick: A penny É Second pick: ... penny É Third pick: ... penny É Therefore, every coin is a penny (prediction) Shortcoming: Sample set may be too small , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

38. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality É Example 2: Considering ancestry: É My grandfather is bald É Jeremie’s grandfather is bald É , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

41. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality É Example 2: Considering ancestry: É My grandfather is bald É Jeremie’s grandfather is bald É Therefore, all grandfathers are bald , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

42. Automated Deduction: Introduction Automated Deduction: What is it not? É Autom. Deduction is not machine learning (ML) as such É Usually associated un-/supervised (deep) learning, neural networks, ... Key difference: ML is related to inductive reasoning É Arguing using empirical evidence É Observations do not necessarily imply actual causality É Example 2: Considering ancestry: É My grandfather is bald É Jeremie’s grandfather is bald É Therefore, all grandfathers are bald Shortcoming: Learning set may be biased , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 8

43. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

44. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

45. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable É Example 1: É If X is human, then X is mortal. É Socrates is human É Sting/Wikimedia Commons, CC BY-SA 2.5 Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

48. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable É Example 1: É If X is human, then X is mortal. É Socrates is human É Question: Therefore, Socrates is mortal? Sting/Wikimedia Commons, CC BY-SA 2.5 Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

49. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable É Example 1: É If X is human, then X is mortal. É Socrates is human É Question: Therefore, Socrates is mortal? AR can derive this conclusion; it is implicitly contained in the rules already Sting/Wikimedia Commons, CC BY-SA 2.5 Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

50. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable É Example 2: É If X is human, then X is mortal. É Socrates is human É Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

53. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable É Example 2: É If X is human, then X is mortal. É Socrates is human É Question: Therefore, Plato is mortal? , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

54. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable É Example 2: É If X is human, then X is mortal. É Socrates is human É Question: Therefore, Plato is mortal? Not necessarily. , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

55. Automated Deduction: Introduction (2) Automated Deduction: What is it then? É Could also call it ”information discovery” É Of course, this is also a form of learning Key difference to ML: It represents deductive reasoning É Arguing using logical inferences É Deductive conclusions are sound and reliable Advantages: No false negatives, no false positives, explainable. Disadvantages: Translation of domain knowledge is expensive Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 9

56. Automated Reasoning How does automated reasoning work? 1. Translate problem to logic É Rules: Knowledge about the world (axioms) É Goal: What is to be to solved (conjecture) 2. Run reasoner for obtaining a solution É Certificate: Describes the solution (proof) — or — É Counter example: Explanation why there is no solution (model) 3. Translate solution to world Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 10

59. Automated Deduction How does automated deduction work? É Usage of formal inference rules É Iterative deduction of facts É Formalization acts as assumptions É A derivation of a goal forms a mathematical proof É Proofs can be verified externally É Act as explicit justification , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 11

60. Automated Deduction How does automated deduction work? É Usage of formal inference rules É Iterative deduction of facts É Formalization acts as assumptions É A derivation of a goal forms a mathematical proof É Proofs can be verified externally É Act as explicit justification , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 11

61. Automated Reasoning: Systems State of the art system output É System output may be (very) large É often unreadable, machine-oriented Small sample proof: (TPTP THF format, widely accepted standard format for proofs) thf(6,axiom,((! [A:(nat > $o)]: (((A @ zero) & ! [B:nat]: ((A @ B) => (A thf(18,plain,((! [A:(nat > $o)]: (((A @ zero) & ! [B:nat]: ((A @ B) => ( thf(24,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A) thf(1,axiom,((? [A:nat]: ~ (p @ A))),file(’oded_2.p’,8)). thf(7,plain,((? [A:nat]: ~ (p @ A))),inference(defexp_and_simp_and_etaex thf(8,plain,((~ (! [A:nat]: (p @ A)))),inference(miniscope,[status(thm)] thf(9,plain,((~ (p @ sk1))),inference(cnf,[status(esa)],[8])). thf(506,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A thf(660,plain,((~ (p @ zero)) | (p @ (sk2 @ (p)))),inference(pre_uni,[st thf(5,axiom,((p @ zero)),file(’oded_2.p’,4)). thf(17,plain,((p @ zero)),inference(defexp_and_simp_and_etaexpand,[statu thf(716,plain,(~ ($true) | (p @ (sk2 @ (p)))),inference(rewrite,[status( thf(717,plain,((p @ (sk2 @ (p)))),inference(simp,[status(thm)],[716])). thf(3,axiom,((! [A:nat]: ((p @ (succ @ A)) = ((p @ A) & (q @ A))))),file thf(13,plain,((! [A:nat]: ((p @ (succ @ A)) = ((p @ A) & (q @ A))))),inf thf(14,plain,(! [A:nat] : (((p @ (succ @ A)) = ((p @ A) & (q @ A))))),in thf(15,plain,(! [A:nat] : ((((p @ A) & (q @ A)) = (p @ (succ @ A))))),in thf(844,plain,(! [A:nat] : (((q @ A) = (p @ (succ @ A))) | ((p @ (sk2 @ thf(845,plain,(((q @ (sk2 @ (p))) = (p @ (succ @ (sk2 @ (p)))))),inferen thf(23,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (~ (A @ (succ @ thf(33,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (~ (A @ (succ @ thf(69,plain,((~ (p @ zero)) | (~ (p @ (succ @ (sk2 @ (p)))))),inference thf(90,plain,(~ ($true) | (~ (p @ (succ @ (sk2 @ (p)))))),inference(rewr thf(91,plain,((~ (p @ (succ @ (sk2 @ (p)))))),inference(simp,[status(thm thf(861,plain,((~ (q @ (sk2 @ (p))))),inference(rewrite,[status(thm)],[8 thf(869,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A thf(882,plain,((~ (q @ zero)) | (q @ (sk2 @ (q)))),inference(pre_uni,[st thf(4,axiom,((q @ zero)),file(’oded_2.p’,5)). thf(16,plain,((q @ zero)),inference(defexp_and_simp_and_etaexpand,[statu thf(901,plain,(~ ($true) | (q @ (sk2 @ (q)))),inference(rewrite,[status( thf(902,plain,((q @ (sk2 @ (q)))),inference(simp,[status(thm)],[901])). thf(2,axiom,((! [A:nat]: ((q @ (succ @ A)) = ((q @ A) | (r @ A))))),file thf(10,plain,((! [A:nat]: ((q @ (succ @ A)) = ((q @ A) | (r @ A))))),inf thf(11,plain,(! [A:nat] : (((q @ (succ @ A)) = ((q @ A) | (r @ A))))),in , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 12

62. Automated Reasoning: Systems State of the art system output É System output may be (very) large É often unreadable, machine-oriented Small sample proof: (TPTP THF format, widely accepted standard format for proofs) thf(6,axiom,((! [A:(nat > $o)]: (((A @ zero) & ! [B:nat]: ((A @ B) => (A thf(18,plain,((! [A:(nat > $o)]: (((A @ zero) & ! [B:nat]: ((A @ B) => ( thf(24,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A) thf(1,axiom,((? [A:nat]: ~ (p @ A))),file(’oded_2.p’,8)). thf(7,plain,((? [A:nat]: ~ (p @ A))),inference(defexp_and_simp_and_etaex thf(8,plain,((~ (! [A:nat]: (p @ A)))),inference(miniscope,[status(thm)] thf(9,plain,((~ (p @ sk1))),inference(cnf,[status(esa)],[8])). thf(506,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A thf(660,plain,((~ (p @ zero)) | (p @ (sk2 @ (p)))),inference(pre_uni,[st thf(5,axiom,((p @ zero)),file(’oded_2.p’,4)). thf(17,plain,((p @ zero)),inference(defexp_and_simp_and_etaexpand,[statu thf(716,plain,(~ ($true) | (p @ (sk2 @ (p)))),inference(rewrite,[status( thf(717,plain,((p @ (sk2 @ (p)))),inference(simp,[status(thm)],[716])). thf(3,axiom,((! [A:nat]: ((p @ (succ @ A)) = ((p @ A) & (q @ A))))),file thf(13,plain,((! [A:nat]: ((p @ (succ @ A)) = ((p @ A) & (q @ A))))),inf thf(14,plain,(! [A:nat] : (((p @ (succ @ A)) = ((p @ A) & (q @ A))))),in thf(15,plain,(! [A:nat] : ((((p @ A) & (q @ A)) = (p @ (succ @ A))))),in thf(844,plain,(! [A:nat] : (((q @ A) = (p @ (succ @ A))) | ((p @ (sk2 @ thf(845,plain,(((q @ (sk2 @ (p))) = (p @ (succ @ (sk2 @ (p)))))),inferen thf(23,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (~ (A @ (succ @ thf(33,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (~ (A @ (succ @ thf(69,plain,((~ (p @ zero)) | (~ (p @ (succ @ (sk2 @ (p)))))),inference thf(90,plain,(~ ($true) | (~ (p @ (succ @ (sk2 @ (p)))))),inference(rewr thf(91,plain,((~ (p @ (succ @ (sk2 @ (p)))))),inference(simp,[status(thm thf(861,plain,((~ (q @ (sk2 @ (p))))),inference(rewrite,[status(thm)],[8 thf(869,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A thf(882,plain,((~ (q @ zero)) | (q @ (sk2 @ (q)))),inference(pre_uni,[st thf(4,axiom,((q @ zero)),file(’oded_2.p’,5)). thf(16,plain,((q @ zero)),inference(defexp_and_simp_and_etaexpand,[statu thf(901,plain,(~ ($true) | (q @ (sk2 @ (q)))),inference(rewrite,[status( thf(902,plain,((q @ (sk2 @ (q)))),inference(simp,[status(thm)],[901])). thf(2,axiom,((! [A:nat]: ((q @ (succ @ A)) = ((q @ A) | (r @ A))))),file thf(10,plain,((! [A:nat]: ((q @ (succ @ A)) = ((q @ A) | (r @ A))))),inf thf(11,plain,(! [A:nat] : (((q @ (succ @ A)) = ((q @ A) | (r @ A))))),in Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 12

63. Automated Reasoning: Systems State of the art system output É System output may be (very) large É often unreadable, machine-oriented Small sample proof: (TPTP THF format, widely accepted standard format for proofs) thf(6,axiom,((! [A:(nat > $o)]: (((A @ zero) & ! [B:nat]: ((A @ B) => (A thf(18,plain,((! [A:(nat > $o)]: (((A @ zero) & ! [B:nat]: ((A @ B) => ( thf(24,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A) thf(1,axiom,((? [A:nat]: ~ (p @ A))),file(’oded_2.p’,8)). thf(7,plain,((? [A:nat]: ~ (p @ A))),inference(defexp_and_simp_and_etaex thf(8,plain,((~ (! [A:nat]: (p @ A)))),inference(miniscope,[status(thm)] thf(9,plain,((~ (p @ sk1))),inference(cnf,[status(esa)],[8])). thf(506,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A thf(660,plain,((~ (p @ zero)) | (p @ (sk2 @ (p)))),inference(pre_uni,[st thf(5,axiom,((p @ zero)),file(’oded_2.p’,4)). thf(17,plain,((p @ zero)),inference(defexp_and_simp_and_etaexpand,[statu thf(716,plain,(~ ($true) | (p @ (sk2 @ (p)))),inference(rewrite,[status( thf(717,plain,((p @ (sk2 @ (p)))),inference(simp,[status(thm)],[716])). thf(3,axiom,((! [A:nat]: ((p @ (succ @ A)) = ((p @ A) & (q @ A))))),file thf(13,plain,((! [A:nat]: ((p @ (succ @ A)) = ((p @ A) & (q @ A))))),inf thf(14,plain,(! [A:nat] : (((p @ (succ @ A)) = ((p @ A) & (q @ A))))),in thf(15,plain,(! [A:nat] : ((((p @ A) & (q @ A)) = (p @ (succ @ A))))),in thf(844,plain,(! [A:nat] : (((q @ A) = (p @ (succ @ A))) | ((p @ (sk2 @ thf(845,plain,(((q @ (sk2 @ (p))) = (p @ (succ @ (sk2 @ (p)))))),inferen thf(23,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (~ (A @ (succ @ thf(33,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (~ (A @ (succ @ thf(69,plain,((~ (p @ zero)) | (~ (p @ (succ @ (sk2 @ (p)))))),inference thf(90,plain,(~ ($true) | (~ (p @ (succ @ (sk2 @ (p)))))),inference(rewr thf(91,plain,((~ (p @ (succ @ (sk2 @ (p)))))),inference(simp,[status(thm thf(861,plain,((~ (q @ (sk2 @ (p))))),inference(rewrite,[status(thm)],[8 thf(869,plain,(! [B:nat,A:(nat > $o)] : ((~ (A @ zero)) | (A @ (sk2 @ (A thf(882,plain,((~ (q @ zero)) | (q @ (sk2 @ (q)))),inference(pre_uni,[st thf(4,axiom,((q @ zero)),file(’oded_2.p’,5)). thf(16,plain,((q @ zero)),inference(defexp_and_simp_and_etaexpand,[statu thf(901,plain,(~ ($true) | (q @ (sk2 @ (q)))),inference(rewrite,[status( thf(902,plain,((q @ (sk2 @ (q)))),inference(simp,[status(thm)],[901])). thf(2,axiom,((! [A:nat]: ((q @ (succ @ A)) = ((q @ A) | (r @ A))))),file thf(10,plain,((! [A:nat]: ((q @ (succ @ A)) = ((q @ A) | (r @ A))))),inf thf(11,plain,(! [A:nat] : (((q @ (succ @ A)) = ((q @ A) | (r @ A))))),in Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 12

64. Applications: Brief look I Classical applications: Mathematics/Computer Science and Verification Mathematics/CS: Analysis of the Pythagorean Triples problem Approx. 200 terabytes in proofs, 16000 CPU hours used , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 13

65. Applications: Brief look I Classical applications: Mathematics/Computer Science and Verification Software verification: Safety-critical and mission-critical components, e.g. É NASA: Verification of mission components using PVS (NASA Langley) É SNCF: Verification of train systems using ProMeLa/Spin É ... , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 13

66. Applications: Brief look II AI and Law: Normative Reasoning and Computational Law É Standards for normative/legal languages (e.g. LegalRuleML) É Tools for business compliance (e.g. Guido Governatori et al.) É Automation of normative logics É Own projects: NAI (jww. T. Libal) and Automated Reasoning with Legal Entities (AuReLeE) , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 14

67. Applications: Brief look II AI and Law: Normative Reasoning and Computational Law É Standards for normative/legal languages (e.g. LegalRuleML) É Tools for business compliance (e.g. Guido Governatori et al.) É Automation of normative logics É Own projects: NAI (jww. T. Libal) and Automated Reasoning with Legal Entities (AuReLeE) (c) Guido Governatori , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 14

71. Applications: Brief look III Automated Reasoning in Philosophy [Benzmüller and Woltzenlogel Paleo, since 2013] , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 15

72. Re: Characteristica Universalis How far have we progressed? Recall: Initial idea of logic as universal language Not yet there: Zoo of logics classical logics, constructive logics, free logics, ... Dynamic logics, epistemic logics, higher-order logics, .. Many-valued logics, fuzzy logics, dynamic logics, constructive logics, temporal logic, ... Multi modal logics, epistemic logics, alethic logics, temporal logics, public announce- ment logics, dynamic logics, deontic logics, paraconsistent logics, paracomplete logics, ... Modal logics, deontic logics, epistemic logics, temporal logics, argumentation logics, dialog logics, I/O logics, paraconsistent logics, ... ... , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 16

73. Summary Conclusion É Logical reasoning core part of AI É Automated reasoning systems É Applications examples (only partly addressed) É Software/hardware verification É Computer Science and Mathematics (also in teaching) É Legal AI, Computational Law, Normative reasoning É Reasoning in Metaphysics, Ethics, ... Thank you! Questions? , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 17

74. Summary Conclusion É Logical reasoning core part of AI É Automated reasoning systems É Applications examples (only partly addressed) É Software/hardware verification É Computer Science and Mathematics (also in teaching) É Legal AI, Computational Law, Normative reasoning É Reasoning in Metaphysics, Ethics, ... Thank you! Questions? , Logic and Automated Reasoning, Elements of AI, Webinar III, 2021 17

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