Logic & Formal Reasoning


AI validates evolution's oldest mathematical model

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Understanding Computation - Programmer Books

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Finally, you can learn computation theory and programming language design in an engaging, practical way. Understanding Computation explains theoretical computer science in a context you'll recognize, helping you appreciate why these ideas matter and how they can inform your day-to-day programming. Rather than use mathematical notation or an unfamiliar academic programming language like Haskell or Lisp, this book uses Ruby in a reductionist manner to present formal semantics, automata theory, and functional programming with the lambda calculus.


A Mathematical Model Unlocks the Secrets of Vision Quanta Magazine

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This is the great mystery of human vision: Vivid pictures of the world appear before our mind's eye, yet the brain's visual system receives very little information from the world itself. Much of what we "see" we conjure in our heads. "A lot of the things you think you see you're actually making up," said Lai-Sang Young, a mathematician at New York University. "You don't actually see them." Yet the brain must be doing a pretty good job of inventing the visual world, since we don't routinely bump into doors.


Introduction to the 35th International Conference on Logic Programming Special Issue

arXiv.org Artificial Intelligence

This volume contains the Regular Papers, Technical Communicationsand the Doctoral Consortium papers of the 35th International Conference on Log ic Programming (ICLP 2019), held in Las Cruces, New Mexico, USA, from September 20-25, 2019. Since the first conference held in Marseille in 1982, ICLP has been the premier international event for presenting research in logic programming. Cont ributions are sought in all areas of logic programming, including but not restricted to: Foundations: Semantics, Formalisms, Nonmonotonic reasoning, Knowledge repre sen-tation.


Achievements in Answer Set Programming

arXiv.org Artificial Intelligence

This paper describes an approach to the methodology of answe r set programming [ Marek and Truszczynski, 1999, Niemel a, 1999] that can facilitate the design of encodings that are easy to u nderstand and provably correct. Under this approach, after appending a rule or a small g roup of rules to the emerging program, the programmer would include a comment that states what has been "achieved" so far, in a certain precise sense. Consider, for instance, the following solution to the 8 quee ns problem, adapted from [ Gebser et al., 2012, Section 3.2 ] .


Bridging Commonsense Reasoning and Probabilistic Planning via a Probabilistic Action Language

arXiv.org Artificial Intelligence

To be responsive to dynamically changing real-world environments, an intelligent agent needs to perform complex sequential decision-making tasks that are often guided by commonsense knowledge. The previous work on this line of research led to the framework called "interleaved commonsense reasoning and probabilistic planning" (icorpp), which used P-log for representing commmonsense knowledge and Markov Decision Processes (MDPs) or Partially Observable MDPs (POMDPs) for planning under uncertainty. A main limitation of icorpp is that its implementation requires non-trivial engineering efforts to bridge the commonsense reasoning and probabilistic planning formalisms. In this paper, we present a unified framework to integrate icorpp's reasoning and planning components. In particular, we extend probabilistic action language pBC+ to express utility, belief states, and observation as in POMDP models. Inheriting the advantages of action languages, the new action language provides an elaboration tolerant representation of POMDP that reflects commonsense knowledge. The idea led to the design of the system pbcplus2pomdp, which compiles a pBC+ action description into a POMDP model that can be directly processed by off-the-shelf POMDP solvers to compute an optimal policy of the pBC+ action description. Our experiments show that it retains the advantages of icorpp while avoiding the manual efforts in bridging the commonsense reasoner and the probabilistic planner.


Solving a Flowshop Scheduling Problem with Answer Set Programming: Exploiting the Problem to Reduce the Number of Combinations

arXiv.org Artificial Intelligence

A distinctive characteristic of combinatorial problems is their massive search space. This huge domain is due to the number of possible solutions that although finit e, grows exponentially with the amount of data. Some typical combinatorial problems are the search fo r the cheapest or shortest paths, internet data packets routing, protein structure prediction, and planni ng and scheduling of resources. In theory it is possible to find the optimal solution for each c ombinatorial problem by conducting an exhaustive search. However, in practice finding an optimal s olution is often an intractable problem, even for problems of modest size. In this paper, Answer Set Programming (ASP) is used to explor e how to solve the scheduling problem for an Automated Wet-etch Station (A WS) of a Semiconduct or Manufacturing System where the optimization objective is the makespan. If a robot is not use d to transfer jobs between baths, the problem can be approximated as a special case of the most general n o-wait scheduling flowshop problem. A flowshop is a multistage production process where all jobs m ust pass through the same stages. There is a set J of jobs with J N jobs in total.


A Syntactic Operator for Forgetting that Satisfies Strong Persistence

arXiv.org Artificial Intelligence

Whereas the operation of forgetting has recently seen a considerable amount of attention in the context of Answer Set Programming (ASP), most of it has focused on theoretical aspects, leaving the practical issues largely untouched. Recent studies include results about what sets of properties operators should satisfy, as well as the abstract characterization of several operators and their theoretical limits. However, no concrete operators have been investigated. In this paper, we address this issue by presenting the first concrete operator that satisfies strong persistence - a property that seems to best capture the essence of forgetting in the context of ASP - whenever this is possible, and many other important properties. The operator is syntactic, limiting the computation of the forgetting result to manipulating the rules in which the atoms to be forgotten occur, naturally yielding a forgetting result that is close to the original program. This paper is under consideration for acceptance in TPLP.


Deduction Theorem: The Problematic Nature of Common Practice in Game Theory

arXiv.org Artificial Intelligence

Deduction Theorem: The Problematic Nature of Common Practice in Game Theory Holger I. MEINHARDT † August 2, 2019 We consider the Deduction Theorem that is used in the literature of game theory to run a purported proof by contradiction. In the context of game theory, it is stated that if we have a proof of φ null ϕ, then we also have a proof of φ ϕ. Hence, the proof of φ ϕ is deduced from a previous known statement. However, we argue that one has to manage to prove that the clauses φ and ϕ exist, i.e., they are known true statements in order to establish that φ null ϕ is provable, and that therefore φ ϕ is provable as well. Thus, we are only allowed to reason with known true statements, i.e., we are not allowed to assume that φ or ϕ exist. Doing so, leads immediately to a wrong conclusion. Apart from this, we stress to other facts why the Deduction Theorem is not applicable to run a proof by contradiction. Finally, we present an example from industrial cooperation where the Deduction Theorem is not correctly applied with the consequence that the obtained result contradicts the well-known aggregation issue. MS Classifications 2010: 03B05, 91A12, 91B24 Keywords: Propositional Logic, Deduction Theorem, Herbrand Theorem, Proof by Contradiction, TU Games, Cooperative Oligopoly Games, Partition Function Approach, γ -Belief, Nash Equilibrium, Aggregation across Firms. 1 Introduction We review a common practice in the literature of game theory of applying the Deduction Theorem (Herbrand Theorem, 1930) on a purported proof by contradiction.


An Implementation of a Non-monotonic Logic in an Embedded Computer for a Motor-glider

arXiv.org Artificial Intelligence

In this article we present an implementation of non-monotonic reasoning in an embedded system. As a part of an autonomous motor-glider, it simulates piloting decisions of an airplane. A real pilot must take care not only about the information arising from the cockpit (airspeed, altitude, variometer, compass...) but also from outside the cabin. Throughout a flight, a pilot is constantly in communication with the control tower to follow orders, because there is an airspace regulation to respect. In addition, if the control tower sends orders while the pilot has an emergency, he may have to violate these orders and airspace regulations to solve his problem (e.g. emergency landing). On the other hand, climate changes constantly (wind, snow, hail...) and can affect the sensors. All these cases easily lead to contradictions. Switching to reasoning under uncertainty, a pilot must make decisions to carry out a flight. The objective of this implementation is to validate a non-monotonic model which allows to solve the question of incomplete and contradictory information. We formalize the problem using default logic, a non-monotonic logic which allows to find fixed-points in the face of contradictions. For the implementation, the Prolog language is used in an embedded computer running at 1 GHz single core with 512 Mb of RAM and 0.8 watts of energy consumption.