Deduction and Induction. Stephan Schulz. The Inference Engine. Machine Learning. A Match Made in Heaven

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1 Deduction and Induction A Match Made in Heaven Stephan Schulz The Inference Engine Machine Learning

2 Deduction and Induction A Match Made in Heaven Stephan Schulz or a Deal with the Devil? The Inference Engine Machine Learning

3 Agenda Search and choice points in saturating theorem proving Basic questions about learning Learning from performance data Classification and heuristic selection Parameters for clause selection Learning from proofs and search graphs Proof extraction Learning clause evaluations (?) Conclusion 2

4 Theorem Proving: Big Picture Real World Problem Formalized Problem 8X : human(x)! mortal(x) 8X : philosopher(x)! human(x) philosopher(socrates)? = mortal(socrates) Proof or Countermodel or Timeout ATP Proof Search 3

Contradiction and Saturation Proof by contradiction Assume negation of conjecture Show that axioms and negated conjecture imply falsity Saturation Convert problem to Clause Normal Form Systematically

5 Contradiction and Saturation Proof by contradiction Assume negation of conjecture Show that axioms and negated conjecture imply falsity Saturation Convert problem to Clause Normal Form Systematically enumerate logical consequences of axioms and negated conjecture Goal: Explicit contradiction (empty clause) Redundancy elimination Use contracting inferences to simplify or eliminate some clauses Formula set Clausifier Equisatisfiable clause set 4

6 Contradiction and Saturation Proof by contradiction Assume negation of conjecture Show that axioms and negated conjecture imply falsity Saturation Convert problem to Clause Normal Form Systematically enumerate logical consequences of axioms and negated conjecture Goal: Explicit contradiction (empty clause) Redundancy elimination Use contracting inferences to simplify or eliminate some clauses Search control problem: How and in which order do we enumerate consequences? Formula set Clausifier Equisatisfiable clause set 4

7 Proof Search and Choice Points First-order logic is semi-decidable Provers search for proof in infinite space... of possible derivations... of possible consequences Major choice points of Superposition calculus: Term ordering (which terms are bigger) (Negative) literal selection Selection of clauses for inferences (with the given clause algorithm) 5

8 Term Ordering and Literal Selection Negative Superposition with selection C s t D u v (C D u [p t] v) σ if σ = mgu(u p, s) and (s t) σ is -maximal in (C s t) σ and s is -maximal in (s t) σ and u v is selected in D u v and u is -maximal in (s t) σ Choice points: is a ground-total rewrite ordering Consistent throughout the proof search I.e. in practice determined up-front Any negative literal can be selected Current practice: Fixed scheme picked up-front 6

9 The Given-Clause Algorithm g=? P (processed clauses) Aim: Move everything from U to P g U (unprocessed clauses) 7

10 The Given-Clause Algorithm g=? Generate g P (processed clauses) Aim: Move everything from U to P Invariant: All generating inferences with premises from P have been performed U (unprocessed clauses) 7

11 The Given-Clause Algorithm g=? g Simplify P (processed clauses) Generate Simplifiable? Aim: Move everything from U to P Invariant: All generating inferences with premises from P have been performed Invariant: P is interreduced U (unprocessed clauses) 7

12 The Given-Clause Algorithm g=? g Simplify P (processed clauses) Generate Simplifiable? U (unprocessed clauses) Cheap Simplify Aim: Move everything from U to P Invariant: All generating inferences with premises from P have been performed Invariant: P is interreduced Clauses added to U are simplified with respect to P 7

13 Choice Point Clause Selection P (processed clauses) g=? Generate Simplifiable? g Cheap Simplify Simplify U (unprocessed clauses) U (unprocessed clauses) 8

14 Choice Point Clause Selection P (processed clauses) g=? Generate Simplifiable? g Cheap Simplify Simplify Choice Point U (unprocessed clauses) U (unprocessed clauses) 8

15 Induction for Deduction Question 1: What to learn from? Performance data (prover is a black box) Proofs (only final result of search is visible) Proof search graphs (most of search is visible) Question 2: What to learn? Here: Learn strategy selection Here: Learn parameterization for clause selection heuristics Here: Learn new clause evaluation functions...? 9

16 Automatic Strategy Selection 10

17 Strategy Selection Definition: A strategy is a collection of all search control parameters Term ordering Literal selection scheme Clause selection heuristic... (minor parameters) 11

18 Strategy Selection Definition: A strategy is a collection of all search control parameters Term ordering Literal selection scheme Clause selection heuristic... (minor parameters) Observation: Different problems are simple for different strategies Question: Can we determine a good heuristic (or set of heuristics) up-front? Original: Manually coded automatic modes Based on developer intuition/insight/experience Limited success, high maintenance State of the art: Automatic generation of automatic modes 11

19 Learning Heuristic Selection TPTP problem library 12

20 Learning Heuristic Selection TPTP problem library 12

21 Learning Heuristic Selection Feature-based classification TPTP problem library 12

22 Learning Heuristic Selection Feature-based classification TPTP problem library Assign strategies to classes based on collected performance data from previous experiments Simplest: Always pick best strategy in class If no data, pick globally best 12

23 Learning Heuristic Selection Feature-based classification TPTP problem library Assign strategies to classes based on collected performance data from previous experiments Simplest: Always pick best strategy in class If no data, pick globally best Example features Number of clausse Arity of symbols Unit/Horn/Non-horn 12

24 Auto Mode Performance E 1.8 Auto E 1.8 Best TPTP CNF&FOF problems 13

25 A Caveat Feature-based classification TPTP problem library Assign strategies to classes based on collected performance data from previous experiments Simplest: Always pick best strategy in class If no data, pick globally best Example features Number of clausse Arity of symbols Unit/Horn/Non-horn 14

26 A Caveat Feature-based classification TPTP problem library Assign strategies to classes based on collected performance data from previous experiments Simplest: Always pick best strategy in class If no data, pick globally best Example features Number of clausse Arity of symbols Unit/Horn/Non-horn Features based on developer intuition insight experience 14

27 : me Current Work: Learning Classification.. Characterize problems by performance vectors Which strategy solved the problem how fast? Unsupervised clustering of problems based om performance Each cluster contains problems on which the Literature would be same strategies perform well Something on clustering Feature extraction: Try to find characterization of clusters Quite ford for a stat! Thanks E.g. based on feature set E.g. using nearest-neighbour approaches My Bachelor Student Ayatallah just started work on this topic - results in 6 months structure in the input data and automatically group them are many clustering algorithms. One of the most popular is K-Means algorithm. More reading and understanding of for the clustering algorithm needed will decide which algor Figure 2: Clustering Example food here, e.g. the E - paper S C 1.8 a 15

28 Learning parameterization for clause selection heuristics 16

29 Reminder: Choice Point Clause Selection P (processed clauses) g=? Generate Simplifiable? g Cheap Simplify Simplify Choice Point U (unprocessed clauses) U (unprocessed clauses) 17

30 Basic Approaches to Clause Selection Symbol counting Pick smallest clause in U {f (X ) a, P(a) $true, g(y ) = f (a)} = 10 FIFO Always pick oldest clause in U Flexible weighting Symbol counting, but give different weight to different symbols E.g. lower weight to symbols from goal! E.g. higher weight for symbols in inference positions Combinations Interleave different schemes 18

31 Given-Clause Selection in E (1) Domain Specific Language (DSL) for clause selection scheme Arbitrary number of priority queues Each queue ordered by: Unparameterized priority function Parameterized heuristic evaluation function Clauses picked using weighted round-robin scheme Example (5 queues): (1*ConjectureRelativeSymbolWeight(SimulateSOS, 0.5, 100, 100, 100, 100, 1.5, 1.5, 1), 4*ConjectureRelativeSymbolWeight(ConstPrio, 0.1, 100, 100, 100, 100,1.5, 1.5, 1.5), 1*FIFOWeight(PreferProcessed), 1*ConjectureRelativeSymbolWeight(PreferNonGoals, 0.5, 100, 100, 100,100, 1.5, 1.5, 1), 4*Refinedweight(SimulateSOS,3,2,2,1.5,2)) 19

32 Given-Clause Selection in E (2) Example clause selection heuristic (1*ConjectureRelativeSymbolWeight(SimulateSOS, 0.5, 100, 100, 100, 100, 1.5, 1.5, 1), 4*ConjectureRelativeSymbolWeight(ConstPrio, 0.1, 100, 100, 100, 100,1.5, 1.5, 1.5), 1*FIFOWeight(PreferProcessed), 1*ConjectureRelativeSymbolWeight(PreferNonGoals, 0.5, 100, 100, 100,100, 1.5, 1.5, 1), 4*Refinedweight(SimulateSOS,3,2,2,1.5,2)) Infinitely many possibilities Several integer and floating point parameters per evaluation function Arbitrary combinations of individual evaluation functions 20

33 Given-Clause Selection in E (2) Example clause selection heuristic (1*ConjectureRelativeSymbolWeight(SimulateSOS, 0.5, 100, 100, 100, 100, 1.5, 1.5, 1), 4*ConjectureRelativeSymbolWeight(ConstPrio, 0.1, 100, 100, 100, 100,1.5, 1.5, 1.5), 1*FIFOWeight(PreferProcessed), 1*ConjectureRelativeSymbolWeight(PreferNonGoals, 0.5, 100, 100, 100,100, 1.5, 1.5, 1), 4*Refinedweight(SimulateSOS,3,2,2,1.5,2)) Infinitely many possibilities Several integer and floating point parameters per evaluation function Arbitrary combinations of individual evaluation functions How do we find good clause selection heuristics (without relying on developer intuition, insight, experience)? 20

34 Genetic Algorithms Optimization based on evolving population of individuals Optimization is organized in generations In each generation, individuals compete to reproduce Each individual is a candidate solution (i.e. search heuristic) Individuals are assigned a fitness score based on performance More fit individuals are more likely to reproduce into the next generation The next generation: Mutation - randomly modify individual Crossover - create new individual from two parents Survivors 21

35 Applying Genetic Algorithms to Clause Selection Encoding: DSL translated into S-Expressions Mutation: Randomly modify parameters of one heuristic Crossover: Compose individual by randomly inserting evaluation functions from both parents If the same generic evaluation function occurs in both, randomly exchange parameters Fitness: How many medium difficulty problems are solved... on smallish sample set... with short time limit Selection: Tournament selection (n 5) 22

Evolution in Action 120 Fitness over generations 115 110 Fitness

36 Evolution in Action 120 Fitness over generations Fitness (solved problems) Generations 23

37 (Very) Preliminary Results Evolution finds good clause selection heuristics from random initial population Convergence in 200 generations Time per generation 45 CPU hours minutes on 24 core server Best evolved heuristic beats best conventional heuristic Evaluation on problems from TPTP second time limit, 2.6GHz Intel Xeon machines, enough memory Evolved: 8814 solutions found Manual: 8750 solutions found Unique solutions: 466 evolved vs. 386 manual 24

38 Current Work: Diversity Beats Ferocity 25

39 Current Work: Diversity Beats Ferocity 25

40 Current Work: Diversity Beats Ferocity 25

41 Current Work: Diversity Beats Ferocity 25

42 Current Work: Diversity Beats Ferocity 25

43 Current Work: Diversity Beats Ferocity 25

44 Current Work: Diversity Beats Ferocity 25

45 Current Work: Diversity Beats Ferocity 25

46 Current Work: Diversity Beats Ferocity 25

47 Current Work: Diversity Beats Ferocity Idea: Modify fitness function Problems are prey Individual heuristics are predators If several predators catch the same prey, they have to share the benefit = problems solved by no or few heuristics are more valuable = Force diversity of the ecosystem 26

48 Current Work: Diversity Beats Ferocity Idea: Modify fitness function Problems are prey Individual heuristics are predators If several predators catch the same prey, they have to share the benefit = problems solved by no or few heuristics are more valuable = Force diversity of the ecosystem My Bachelor Student Ahmed just started work on this topic - results in 6 months 26

49 Proof Extraction and Learning 27

50 c8 c9 c30 c3 c24 c6 c21 c2 c13 c28 c10 c26 c29 c1 c17 c27 c0 c16 c25 c19 c22 c12 c4 c14 c18 c20 c11 c23 c5 c15 c7 Learning from Proofs and Proof Search Graphs Intuition: Previous proof searches are useful to guide new proof attempts Naive approach: Clauses in the proof tree are positive examples (All other clauses are negative examples) Initial attempts DISCOUNT (Schulz 1995, Schulz&Denzinger 1996) - UEQ, patterns E (Schulz 2000, 2001) - CNF, patterns Overall, modest successes Mostly with positive examples only - compare Otter s hints 28

51 Problems and Solutions Problem: Search protocol size Initial approach: Store all intermediate steps Bad time and space performance Borderline impossible in 2000, still hard today Problem: Not all examples represent search decisions Many intermediate results Also: Vastly unbalanced ratio of positive/negative examples Common solution: Internal proof object (re-)construction Compact representation of the search graph Actually evaluated and picked clauses are recorded Minimal overhead (0.24%) in time Small overhead in memory (due to structure sharing and early discarding of many redundant clauses) 29

52 Proof Generation with Limited Archiving DISCOUNT loop: Only clauses in P are used for inferences U is subject to simplification, but is passive Only clauses in P need to be available in the proof tree g= g? Simplify P (processed clauses) Generate Simplifiable? Cheap Simplify U (unprocessed clauses) 30

53 g=? Simplify Generate Simplifiable? Cheap Simplify Proof Generation with Limited Archiving DISCOUNT loop: Only clauses in P are used for inferences U is subject to simplification, but is passive Only clauses in P need to be available in the proof tree Backward simplification is rare Only clauses in P can be backwards-simplified (and P is small) Heuristically, newer clauses are larger (and big clauses rarely simplify small clauses) g P (processed clauses) U (unprocessed clauses) U (unprocessed clauses) 30

54 g=? Simplify Generate Simplifiable? Cheap Simplify Proof Generation with Limited Archiving DISCOUNT loop: Only clauses in P are used for inferences U is subject to simplification, but is passive Only clauses in P need to be available in the proof tree Backward simplification is rare Only clauses in P can be backwards-simplified (and P is small) Heuristically, newer clauses are larger (and big clauses rarely simplify small clauses) Solution: Non-destructive backwards-simplification Clauses in P are archived on simplification Simplified new clause is build from fresh copy g P (processed clauses) U (unprocessed clauses) U (unprocessed clauses) 30

55 Proof Generation P (processed clauses) g=? Generate Simplifiable? g Cheap Simplify Simplify U (unprocessed clauses) 31

56 Proof Generation P (processed clauses) g=? Generate Simplifiable? A (archive) g Cheap Simplify Simplify U (unprocessed clauses) 31

57 cnf(c_0_8, negated_conjecture, (multiply(b,a)!=c), file('/users/schulz/eprover/tptp_6.0.0_flat/rng008-7.p', prove_commutativity)). cnf(c_0_9, negated_conjecture, (multiply(a,b)=c), file('/users/schulz/eprover/tptp_6.0.0_flat/rng008-7.p', a_times_b_is_c)). cnf(c_0_30, negated_conjecture, ($false), inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[c_0_8, c_0_29]), c_0_9])])). cnf(c_0_3, axiom, (add(x1,add(x2,x3))=add(add(x1,x2),x3)), file('/users/schulz/eprover/tptp_6.0.0_flat/axioms/rng005-0.ax', associativity_for_addition)). cnf(c_0_24, plain, (add(x1,add(x2,x3))=add(x3,add(x1,x2))), inference(pm,[status(thm)],[c_0_2, c_0_3])). cnf(c_0_6, axiom, (multiply(add(x1,x2),x3)=add(multiply(x1,x3),multiply(x2,x3))), file('/users/schulz/eprover/tptp_6.0.0_flat/axioms/rng005-0.ax', distribute2)). cnf(c_0_21, hypothesis, (multiply(add(x1,x2),x1)=add(x1,multiply(x2,x1))), inference(pm,[status(thm)],[c_0_6, c_0_1])). cnf(c_0_2, axiom, (add(x1,x2)=add(x2,x1)), file('/users/schulz/eprover/tptp_6.0.0_flat/axioms/rng005-0.ax', commutativity_for_addition)). cnf(c_0_13, hypothesis, (multiply(add(x1,x2),x2)=add(x2,multiply(x1,x2))), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_6, c_0_1]), c_0_2])). cnf(c_0_26, hypothesis, (add(x1,add(x2,x1))=x2), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_24, c_0_25]), c_0_7])). cnf(c_0_28, hypothesis, (add(x1,multiply(x1,x2))=add(x1,multiply(x2,x1))), inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_13, c_0_26]), c_0_17]), c_0_10]), c_0_3]), c_0_27])). cnf(c_0_29, hypothesis, (multiply(x1,x2)=multiply(x2,x1)), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_27, c_0_28]), c_0_27])). cnf(c_0_1, hypothesis, (multiply(x1,x1)=x1), file('/users/schulz/eprover/tptp_6.0.0_flat/rng008-7.p', boolean_ring)). cnf(c_0_10, hypothesis, (multiply(x1,add(x2,x1))=add(x1,multiply(x1,x2))), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_0, c_0_1]), c_0_2])). cnf(c_0_17, hypothesis, (multiply(x1,add(x1,x2))=add(x1,multiply(x1,x2))), inference(pm,[status(thm)],[c_0_0, c_0_1])). cnf(c_0_27, hypothesis, (add(x1,add(x1,x2))=x2), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_3, c_0_25]), c_0_5])). cnf(c_0_0, axiom, (multiply(x1,add(x2,x3))=add(multiply(x1,x2),multiply(x1,x3))), file('/users/schulz/eprover/tptp_6.0.0_flat/axioms/rng005-0.ax', distribute1)). cnf(c_0_16, hypothesis, (add(x1,multiply(additive_identity,x1))=x1), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_13, c_0_5]), c_0_1])). cnf(c_0_25, hypothesis, (add(x1,x1)=additive_identity), inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_21, c_0_4]), c_0_22]), c_0_23])). cnf(c_0_19, hypothesis, (multiply(additive_identity,additive_inverse(x1))=additive_identity), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_11, c_0_16]), c_0_4])). cnf(c_0_22, hypothesis, (multiply(additive_identity,x1)=additive_identity), inference(pm,[status(thm)],[c_0_19, c_0_15])). cnf(c_0_4, axiom, (add(x1,additive_inverse(x1))=additive_identity), file('/users/schulz/eprover/tptp_6.0.0_flat/axioms/rng005-0.ax', right_additive_inverse)). cnf(c_0_12, hypothesis, (add(x1,multiply(x1,additive_identity))=x1), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_10, c_0_5]), c_0_1])). cnf(c_0_14, hypothesis, (multiply(additive_inverse(x1),additive_identity)=additive_identity), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_11, c_0_12]), c_0_4])). cnf(c_0_18, hypothesis, (multiply(x1,additive_identity)=additive_identity), inference(pm,[status(thm)],[c_0_14, c_0_15])). cnf(c_0_20, hypothesis, (add(x1,multiply(x1,additive_inverse(x1)))=additive_identity), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_17, c_0_4]), c_0_18])). cnf(c_0_5, axiom, (add(additive_identity,x1)=x1), file('/users/schulz/eprover/tptp_6.0.0_flat/axioms/rng005-0.ax', left_additive_identity)). cnf(c_0_11, plain, (add(x1,add(additive_inverse(x1),x2))=x2), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_3, c_0_4]), c_0_5])). cnf(c_0_15, plain, (additive_inverse(additive_inverse(x1))=x1), inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_11, c_0_4]), c_0_7])). cnf(c_0_23, hypothesis, (multiply(additive_inverse(x1),x1)=x1), inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(pm,[status(thm)],[c_0_11, c_0_20]), c_0_7]), c_0_15])). cnf(c_0_7, axiom, (add(x1,additive_identity)=x1), file('/users/schulz/eprover/tptp_6.0.0_flat/axioms/rng005-0.ax', right_additive_identity)). The Structure of Proofs: Commutative Rings 32

58 cnf(c_0_8, negated_conjecture, (multiply(b,a)!=c)). cnf(c_0_9, negated_conjecture, (multiply(a,b)=c)). cnf(c_0_30, negated_conjecture, ($false)). cnf(c_0_3, axiom, (add(x1,add(x2,x3))=add(add(x1,x2),x3))). cnf(c_0_24, plain, (add(x1,add(x2,x3))=add(x3,add(x1,x2)))). cnf(c_0_6, axiom, (multiply(add(x1,x2),x3)=add(multiply(x1,x3),multiply(x2,x3)))). cnf(c_0_21, hypothesis, (multiply(add(x1,x2),x1)=add(x1,multiply(x2,x1)))). cnf(c_0_13, hypothesis, (multiply(add(x1,x2),x2)=add(x2,multiply(x1,x2)))). cnf(c_0_26, hypothesis, (add(x1,add(x2,x1))=x2)). cnf(c_0_28, hypothesis, (add(x1,multiply(x1,x2))=add(x1,multiply(x2,x1)))). cnf(c_0_2, axiom, (add(x1,x2)=add(x2,x1))). cnf(c_0_29, hypothesis, (multiply(x1,x2)=multiply(x2,x1))). cnf(c_0_17, hypothesis, (multiply(x1,add(x1,x2))=add(x1,multiply(x1,x2)))). cnf(c_0_27, hypothesis, (add(x1,add(x1,x2))=x2)). cnf(c_0_1, hypothesis, (multiply(x1,x1)=x1)). cnf(c_0_10, hypothesis, (multiply(x1,add(x2,x1))=add(x1,multiply(x1,x2)))). cnf(c_0_25, hypothesis, (add(x1,x1)=additive_identity)). cnf(c_0_0, axiom, (multiply(x1,add(x2,x3))=add(multiply(x1,x2),multiply(x1,x3)))). cnf(c_0_16, hypothesis, (add(x1,multiply(additive_identity,x1))=x1)). cnf(c_0_19, hypothesis, (multiply(additive_identity,additive_inverse(x1))=additive_identity)). cnf(c_0_22, hypothesis, (multiply(additive_identity,x1)=additive_identity)). cnf(c_0_12, hypothesis, (add(x1,multiply(x1,additive_identity))=x1)). cnf(c_0_4, axiom, (add(x1,additive_inverse(x1))=additive_identity)). cnf(c_0_14, hypothesis, (multiply(additive_inverse(x1),additive_identity)=additive_identity)). cnf(c_0_18, hypothesis, (multiply(x1,additive_identity)=additive_identity)). cnf(c_0_20, hypothesis, (add(x1,multiply(x1,additive_inverse(x1)))=additive_identity)). cnf(c_0_5, axiom, (add(additive_identity,x1)=x1)). cnf(c_0_11, plain, (add(x1,add(additive_inverse(x1),x2))=x2)). cnf(c_0_23, hypothesis, (multiply(additive_inverse(x1),x1)=x1)). cnf(c_0_15, plain, (additive_inverse(additive_inverse(x1))=x1)). cnf(c_0_7, axiom, (add(x1,additive_identity)=x1)). The Structure of Proofs: Commutative Rings 32

59 The Structure of Proofs: Commutative Rings c8 c9 c3 c6 c2 c1 c0 c4 c5 c7 c24 c21 c13 c10 c16 c12 c11 c19 c14 c15 c17 c22 c18 c20 c23 c25 c26 c27 c28 c29 c30 32

60 The Structure of Proofs: Commutative Rings c6 c2 c0 c13 c10 c1 c21 c24 c28 c16 c12 c17 c29 c9 c30 c27 c3 c5 c26 c25 c19 c14 c8 c20 c11 c4 c22 c23 c18 c7 c15 32

61 The Structure of Proofs: Commutative Rings c0 c11 c18 c49 c74 c113 c116 c125 c126 c129 c140 c141 c142 c146 c150 c151 c152 c153 c154 c155 c156 c157 c158 c159 c160 c161 c162 c163 c164 c165 c166 c167 c168 c169 c170 c1 c13 c14 c17 c22 c29 c38 c2 c25 c31 c36 c99 c100 c101 c102 c104 c105 c106 c107 c108 c3 c12 c30 c35 c37 c43 c44 c45 c46 c47 c50 c52 c53 c55 c56 c57 c58 c61 c62 c63 c64 c65 c66 c67 c68 c69 c70 c71 c72 c75 c77 c79 c80 c81 c82 c85 c86 c87 c88 c89 c90 c115 c124 c133 c134 c135 c136 c137 c138 c139 c171 c172 c173 c174 c175 c176 c177 c178 c179 c180 c182 c184 c185 c186 c187 c188 c190 c192 c193 c194 c195 c196 c199 c202 c205 c206 c207 c208 c211 c212 c213 c214 c215 c216 c217 c218 c219 c220 c221 c222 c223 c224 c4 c15 c16 c20 c21 c26 c5 c6 c48 c73 c91 c92 c93 c94 c95 c96 c97 c98 c103 c109 c110 c111 c112 c114 c117 c7 c24 c27 c28 c233 c8 c33 c39 c127 c143 c225 c226 c227 c228 c231 c232 c9 c32 c234 c10 c41 c144 c145 c147 c148 c149 c19 c23 c42 c60 c84 c118 c119 c120 c121 c122 c123 c128 c130 c131 c132 c197 c198 c200 c201 c34 c59 c83 c40 c181 c183 c189 c191 c229 c51 c54 c76 c78 c203 c204 c209 c210 c230 32

62 Classification of Search Decisions Proof state at success: All proof clauses are in P A Clauses in U never contribute All clauses in P A have been selected for processing Positive examples: Proof clauses Negative examples: Non-proof clauses g= g? Simplify P (processed clauses) Generate Simplifiable? U (unprocessed clauses) Cheap Simplify A (archive) 33

63 Classification of Search Decisions Proof state at success: All proof clauses are in P A Clauses in U never contribute All clauses in P A have been selected for processing Positive examples: Proof clauses Negative examples: Non-proof clauses Idea: Apply Machine Learning g= g? Simplify P (processed clauses) Generate Simplifiable? U (unprocessed clauses) Cheap Simplify A (archive) 33

64 Example: Proof Objects RNG008-7 c18). c14). c20). c8). c15). c7). c23). c4). c27). c11). c19). c5). c3). c17). c12). c10). c1). c0). c9). c30). c29). c22). c25). c26). c28). c16). c24). c13). c2). c21). c6). 34

65 Example: Proof Objects RNG008-7 c4). c23). c5). c26). c7). c20). c30). c31). c14). c3). c18). c22). c8). c27). c28). c13). c33). c15). c37). c34). c35). c19). c9). c39). c40). c38). c32). c36). c24). c25). c29). c16). c11). c10). c0). c21). c12). c1). c17). c2). c6). 34

66 Example: Proof Objects RNG008-7 c0). c11). c18). c49). c74). c113). c115). c125). c126). c129). c140). c141). c144). c146). c150). c151). c152). c153). c154). c155). c156). c157). c158). c159). c160). c161). c162). c163). c164). c165). c166). c167). c168). c169). c170). c1). c13). c14). c17). c22). c29). c38). c2). c25). c31). c36). c99). c100). c101). c102). c104). c105). c106). c107). c108). c3). c12). c28). c30). c35). c37). c43). c44). c45). c46). c47). c51). c52). c53). c55). c56). c57). c60). c61). c62). c63). c64). c65). c66). c67). c68). c69). c70). c71). c72). c76). c77). c79). c80). c81). c84). c85). c86). c87). c88). c89). c90). c117). c124). c133). c134). c135). c136). c137). c138). c139). c171). c172). c173). c174). c175). c176). c177). c178). c179). c180). c182). c184). c185). c186). c187). c188). c190). c192). c193). c194). c195). c196). c199). c202). c205). c206). c207). c208). c211). c212). c213). c214). c215). c216). c217). c218). c219). c220). c221). c222). c223). c224). c4). c15). c16). c20). c21). c26). c5). c6). c48). c73). c91). c92). c93). c94). c95). c96). c97). c98). c103). c109). c110). c111). c112). c114). c116). c7). c24). c27). c233). c8). c33). c39). c128). c145). c225). c226). c227). c228). c231). c232). c9). c32). c234). c10). c41). c142). c143). c147). c148). c149). c19). c23). c42). c59). c83). c118). c119). c120). c121). c122). c123). c127). c130). c131). c132). c197). c198). c200). c201). c34). c58). c82). c40). c181). c183). c189). c191). c229). c50). c54). c75). c78). c203). c204). c209). c210). c230). 34

67 Example: Proof Objects RNG008-7 c0). c11). c1). c12). c2). c13). c3). c14). c4). c15). c5). c16). c6). c18). c7). c21). c8). c35). c9). c39). c10). c51). c17). c26). c60). c85). c124). c126). c136). c137). c140). c151). c152). c155). c157). c161). c162). c163). c164). c165). c166). c167). c168). c169). c170). c171). c172). c173). c174). c175). c176). c177). c178). c179). c180). c181). c20). c22). c25). c30). c38). c48). c33). c41). c46). c110). c111). c112). c113). c115). c116). c117). c118). c119). c19). c37). c40). c45). c47). c54). c55). c56). c57). c58). c61). c63). c64). c66). c67). c68). c70). c71). c73). c74). c75). c76). c77). c78). c79). c80). c81). c82). c83). c86). c88). c90). c91). c92). c94). c95). c97). c98). c99). c100). c101). c128). c135). c144). c145). c146). c147). c148). c149). c150). c182). c183). c184). c185). c186). c187). c188). c189). c190). c191). c193). c195). c196). c197). c198). c199). c201). c203). c204). c205). c206). c207). c210). c213). c216). c217). c218). c219). c222). c223). c224). c225). c226). c227). c228). c229). c230). c231). c232). c233). c234). c235). c23). c24). c28). c29). c34). c52). c153). c154). c156). c158). c159). c160). c59). c84). c102). c103). c104). c105). c106). c107). c108). c109). c114). c120). c121). c122). c123). c125). c127). c32). c244). c36). c27). c31). c53). c69). c93). c129). c130). c131). c132). c133). c134). c138). c139). c141). c142). c143). c208). c209). c211). c212). c43). c49). c236). c237). c238). c239). c242). c243). c44). c72). c96). c42). c50). c245). c192). c194). c200). c202). c241). c62). c65). c87). c89). c214). c215). c220). c221). c240). 34

68 Some Initial Results Training examples can be cheaply extracted Ratio of utilized to useless given clauses (GCU-ratio) is a good predictor of Heuristic perfomance (Schulz/Möhrmann, IJCAR 2016) Positive training examples can be automatically written into a watch list and used as hints Clauses on the watchlist are preferred over all other clauses First experiments Reproving with much better GCU-ratio (and much faster) Some improvement even for related problems 35

69 Open Questions Abstractions Are concrete function symbols relevant? Is the concrete term structure relevant? Learning methods Folding architecture networks? Feature-based numerical methods?? Pattern-based learning? Deep learning with convoluted networks? Trade-offs Power vs. convenience Speed vs. quality Online vs. offline costs 36

Feature-based numerical methods?? Pattern-based learning?

70 Open Questions Abstractions Are concrete function symbols relevant? Is the concrete term structure relevant? Learning methods Folding architecture networks? Feature-based numerical methods?? Pattern-based learning? Deep learning with convoluted networks? Trade-offs Power vs. convenience Speed vs. quality Online vs. offline costs Work in Progress 36

71 (Nearly) The End 37

72 Conclusion Controlling proof search for theorem provers is a rich application for machine learning techniques Inductive techniques can be applied at several different levels of search control Explicit proofs can be generated efficiently... and mined for training examples! Proofs are beautiful and informative Learning from proofs may be the future 38

Conclusion Controlling proof search for theorem provers is a rich application for machine learning techniques Inductive techniques can be applied at several different levels of search

73 Conclusion Controlling proof search for theorem provers is a rich application for machine learning techniques Inductive techniques can be applied at several different levels of search control Explicit proofs can be generated efficiently... and mined for training examples! Proofs are beautiful and informative Learning from proofs may be the future Thank you! Questions? 38

74 Image Credit Clipart via Hieronimous Bosch via 39

First-Order Theorem Proving and Vampire. Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester)

First-Order Theorem Proving and Vampire Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester) Outline Introduction First-Order Logic and TPTP Inference Systems