Key Points Karl Lieberherr

Challenge: old high-level description Price Set of problems 1/5/20162Summary

Challenge: new high-level description Challenge(X): – Price – A constructive belief involving algorithms in domain X and a set of problems in X A constructive belief (X) – A claim about algorithms in domain X A constructive belief defines a protocol to demonstrate that the claim is not supported Bryan: A challenge represents a constructive belief 1/5/20163Summary

Algorithms in a domain The algorithms are about – Solve problems in the domain – Hide secrets in problems of the domain – Find hard problems 1/5/20164Summary

Market Design The market is about constructive beliefs and constructive support for those beliefs. Constructive means that the objects that are claimed to exist must be constructed by an algorithm (constructive mathematics). Reference to Specker’s Theorem. Constructive support involves an interactive protocol (reference to interactive proofs). If the constructive support of a belief fails, the belief is called unsupported. 1/5/20165Summary

Constructive support Demonstrations for unsupported claims must be efficiently checkable. 1/5/20166Summary

Unsupported beliefs and counterexamples An unsupported belief is not a counter example. When a belief is unsupported, it might still be a true fact. Alice might have failed to find the right f in F. When Alice is perfect, she will always be able to support her belief if it is a true fact. When Bob is perfect, he will always be able to show that the belief is unsupported if it is a wrong fact. 1/5/20167Summary

Why beliefs? Beliefs are easier to work with than theorems. 1/5/20168Summary

Project Summary to Slides Traditional benchmarks set standards for the evaluation of software innovation in many disciplines within computer science in both industry and academia. Although necessary, developing benchmarks is challenging and once developed, benchmarks are inflexible, representing only a sample of the application space, and are difficult to maintain as applications typically evolve quickly. The objective of this proposal is to address benchmark problems by creating a dynamic strategy based on {\it artificial markets} of computational challenges. Dynamic market evaluation fosters more innovation through effective feedback, aiding in the discovery of better algorithms and the production of high quality software --- ultimately increasing the overall algorithmic knowledge of a given domain. 1/5/20169Summary

Given a concise description of a computational problem domain, $X$, it will be possible to create an {\it artificial market} around $X$, \market{X}. Software responsible for the creation of the market will generate a starting {\it agent} and a trust-worthy {\it market administrator}. Teams and individuals competing within the problem domain will improve their agent by participating in regular, administrator run competitions. Throughout the course of a competition, each agent offers and solves computational {\it challenges}, and is rewarded for both offering hard problems in $X$, and solving other agent's challenges effectively. Agents are ranked based on their performance within competitions and the corresponding teams can use the log of a competition to discover hard problems for their agent to offer, and gather ideas to improve their agent's algorithms. 1/5/201610Summary

Additional Fundamental to the \market(X) design is the concept of a constructive belief and the concept of supported or unsupported constructive belief. We build on ideas from interactive proofs and program checking (a subarea of interactive proofs). Each challenge represents a constructive belief. Demonstrating that a constructive belief is unsupported does not imply that the belief is wrong. It only implies that the current holder of the belief cannot constructively support the belief. Another holder might be able to support it. 1/5/201611Summary

Additional 2 A constructive belief is specific to a holder. It is like “I, Alice, believe, theorem z holds” and then Alice must interact with a verifier (Bob, who does not believe theorem z holds) to support her belief. 1/5/201612Summary

Additional 3 Outcomes – Z holds Supported: Alice understands Z and she makes it impossible for Bob to attack the belief. Unsupported: Alice does not understand Z and she makes it possible for Bob to attack her belief. Alice is buggy. – Z does not hold Supported: Bob does not understand why Z does not hold and cannot find an attack. Bob is buggy. Unsupported: Bob understands why Z does not hold and he can find an attack. 1/5/201613Summary

Additional 4 True statements may be shown to not be supported. The holder of the true statement (Alice) is buggy. The holder is likely to offer the challenge at a price that is too low. Wrong statements may be shown to be supported. The challenger of the wrong statement (Bob) is buggy. The challenger is likely to accept the challenge at a price that is too high. 1/5/201614Summary

Supporting is weaker than proving Supporting a belief against a perfect challenger is like proving it. To show that a belief is unsupported against a perfect holder is like finding a counterexample. A perfect holder or challenger has unlimited resources and the perfect algorithm. In reality, holders and challengers are not perfect. They will become better by playing the game for a while. 1/5/201615Summary

Witnesses Supporting a belief is like finding a positive witness. Replace “for all” by an object and show that prop holds. To show that a belief is unsupported is like finding a negative witness. 1/5/201616Summary

Algorithms Must be SCG will improve all those properties. Correct No programming errors No conceptual errors Fast Use state-of-the-art techniques 1/5/201617Summary

From teaching to Research Undergraduates need to deal with programming errors and conceptual problems. Researchers need to deal with finding state-of- the-art techniques. 1/5/201618Summary

Beliefs Mathematical – Must be of the form: EAEA… Secret (time t) – You cannot find my secret – You cannot approximate my secret Duel – I solve your problems better than you solve my problems 1/5/201619Summary

Beliefs With beliefs we can express claims and show that they are supported or unsupported even if we don’t have enough information or knowledge to formally prove the claim. A belief, when challenged, is either supported or unsupported. When it is supported against a strong challenger the belief is more likely to hold. When it is unsupported against a strong holder, the belief is more likely to be wrong. 1/5/201620Summary

{\bf Intellectual Merit:} The main objectives of this project are (1) to study the design of artificial markets of computational challenges with specific objectives; (2) show that artificial markets of computational challenges are a useful dynamic benchmark tool for software; and (3) show that they drive innovation, helping to both find better algorithms and improve understanding of their development. The overall goal is to help program officers, managers, and professors with the evaluation of research prototypes, competing software packages, and student programs, respectively, and to help designers improve their algorithms using feedback from the artificial market. The design of particular artificial markets will drive innovation in specific objective areas, ultimately providing targeted feedback. The development of complex agents, administrators, and game generators will be supported by a number of generic programming tools, libraries, and design patterns that push the state-of-the-art. The resulting languages and methodologies are independent of the problem domain and contribute to programmer productivity and software quality. 1/5/201621Summary

{\bf Broader Impacts:} The impact of this project will be through advancing discovery and understanding of how to solve computational problems and providing an exciting platform for interdisciplinary teaching and learning. A generic artificial market supports the improvement of algorithmic techniques for a wide variety of problems through an experiential teaching curriculum. Students become engrossed in learning the art of software development, algorithmic analysis, and even modeling, through the refinement of their agent and frequent competitions. Distributed competitions will be run over the Internet allowing agents from different schools, companies, and even countries to compete within a common domain. The resulting algorithms, improved software, and useful tools, libraries and techniques will benefit software developers, scientific communities, and the general public as they make their way into various products and services. 1/5/201622Summary

Compare belief/theorem Belief B Supporting B is short. Supporting !B is short. Theorem T Proof of T is long. Proof for !T is short. 1/5/2016Summary23

Definition: Proof System Proof system for a language L – P-proof x for y: P(x,y) for y in L. – Soundness: if y has a P-proof, y is in L. – Completeness: If y in L, y has a P-proof. – Effectiveness: P(x,y) is in P / zSzdehn.pdf/dehn-function-and-length.pdf 1/5/2016Summary24

Terminology Adversary (instead of verifier) Supporter (instead of prover) BSS: questioner / supporter Adversary Disprover Questioner Challenger AM: agents challenges Challenging to disprove Challenging beliefs that the challenge represents Questioning challenge 1/5/2016Summary25

Assumptions All beliefs are constructive beliefs. The challenging of a belief is constructive. The beliefs are about properties of algorithms but could be about other topics. 1/5/2016Summary26

Definition: Constructive Belief Support System For statements y of the form (EA)+ in first-order predicate calculus. – Derive interactive protocol from y. x is the outcome of the protocol for y. – P-support x for y: P(x,y). x is the support for y. – P-unsupport x for y: Pun(x,y). x shows that y has no support. – Soundness related Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q. – Completeness related Contra-Weak completeness: If y has P-unsupport with supporter of strength q, y is more likely to be false with confidence q. Completeness: if y holds, y has P-support x, P(x,y), for some x. – Effectiveness: P(x,y) and Pun(x,y) are in P. 1/5/2016Summary27

Properties: Constructive Belief Support System For statements y of the form (EA)+ in first-order predicate calculus. – Derive interactive protocol from y. x is the outcome of the protocol for y. – P-support x for y: P(x,y). x is the support for y. – P-unsupport x for y: Pun(x,y). x shows that y has no support. – For all y there is an x so that either P(x,y) or Pun(x,y) and such an x can be found efficiently by following the protocol. – Soundness related Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q. Perfect soundness: If y has P-support with perfect challenger, y holds. – Completeness related Contra-Weak completeness: If y has P-unsupport with supporter of strength q, y is more likely to be false with confidence q. Contra-Perfect completeness: If y has P-unsupport with perfect supporter, y is false. Completeness: if y holds, y has P-support x, P(x,y), for some x. – Effectiveness: P(x,y) and Pun(x,y) are in P. 1/5/2016Summary28 Supporting a belief against a perfect challenger is like proving it.

Why? To show that a belief is unsupported against a perfect supporter is like finding a counterexample. – Because the perfect supporter will find the hardest object. If the challenger can find an object that negates the predicate, we must have a counterexample. Supporting a belief against a perfect challenger is like proving it. – Because the perfect challenger will find the best object and if this object does not negate the predicate, the supporter must have found the object that the statement claims to exist. 1/5/2016Summary29

Definition: Constructive Belief Support System, secret form For statements y of the secret form: You cannot recover my secret solution for objects satisfying predicate pred. – Derive interactive protocol from y. x is the outcome of the protocol for y. – P-support x for y: P(x,y). x is the support for y. – P-unsupport x for y: Pun(x,y). x shows that y has no support. – Soundness related Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q. – Completeness related Contra-Weak completeness: If y has P-unsupport with supporter of strength q, y is more likely to be false with confidence q. Completeness: if y holds, y has P-support x, P(x,y), for some x. – Effectiveness: P(x,y) and Pun(x,y) are in P. 1/5/2016Summary30

Definition: Constructive Belief Support System, duel form For statements y of the duel form: If you give me k problems (satisfying pred) and I give you k problems satisfying pred, I will solve your problems better than you solve mine. – Derive interactive protocol from y. x is the outcome of the protocol for y. – P-support x for y: P(x,y). x is the support for y. – P-unsupport x for y: Pun(x,y). x shows that y has no support. – Soundness related Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q. – Completeness related Contra-Weak completeness: If y has P-unsupport with supporter of strength q, y is more likely to be false with confidence q. Completeness: if y holds, y has P-support x, P(x,y), for some x. – Effectiveness: P(x,y) and Pun(x,y) are in P. 1/5/2016Summary31

Terminology Challenger supports y. – Challenger cannot find an object negating claim y. Challenger unsupports y. – Challenger finds an object negating claim y. Finding unsupport. Find an object which negates the claim. 1/5/2016Summary32

Properties: Constructive Belief Support System, testing perspective For statements y of the form (EA)+ in first-order predicate calculus. – Derive interactive protocol from y. x is the outcome of the protocol for y. – P-support x for y: P(x,y). x is the support for y. – P-unsupport x for y: Pun(x,y). x shows that y has no support. – For all y there is an x so that either P(x,y) or Pun(x,y) and such an x can be found efficiently by following the protocol. – Soundness related Weak soundness: if y has P-support with challenger of strength q, y is more likely to hold with confidence q. Perfect soundness: If y has P-support with perfect challenger, y holds. – Completeness related Contra-Weak completeness: If y has P-unsupport with supporter of strength q, y is more likely to be false with confidence q. Contra-Perfect completeness: If y has P-unsupport with perfect supporter, y is false. Completeness: if y holds, y has P-support x, P(x,y), for some x. – Testing related If y holds, and challenger unsupports y, supporter is buggy. If y does not hold, and challenger supports y, challenger is buggy. – Effectiveness: P(x,y) and Pun(x,y) are in P. 1/5/2016Summary33 Supporting a belief against a perfect challenger is like proving it.

Advantages of Belief Support System Given a claim, we can quickly find support or unsupport. The Belief Support system feeds the game: – Beliefs which cannot be supported by the supporter, cost money. – Beliefs must be made and put on the market. – Beliefs which are challenged must be challenged successfully, otherwise they cost money. – The stronger the agents become, the better the beliefs become: unchallenged beliefs become truths and challenged beliefs turn into counterexamples. – The game acts as a truth or belief maintenance system for constructive beliefs about algorithms. Bad beliefs are eliminated because agents having bad beliefs lose money and drop out from the game. 1/5/2016Summary34

Challenge / Belief Mathematical challenge: – Constructive mathematical belief Implies protocol for support/unsupport – Price Etc. for other challenges 1/5/2016Summary35

About beliefs y?doi= y?doi= ce_system ce_system 1/5/2016Summary36

Focus of proposal algorithmic competitions for dynamic benchmarks The dynamic benchmarks are created by mathematical beliefs, secret beliefs and duel beliefs. – Mathematical beliefs, optimization problems, EA Specific example: CSP – Secret beliefs, optimization problems Reproduce Approximate within bound Specific example: CSP – Duel beliefs, optimization problems Specific example: CSP 1/5/2016Summary37

Artificial Market and Constructive Belief Support A constructive belief support system is the foundation for the artificial market. – Money is made when a belief is supported: encourages agents to offer only beliefs they can support. Pushes them towards offering “correct” beliefs. – Money is made when a belief is successfully challenged: encourages agents to only challenge beliefs they can successfully challenge. Pushes them towards challenging only “wrong” beliefs. – Efficiency (administrator must be efficient) It must be easy to express successful support and it must be easy to check successful support. It must be easy to express successful challenge and must be easy to check successful support. 1/5/2016Summary38

Difficult Problems for Dynamic Benchmarks The beliefs are about what an algorithm can do. To support the belief means to have a good algorithm. To challenge the belief means to give the algorithm difficult problems. 1/5/2016Summary39

Explaining SCG(X) 1/5/2016Summary40 ConstructiveBeliefSupportSystem(X) ArtificialMarket(X) No prices No game rules Supported and Unsupported Beliefs SCG math (X) SCG secret (X)SCG duel (X) Beliefs about optimization problems Add market rules ArtificialMarket optimization (X)

Market Design Belief Support Systems Artificial Markets 1/5/2016Summary41

Bryan – ? \not\exists x. !sup(x,y) => valid(y) – ? \forall x. sup(x,y) => valid(y) – \exists x. !sup(x,y) => !valid(y) if confidence = 1.0 Karl – \exists x.sup(x,y) = there is support for y 1/5/2016Summary42

For interesting y Sup(x,y) = x supports claim y Even if y is true \exists x: !sup(x,y) Even if y is false \exists x: sup(x,y) 1/5/2016Summary43

There exists a y so that Even if y is true \exists x: !sup(x,y) There exists a y so that Even if y is false \exists x: sup(x,y) 1/5/2016Summary44

New terminology Constructive Belief Support System – Belief: supporter questioner = adversary Artificial Market – Challenge: supporter = challenger questioner = adversary = acceptor 1/5/2016Summary45

Research on Constructive Belief Support Systems Belief support systems for other claims than mathematical (EA)+, secret and duel. Formally defining confidence. Deriving the protocol from the claim form. Further formal properties of belief support systems. 1/5/2016Summary46

Research on Constructive Belief Support Systems and Artificial Markets Different ways of embedding belief support systems into artificial markets. 1/5/2016Summary47

Research on Applicability Range of SCG What is needed for winning? Programming Skills Currently: high, although a baby agent with basic communication skills is generated. Still solid programming skills needed. Future: would like to make it easier to transfer knowledge to agent. Algorithmic Knowledge Good knowledge about successfully supporting and questioning beliefs. For appropriate beliefs turns into: good knowledge about how pose hard problems and how to solve hard problems 1/5/2016Summary48

Special case: Belief form: EA q Beliefs Alice supports belief Bob questions belief Problems Alice poses hard problem Bob solves hard problem 1/5/2016Summary49

Interactive Proof System Constructive Belief Support System Interactive Proof System Prover, all-powerful Verifier, limited resources True statement False statement Completeness – If true, verifier will be convinced Soundness – If false, verifier will not be convinced (small prob.) Constructive Belief Support Supporter, limited resources Questioner, limited resources Supported statement Unsupported statement Completeness – If true, questioning will not succeed Soundness – If false, questioner will succeed 1/5/2016Summary50

How to influence the hearts of computer science students When the students see their agent suffer when playing with the agents of their peers, they want to help their agent. While doing so they learn a lot of computer science skills while transferring their knowledge of the subject domain into their agent. 1/5/2016Summary51

Relevant reference y?doi= y?doi= Reasoning about Beliefs and Actions under Computational Resource Constraints (1987) by Eric J. Horvitz In Proceedings of the 1987 Workshop on Uncertainty in Artificial Intelligence 1/5/2016Summary52

Questions based on Reasoning about Beliefs and Actions under Computational Resource Constraints What is the comprehensive value of computation? – Does more computation help me to create a belief that I can support? – Does more computation help me to successfully question this belief. – Cost/benefit tradeoffs. 1/5/2016Summary53

Value of partial solution The parameter p in belief B(p) must be above p_low and below p_high. There is one right answer. How can we refine this result with increasing amounts of computation? 1/5/2016Summary54

Dempster-Shafer Theory ??? Epistemic or subjective uncertainty (lack of knowledge about a system) 1/5/2016Summary55

Implications between beliefs If x supports belief y1 then x supports belief y2 – E.g.: Ef AJ < 0.6 implies Ef AJ < 0.7 A claim that is never successfully questioned is true? 1/5/2016Summary56

Measure confidence Accumulate evidence What can we learn from a long game history about the confidence in agents and statements? How can we measure the confidence of an agent? Confidence in agents: use something similar to the FIDE rating approach for Chess. 1/5/2016Summary57

Difficult Problems for Dynamic Benchmarks The beliefs are about what an algorithm can do. To support the belief means to have a good algorithm. To challenge the belief means to give the algorithm difficult problems. 1/5/2016Summary58 THIS IS WRONG: SEE NEXT SLIDE

Difficult Problems for Dynamic Benchmarks The beliefs are about what an algorithm can do. Belief is of the form: Exists f in F For all J in fs(f) q(f,J) < c To support the belief means to find hard problems f. To challenge the belief means to efficiently solve hard problems f. 1/5/2016Summary59

extra 1/5/2016Summary60

Relation numbers AND = 128 NAND = 127 OR = 254 NOR = 1 1/5/2016Summary61

Alternative names Hypothesis management system Hypothesis support system 1/5/2016Summary62

Artificial Markets of Computational Challenges Constructive beliefs: supported or unsupported Challenge: (belief(p), price p): I believe I can support belief belief(p) and you cannot successfully question belief(p). If you can, you win 2*p, if you cannot you get 0. In both cases you had to pay p to accept the challenge. 1/5/2016Summary63

Interesting Belief(price,predicate,quality) Computational domain. Predicate selects set of problems. Challenge(Belief(price,predicate,quality), price) If I give you problem p satisfying predicate, you cannot find a solution of quality. If you can, you get 2*price, otherwise 0. The cost of accepting the challenge is price. Belief form: Exists problem: “your algorithm cannot do a task” 1/5/2016Summary64

Belief support system Does not need the notion of a computational domain. Just supported and unsupported beliefs. 1/5/2016Summary65

Artificial Markets Based on beliefs. Creating challenges based on beliefs that you can support, makes money. Accepting challenges based on beliefs that you can successfully question, makes money. Supporter (prover) and questioner (verifier) have both limited resources. If a belief is supported, there is evidence that the belief is a theorem. The stronger the questioner, the stronger the evidence. If a belief is unsupported, there is evidence that the belief is not a theorem. The stronger the supporter, the stronger the evidence. 1/5/2016Summary66

Constructive Belief math Problem domain. Define feasible solutions fs(f) for a problem f. Define quality function q(f,J). Predicate F: defining a subset of problems in domain. Belief math(F, q(f,J) < c, c): Alice believes that if she chooses f in F and Bob chooses J in fs(f), q(f,J) < c (belief is supported by (f,J)). If !(q(f,J)<c), belief is not supported by (f,J). 1/5/2016Summary67

Example Domain: MAX CSP, relations of arity 3 Belief math(F,q(f,J)<c,c): – F: relation one in three – Second parameter: fsat(f,J) < c – c = 0.4 Bob can show that this belief is not supported. If c = 0.5, Alice can prevent Bob from showing that belief is unsupported break-even price 1/5/2016Summary68

Constructive Belief secret Problem domain. Define feasible solutions fs(f) for a problem f. Predicate F: defining a subset of problems in domain. secret(f): maps f to a secret feasible solution. Belief secret(F,q(f,J,Js)): I believe if Alice chooses f in F and secret(f) and Bob chooses J in fs(f) in time t, q(f, J,secret(f)) (belief is supported by (f,J)). If ! q(f, secret(f), J), belief is not supported. 1/5/2016Summary69

Example: secret Belief Domain: MAX CSP, relations of arity 3 Belief secret(F,q(f,J)<c,c): – F: relation one in three – Second parameter: fsat(f,J) < c – c = 0.4 Bob can show that this belief is not supported. If c = 0.5, Alice can prevent Bob from showing that belief is unsupported break-even price 1/5/2016Summary70

Transition to Challenge Belief math(F,q(f,J,c),c) is combined with profit: Alice offers challenge math (belief math (F,q(f,J,c),c), price) – If Bob accepts challenge, he must show that belief is not supported. If he succeeds, he gains g(price, quality(f,J)) If he fails, he looses price. 1/5/2016Summary71

Proposed Research 4 Design 1/5/2016Summary72