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The Logic of Intelligence Pei Wang Department of Computer and Information Sciences Temple University

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Artificial General Intelligence Mainstream AI treats “Intelligence” as a collection of problem-specific and domain-specific parts Artificial General Intelligence (AGI) takes “Intelligence” as a general-purpose capability that should be treated as a whole AGI research still includes different research objectives and strategies

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Artificial Intelligence and Logic “Intelligence” can be understood as “rationality” and “validity” --- “do the right thing” In general, “logic” is the study of valid reasoning, or the regularity in thinking Therefore, an AI system may be built according to a logic, by converting various thinking processes into reasoning processes

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Reasoning System A reasoning system typically consists of the following major components: a formal language a semantic theory a set of inference rules a memory structure a control mechanism The first three are usually called a “logic”

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Traditional Theories Language and inference rules: first-order predicate calculus Semantics: model theory Memory: relational or object-oriented data structures and database Inference control: theory of computation (algorithm, computability, and computational complexity)

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Problems of Traditional Theories Uncertainty: fuzzy concepts, changing meanings and truth values, plausible results, conflicting evidence, nondeterministic inference process, … Semantic justification of non-deductive inference: induction, abduction, analogy, … Counter-intuitive results: sorites paradox, implication paradox, confirmation paradox, Wason’s selection task, … Computability and complexity: termination problem, combinatorial explosion, …

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Proposed Solutions non-monotonic logic paraconsistent logic relevance logic probabilistic logic fuzzy logic inductive logic temporal logic modal logic situation calculus possible world theory mental logic mental model case-based reasoning Bayesian network neural network genetic algorithm heuristic algorithm learning algorithm anytime algorithm … …

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Common Root of the Problems The traditional theories were developed in the study of the foundation of mathematics, while the problems appear outside math The logic of mathematics may be different from the logic of cognition In mathematical reasoning, the knowledge and resources are assumed to be sufficient (with respect to the tasks)

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Different Types of Systems “Pure-axiomatic system”: the system’s knowledge and resources are assumed to be sufficient “Semi-axiomatic system”: certain aspects (but not all) of the knowledge and resources are assumed to be sufficient “Non-axiomatic system”: the knowledge and resources of the system are assumed to be generally insufficient

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NARS (Non-Axiomatic Reasoning System) NARS uses a formal logic (language, semantics, inference rules) and is implemented in a computer system NARS is fully based on the assumption of insufficient knowledge and resources, in the sense of being a finite, real time, open, and adaptive system NARS is different from traditional theories in all major components

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Inheritance Based Representation S P : there is an inheritance relation from term S to term P S is a specialization of P P is a generalization of S Inheritance is reflexive and transitive birdanimal

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Extension and Intension For a given term T, its extension T E = {x | x T} its intension T I = {x | T x} T TETE TITI Theorem: (S P) (S E P E ) (P I S I ) Therefore, “Inheritance” means “inheritance of extension/intension”

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Evidence Positive evidence of S P : {x | x (S E P E ) (P I S I )} Negative evidence of S P : {x | x (S E – P E ) (P I – S I )} SP Amount of evidence: positive: w + = | S E P E | + | P I S I | negative: w – = | S E – P E | + | P I – S I | total: w = w + + w – = | S E | + | P I |

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Truth Value In NARS, the truth value of a statement is a pair of numbers, and measures the evidential support to the statement. SP [f, c] S P [f, c] f: frequency, w + /w c: confidence, w / (w +1)

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Experience-Grounded Semantics The truth value of a statement is defined according to certain “idealized experience”, consisting of a set of binary inheritance statements The meaning of a term is defined by its extension and intension, according to certain “idealized experience” So meaning and truth-value changes according to the system’s experience

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Syllogistic Inference Rules A typical syllogistic inference rule takes a pair of premises with a common term, and produces a conclusion The truth value of the conclusion is calculated by a truth-value function Different combinations of premises trigger different rules (with different truth-value functions)

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To Design a Truth-value Function 1. Treat all involved variables as Boolean (binary) variables 2. For each value combination in premises, decide the values in conclusion 3. Build Boolean functions among the variables 4. Extend the functions to real-number: not (x) = 1 – x and (x, y) = x * y or (x, y) = 1 – (1 – x) * (1 – y)

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Deduction bird animal [1.00, 0.90] robin bird [1.00, 0.90] robin animal [1.00, 0.81] M P [f 1, c 1 ] S M [f 2, c 2 ] S P [f, c] f = f 1 * f 2 c = c 1 * c 2 * f 1 * f 2 M SP

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Induction swan bird [1.00, 0.90] swan swimmer [1.00, 0.90] bird swimmer [1.00, 0.45] M P [f 1, c 1 ] M S [f 2, c 2 ] S P [f, c] f = f 1 c = f 2 * c 1 * c 2 / (f 2 * c 1 * c 2 + 1) S M P

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Abduction seabird swimmer [1.00, 0.90] gull swimmer [1.00, 0.90] gull seabird [1.00, 0.45] P M [f 1, c 1 ] S M [f 2, c 2 ] S P [f, c] f = f 2 c = f 1 * c 1 * c 2 / (f 1 * c 1 * c 2 + 1) S M P

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Revision bird swimmer [1.00, 0.62] bird swimmer [0.00, 0.45] bird swimmer [0.67, 0.71] S P [f 1, c 1 ] S P [f 2, c 2 ] S P [f, c] f = f 1 * c 1 * (1 - c 2 ) + f 2 * c 2 * (1 - c 1 ) c 1 * (1 - c 2 ) + c 2 * (1 - c 1 ) c 1 * (1 - c 2 ) + c 2 * (1 - c 1 ) + (1 - c 2 ) * (1 - c 1 ) c = S P

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Other Inference Rules M P [f 1, c 1 ] S M [f 2, c 2 ] S P [f, c] analogy union implication P M [f 1, c 1 ] S M [f 2, c 2 ] (S P) M [f, c] B C [f 1, c 1 ] A B [f 2, c 2 ] A C [f, c]

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Other Relations and Inheritance An arbitrary statement R(a, b, c) can be rewritten as inheritance relations with compound terms: (*, a, b, c) R “The relation among a, b, c is a kind of R.” a (/, R, _, b, c) “a is such an x that satisfies R(x, b, c ).” b (/, R, a, _, c) “b is such an x that satisfies R(a, x, c ).” c (/, R, a, b, _) “c is such an x that satisfies R(a, b, x ).”

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Memory as a Belief Network bird gull swan robin swimmer crow feathered_creature [1.00, 0.90] [0.00, 0.90] [1.00, 0.90] The knowledge of the system is a network of beliefs among terms. A term with all of its beliefs is a concept C bird

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Inference Tasks NARS accepts several types of inference tasks: Knowledge to be absorbed Questions to be answered Goals to be achieved A task is stored in the corresponding concepts To process each task means letting it interacts with the available beliefs in the concept This process usually generates new tasks, beliefs, and concepts, recursively

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Inference Process NARS runs by repeating the following cycle: 1. Choose a concept within the memory 2. Choose a task within the concept 3. Choose a belief within the concept 4. Use inference rules to produce new tasks 5. Return the used items to memory 6. Add the new tasks into the memory and provide an answer if available

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Control Strategy NARS maintains priority distributions among data items, uses them to make choice, and adjusts them after each step Factors influence priority: quality of the item usefulness of the item in history relevance of the item to the current context

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Architecture and Working Cycle

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Design and Implementation The conceptual design of NARS has been described in a series of publications Most parts of the design have been implemented in several prototypes, and the current version is open source in Java Working examples exist as proof of concept, and only cover single-step inference or short inference processes The project is on-going, though has produced novel and interesting results

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Unified Solutions The truth value uniformly represents various kinds of uncertainty The truth value depends on both positive and negative evidence The non-deductive inference rules is justified according to the semantics The meaning of a term is determined by its experienced relations with other terms With syllogistic rules, the premises and conclusions must be semantically related The inference processes in NARS does not follow predetermined algorithms

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Conclusions It is possible to build a reasoning system that adapts to its environment, and works with insufficient knowledge and resources Such a system provides a unified solution to many problems in A(G)I There is a logic of intelligence, though it is fundamentally different from the logic of mathematics

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Formal Models in AGI Research Pei Wang Temple University Philadelphia, USA.

Formal Models in AGI Research Pei Wang Temple University Philadelphia, USA.

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