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Fast and Loose Semantics for Computational Cognition

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Fast and Loose Semantics for Computational Cognition Logical Formalizations of Commonsense Reasoning: Papers from the 2015 AAAI Spring Symposium Fast and Loose Semantics for Computational Cognition Loizos Michael Open University of Cyprus [email protected] Abstract tics. The questions that arise, the alternatives that one could Psychological evidence supporting the profound effort- have adopted, and the prior works that are relevant to such an lessness (and often substantial carelessness) with which endeavor, are much too numerous to be reasonably covered human cognition copes with typical daily life situations in the constraints of one paper. We view this work as a start- abounds. In line with this evidence, we propose a formal ing point for a fuller investigation of this line of research. semantics for computational cognition that places em- phasis on the existence of naturalistic and unpretentious Perception Semantics algorithms for representing, acquiring, and manipulat- ing knowledge. At the heart of the semantics lies the re- We assume that the environment determines at each moment alization that the partial nature of perception is what ul- in time a state that fully specifies what holds. An agent never timately necessitates — and hinders — cognition. Inex- fully perceives these states. Instead, the agent uses some pre- orably, this realization leads to the adoption of a unified specified language to assign finite names to atoms, which are treatment for all considered cognitive processes, and to used to represent concepts related to the environment. The the representation of knowledge via prioritized implica- set of all atoms is not explicitly provided upfront. Atoms are tion rules. Through discussion and the implementation encountered through the agent’s interaction with its environ- of an early prototype cognitive system, we argue that ment, or introduced through the agent’s cognitive processing such fast and loose semantics may offer a good basis for mechanism. At the neural level, each atom might be thought the development of machines with cognitive abilities. of as a set of neurons that represent a concept (Valiant 2006). A scene s is a mapping from atoms to {0, 1, ∗}. We write Introduction s[α] to mean the value associated with atom α, and call atom The founding statement of Artificial Intelligence (McCarthy α specified in scene s if s[α] ∈{0, 1}. Scenes s1,s2 agree et al. 1955) proceeds on the basis that “every [...] feature of on atom α if s1[α]=s2[α]. Scene s1 is an expansion of intelligence can in principle be so precisely described that a scene s2 if s1,s2 agree on every atom specified in s2. Scene machine can be made to simulate it” and proposes to “find s1 is a reduction of scene s2 if s2 is an expansion of s1.A how to make machines [...] solve kinds of problems now re- scene s is the greatest common reduct of a set S of scenes if served for humans”. Philosophical considerations relating s is the only scene among its expansions that is a reduction to the debate on cognitivism vs. behaviorism aside, it is hard of each scene in S. A set S of scenes is coherent if there not to acknowledge that modern-day Artificial Intelligence exists a scene that is an expansion of each scene in S. research has pushed forward on the latter front more than on In simple psychological terms, a scene can be thought of the former. As readily seen from the exceptionally power- as corresponding to the contents of an agent’s working mem- ful machine learning algorithms and the ingeniously crafted ory, where the agent’s perception of the environment state, reasoning algorithms, more emphasis has been placed on the and any inferences relevant to this perception, are recorded performance of the algorithms, and less emphasis on their for further processing. In line with psychological evidence, design to replicate certain features of the human intelligence. scenes actually used by an agent can be assumed to have a Making progress on the former front presumably necessi- severely-limited number of specified atoms (Miller 1956). tates the abandonment of rigid semantics and convoluted al- A formula ψ is true (resp., false) and specified in s if ψ gorithms, and the investigation of naturalistic and unpreten- (resp., ¬ψ) is classically entailed by the conjunction of each tious solutions, guided by psychological evidence on human atom α such that s[α]=1 and the negation of each atom α cognition. At the same time, and in contrast to the specializa- such that s[α]=0; otherwise, ψ is unspecified in s. tion of much modern-day Artificial Intelligence research, a When convenient, we represent a scene by the set of all holistic view of cognition should be taken, whereby the pro- literals (atoms or their negations) that are true in it, which cesses of perception, reasoning, and learning are considered suffices to fully capture the information available in a scene. in a unified framework that facilitates their close interaction. To formalize the agent’s interaction with its environment, In this spirit, we present a simple unified semantics and an let E denote the set of environments of interest, and let a par- early prototype cognitive system built on top of this seman- ticular environment dist, perc∈E determine: a proba- 114 bility distribution dist over states, capturing the possibly qualified on si by s, and (iii) is not endogenously qualified complex and unknown dynamics with which states are pro- by a rule in κ on si; such a rule r is called dominant in the duced; a stochastic perception process perc determining for step. each state a probability distribution over a coherent subset Intuitively, the truth-values of atoms specified in percept dist, perc of scenes. The agent has only oracle access to s remain as perceived, since they are not under dispute.1 The such that: in unit time the agent senses its environment and truth-values of other atoms in si are updated to incorporate obtains a percept s, resulting by an unknown state t being in si+1 the inferences drawn by dominant rules, and also dist s perc(t) drawn from , and scene then drawn from .A updated to drop any inferences that are no longer supported. s t percept represents, thus, what the agent senses from state . The inferences of a knowledge base on a percept are deter- An agent’s ultimate goal is to act in a manner appropriate mined by the set of scenes that one reaches, and from which for the current state of the environment. We shall not attempt one cannot escape, by repeatedly applying the step operator. to specify here how this appropriateness is defined or quan- tified. Suffices to say that this decision making problem is Definition 3 (Inference Trace and Frontier). The infer- κ s sufficiently arduous, and that the agent can be aided if more ence trace of a knowledge base on a percept is the in- trace(κ,s)=s ,s ,s ,... information on the current state of the environment becomes finite sequence 0 1 2 of scenes, with s = s s κ,s s i ≥ 0 available. Reasoning serves precisely this role: to complete 0 and i i+1 for each integer . The inference κ s information that is not explicitly available in a percept, and frontier of a knowledge base on a percept is the minimal front(κ,s) trace(κ,s) to facilitate, in this manner, the decision making process. To set of scenes found in a suffix of . do so, it utilizes knowledge that the agent has been given, or Lemma 1 (Properties of Inference Frontier). front(κ,s) has acquired, on certain regularities in the environment. is unique and finite, for any knowledge base κ and percept s. Reasoning Semantics In general, the inference frontier includes multiple scenes, Since reasoning aims to complete information, it should be and one can define many natural entailment notions. computationally easy to determine what inferences follow Definition 4 (Entailment Notions). A knowledge base κ from the agent’s knowledge, and inferences should follow applied on a percept s entails a formula ψ if ψ is: often. We present our chosen representation for this knowl- (E1) true in a scene in front(κ,s); edge, and defer the discussion of its properties for later. (E2) true in a scene in front(κ,s) and not false in others; A rule is an expression of the form ϕ λ, where formula (E3) true in every scene in front(κ,s); ϕ is the body of the rule, and literal λ is the head of the (E4) true in the greatest common reduct of front(κ,s). rule, with ϕ and λ not sharing any atoms, and with ϕ being read-once (no atom appears more than once). The intuitive Going from the first to the last notion, entailment becomes reading of a rule is that when the rule’s body holds in a scene, more skeptical. Only the first notion of entailment captures an agent has evidence that the rule’s head should also hold. what one would typically call credulous entailment, in that A collection of rules could happen to simultaneously pro- ψ is possible, but ¬ψ might also be possible. The following vide evidence for conflicting conclusions. To resolve such result clarifies the relationships between these notions. conflicts, we let rules be qualified based on their priorities. Proposition 2 (Relationships Between Entailment No- A knowledge base κ = , over a set R of rules com- tions). A knowledge base κ applied on a percept s entails ψ prises a finite collection ⊆Rof rules, and an irreflexive under Ei if it entails ψ under Ej, for every pair of entailment antisymmetric priority relation that is a subset of × .
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