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Detailed requirements for the real-time computer program to be employed in support of the Apollo missions beginning with the G Mission (CSM 107/LM-6).

Detailed requirements for the real-time computer program to be employed in support of the Apollo missions beginning with the G Mission (CSM 107/LM-6).

This preprint has been withdrawn.

This preprint has been withdrawn.

The desire to understand things is what drives Rod Downey in his work in computational mathematics. In this interview he talks about the use of mathematics in areas as diverse as industrial smelting, computer chip manufacturing and tumble drier sensors.

Here's a summary of logic: What is Logic? Logic is the systematic study of the principles of valid inference and reasoning. It involves the use of language, symbols, and rules to evaluate arguments and arrive at conclusions. Key Concepts: 1. Propositions: Statements that can be true or false. 2. Arguments: Sets of propositions used to support a conclusion. 3. Inference: The process of deriving a conclusion from premises. 4. Validity: An argument is valid if its conclusion follows logically from its premises. 5. Soundness: An argument is sound if it is both valid and has true premises. Types of Logic: 1. Deductive Logic: Concerned with certain and absolute conclusions. 2. Inductive Logic: Concerned with probable conclusions based on evidence. 3. Abductive Logic: Concerned with making educated guesses or hypotheses. 4. Modal Logic: Concerned with possibility, necessity, and obligation. 5. Non-Classical Logic: Includes fuzzy logic, intuitionistic logic, and others. Logical Operators: 1. Conjunction (AND) 2. Disjunction (OR) 3. Negation (NOT) 4. Implication (IF-THEN) 5. Equivalence (IF AND ONLY IF) Rules of Inference: 1. Modus Ponens 2. Modus Tollens 3. Hypothetical Syllogism 4. Disjunctive Syllogism 5. Conjunctive Syllogism Fallacies: 1. Ad Hominem 2. Straw Man 3. False Dilemma 4. Slippery Slope 5. Begging the Question   Early 19th Century (1800-1840)   - George Boole (1815-1864):     - Published "The Mathematical Analysis of Logic" (1847)     - Developed symbolic logic, using algebraic methods to represent logical operations     - Introduced the concept of a "logical algebra" - Augustus De Morgan (1806-1871):     - Published "Formal Logic" (1847)     - Developed relational logic, emphasizing the importance of relations between terms     - Introduced the concept of "De Morgan's laws"   Mid-19th Century (1840-1870)   - William Hamilton (1788-1856):     - Published "Lectures on Logic" (1833)     - Developed the concept of "quantification of the predicate"     - Influenced the development of mathematical logic - Charles Babbage (1791-1871):     - Proposed the idea of a mechanical computer, the Analytical Engine     - Recognized the importance of logic in computer design   Late 19th Century (1870-1900)   - Gottlob Frege (1848-1925):     - Published "Begriffsschrift" (1879)     - Developed modern predicate logic, using quantifiers and variables     - Introduced the concept of "sense and reference" - Ernst Schröder (1841-1902):     - Published "Vorlesungen über die Algebra der Logik" (1890-1905)     - Developed the algebra of logic, using lattice theory     - Influenced the development of mathematical logic - Giuseppe Peano (1858-1932):     - Published "Arithmetices principia, nova methodo exposita" (1889)     - Developed the Peano axioms for arithmetic     - Introduced the concept of "formal language"  

This article is from Nucleic Acids Research , volume 41 . Abstract Biological computing circuits can enhance our ability to control cellular functions and have potential applications in tissue engineering and medical treatments. Transcriptional activator-like effectors (TALEs) represent attractive components of synthetic gene regulatory circuits, as they can be designed de novo to target a given DNA sequence. We here demonstrate that TALEs can perform Boolean logic computation in mammalian cells. Using a split-intein protein-splicing strategy, we show that a functional TALE can be reconstituted from two inactive parts, thus generating two-input AND logic computation. We further demonstrate three-piece intein splicing in mammalian cells and use it to perform three-input AND computation. Using methods for random as well as targeted insertion of these relatively large genetic circuits, we show that TALE-based logic circuits are functional when integrated into the genome of mouse embryonic stem cells. Comparing construct variants in the same genomic context, we modulated the strength of the TALE-responsive promoter to improve the output of these circuits. Our work establishes split TALEs as a tool for building logic computation with the potential of controlling expression of endogenous genes or transgenes in response to a combination of cellular signals.

Recent experimental advances have demonstrated technologies capable of supporting scalable quantum computation. A critical next step is how to put those technologies together into a scalable, fault-tolerant system that is also feasible. We propose a Quantum Logic Array (QLA) microarchitecture that forms the foundation of such a system. The QLA focuses on the communication resources necessary to efficiently support fault-tolerant computations. We leverage the extensive groundwork in quantum error correction theory and provide analysis that shows that our system is both asymptotically and empirically fault tolerant. Specifically, we use the QLA to implement a hierarchical, array-based design and a logarithmic expense quantum-teleportation communication protocol. Our goal is to overcome the primary scalability challenges of reliability, communication, and quantum resource distribution that plague current proposals for large-scale quantum computing.

Includes bibliographical references (p. 257-260) and index

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Process algebra and temporal logic are two popular paradigms for the specification, verification and systematic development of reactive and concurrent systems. These two approaches take different standpoint for looking at specifications and verifications, and offer complementary advantages. In order to mix algebraic and logic styles of specification in a uniform framework, the notion of a logic labelled transition system (LLTS) has been presented and explored by Luttgen and Vogler. This paper intends to propose a LLTS-oriented process calculus which, in addition to usual process-algebraic operators, involves logic connectives (conjunction and disjunction) and standard temporal operators (always and unless). This calculus preserves usual properties of these logic operators, allows one to freely mix operational and logic operators, and supports compositional reasoning. Moreover, the links between this calculus and Action-based Computation Tree Logic (ACTL) including characteristic formulae of process terms, characteristic processes of ACTL formulae and Galois connection are explored.

Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics: Knot Theory, Categorical Quantum Mechanics, and Logic and Computation. We shall focus in particular on the following two topics: - The Temperley-Lieb algebra has always hitherto been presented as a quotient of some sort: either algebraically by generators and relations as in Jones' original presentation, or as a diagram algebra modulo planar isotopy as in Kauffman's presentation. We shall use tools from Geometry of Interaction, a dynamical interpretation of proofs under Cut Elimination developed as an off-shoot of Linear Logic, to give a direct description of the Temperley-Lieb category -- a "fully abstract presentation", in Computer Science terminology. This also brings something new to the Geometry of Interaction, since we are led to develop a planar version of it, and to verify that the interpretation of Cut-Elimination (the "Execution Formula", or "composition by feedback") preserves planarity. - We shall also show how the Temperley-Lieb algebra provides a natural setting in which computation can be performed diagrammatically as geometric simplification -- "yanking lines straight". We shall introduce a "planar lambda-calculus" for this purpose, and show how it can be interpreted in the Temperley-Lieb category.

Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics: Knot Theory, Categorical Quantum Mechanics, and Logic and Computation. We shall focus in particular on the following two topics: - The Temperley-Lieb algebra has always hitherto been presented as a quotient of some sort: either algebraically by generators and relations as in Jones' original presentation, or as a diagram algebra modulo planar isotopy as in Kauffman's presentation. We shall use tools from Geometry of Interaction, a dynamical interpretation of proofs under Cut Elimination developed as an off-shoot of Linear Logic, to give a direct description of the Temperley-Lieb category -- a "fully abstract presentation", in Computer Science terminology. This also brings something new to the Geometry of Interaction, since we are led to develop a planar version of it, and to verify that the interpretation of Cut-Elimination (the "Execution Formula", or "composition by feedback") preserves planarity. - We shall also show how the Temperley-Lieb algebra provides a natural setting in which computation can be performed diagrammatically as geometric simplification -- "yanking lines straight". We shall introduce a "planar lambda-calculus" for this purpose, and show how it can be interpreted in the Temperley-Lieb category.

Much work has been done on extending the well-founded semantics to general disjunctive logic programs and various approaches have been proposed. However, these semantics are different from each other and no consensus is reached about which semantics is the most intended. In this paper we look at disjunctive well-founded reasoning from different angles. We show that there is an intuitive form of the well-founded reasoning in disjunctive logic programming which can be characterized by slightly modifying some exisitng approaches to defining disjunctive well-founded semantics, including program transformations, argumentation, unfounded sets (and resolution-like procedure). We also provide a bottom-up procedure for this semantics. The significance of our work is not only in clarifying the relationship among different approaches, but also shed some light on what is an intended well-founded semantics for disjunctive logic programs.

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.

Generalized Quantifiers and Computation: 9th European Summer School in Logic, Language, and Information ESSLLI’97Workshop Aix-en-Provence, France, August 11–22, 1997 Revised Lectures Author: Jouko Väänänen Published by Springer Berlin Heidelberg ISBN: 978-3-540-66993-7 DOI: 10.1007/3-540-46583-9 Table of Contents: Generalized Quantifiers, an Introduction Counting and Locality over Finite Structures A Survey A Perspective on Lindström Quantifiers and Oracles Generalized Quantifiers in Logic Programs A Generalized Quantifier Concept in Computational Complexity Theory Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers

Generalized Quantifiers and Computation: 9th European Summer School in Logic, Language, and Information ESSLLI’97Workshop Aix-en-Provence, France, August 11–22, 1997 Revised Lectures Author: Jouko Väänänen Published by Springer Berlin Heidelberg ISBN: 978-3-540-66993-7 DOI: 10.1007/3-540-46583-9 Table of Contents: Generalized Quantifiers, an Introduction Counting and Locality over Finite Structures A Survey A Perspective on Lindström Quantifiers and Oracles Generalized Quantifiers in Logic Programs A Generalized Quantifier Concept in Computational Complexity Theory Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers

Generalized Quantifiers and Computation: 9th European Summer School in Logic, Language, and Information ESSLLI’97Workshop Aix-en-Provence, France, August 11–22, 1997 Revised Lectures Author: Jouko Väänänen Published by Springer Berlin Heidelberg ISBN: 978-3-540-66993-7 DOI: 10.1007/3-540-46583-9 Table of Contents: Generalized Quantifiers, an Introduction Counting and Locality over Finite Structures A Survey A Perspective on Lindström Quantifiers and Oracles Generalized Quantifiers in Logic Programs A Generalized Quantifier Concept in Computational Complexity Theory Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers

Generalized Quantifiers and Computation: 9th European Summer School in Logic, Language, and Information ESSLLI’97Workshop Aix-en-Provence, France, August 11–22, 1997 Revised Lectures Author: Jouko Väänänen Published by Springer Berlin Heidelberg ISBN: 978-3-540-66993-7 DOI: 10.1007/3-540-46583-9 Table of Contents: Generalized Quantifiers, an Introduction Counting and Locality over Finite Structures A Survey A Perspective on Lindström Quantifiers and Oracles Generalized Quantifiers in Logic Programs A Generalized Quantifier Concept in Computational Complexity Theory Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers

Flowcharts for the software for the lunar landing mission real-time (ground control) computer system.

Flowcharts for the software for the lunar landing mission real-time (ground control) computer system.

We present a nondeterministic model of computation based on reversing edge directions in weighted directed graphs with minimum in-flow constraints on vertices. Deciding whether this simple graph model can be manipulated in order to reverse the direction of a particular edge is shown to be PSPACE-complete by a reduction from Quantified Boolean Formulas. We prove this result in a variety of special cases including planar graphs and highly restricted vertex configurations, some of which correspond to a kind of passive constraint logic. Our framework is inspired by (and indeed a generalization of) the ``Generalized Rush Hour Logic'' developed by Flake and Baum. We illustrate the importance of our model of computation by giving simple reductions to show that several motion-planning problems are PSPACE-hard. Our main result along these lines is that classic unrestricted sliding-block puzzles are PSPACE-hard, even if the pieces are restricted to be all dominoes (1x2 blocks) and the goal is simply to move a particular piece. No prior complexity results were known about these puzzles. This result can be seen as a strengthening of the existing result that the restricted Rush Hour puzzles are PSPACE-complete, of which we also give a simpler proof. Finally, we strengthen the existing result that the pushing-blocks puzzle Sokoban is PSPACE-complete, by showing that it is PSPACE-complete even if no barriers are allowed.

We study the following problem: given a class of logic programs C, determine the maximum number of stable models of a program from C. We establish the maximum for the class of all logic programs with at most n clauses, and for the class of all logic programs of size at most n. We also characterize the programs for which the maxima are attained. We obtain similar results for the class of all disjunctive logic programs with at most n clauses, each of length at most m, and for the class of all disjunctive logic programs of size at most n. Our results on logic programs have direct implication for the design of algorithms to compute stable models. Several such algorithms, similar in spirit to the Davis-Putnam procedure, are described in the paper. Our results imply that there is an algorithm that finds all stable models of a program with n clauses after considering the search space of size O(3^{n/3}) in the worst case. Our results also provide some insights into the question of representability of families of sets as families of stable models of logic programs.

An MSC (NASA Manned Spacecraft Center) internal note that "presents the logic and equations defining certain subroutines which are called by the real-time computer programs for the [Apollo] AS-207/208 mission.  The subroutines compute the launch vehicle insertion elements for the launch targeting processor and compute several specialized maneuvers for the rendezvous planning processors.  Detailed flow charts are included in the appendix."

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q, though Tarski proved that this theory is decidable in the domains of R and C. The theory of multiplication and order (x,

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q, though Tarski proved that this theory is decidable in the domains of R and C. The theory of multiplication and order (x,

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q, though Tarski proved that this theory is decidable in the domains of R and C. The theory of multiplication and order (x,

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of addition and multiplication is undecidable in the domains of N, Z and Q, though Tarski proved that this theory is decidable in the domains of R and C. The theory of multiplication and order (x,

The composite rotation approach has been used to develop a range of robust quantum logic gates, including single qubit gates and two qubit gates, which are resistant to systematic errors in their implementation. Single qubit gates based on the BB1 family of composite rotations have been experimentally demonstrated in a variety of systems, but little study has been made of their application in extended computations, and there has been no experimental study of the corresponding robust two qubit gates to date. Here we describe an application of robust gates to Nuclear Magnetic Resonance (NMR) studies of approximate quantum counting. We find that the BB1 family of robust gates is indeed useful, but that the related NB1, PB1, B4 and P4 families of tailored logic gates are less useful than initially expected.

We establish a connection between measurement-based quantum computation and the field of mathematical logic. We show that the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which can yield a computational speed-up with respect to classical computation, the underlying graphs--describing the quantum correlations of the states--are associated with undecidable logic theories. Here undecidability is to be interpreted in a sense similar to Goedel's incompleteness results, meaning that there exist propositions, expressible in the above classical formal logic, which cannot be proven or disproven.

The difficulty of explaining non-local correlations in a fixed causal structure sheds new light on the old debate on whether space and time are to be seen as fundamental. Refraining from assuming space-time as given a priori has a number of consequences. First, the usual definitions of random- ness depend on a causal structure and turn meaningless. So motivated, we propose an intrinsic, physically motivated measure for the randomness of a string of bits: its length minus its normalized work value, a quantity we closely relate to its Kolmogorov complexity (the length of the shortest program making a universal Turing machine output this string). We test this alternative concept of randomness for the example of non-local correlations, and we end up with a reasoning that leads to similar conclusions as, but is more direct than, in the probabilistic view since only the outcomes of measurements that can actually all be carried out together are put into relation to each other. In the same context-free spirit, we connect the logical reversibility of an evolution to the second law of thermodynamics and the arrow of time. Refining this, we end up with a speculation on the emergence of a space-time structure on bit strings in terms of data-compressibility relations. Finally, we show that logical consistency, by which we replace the abandoned causality, it strictly weaker a constraint than the latter in the multi-party case.

The satisfiability problem of the branching time logic CTL is studied in terms of computational complexity. Tight upper and lower bounds are provided for each temporal operator fragment. In parallel, the minimal model size is studied with a suitable notion of minimality. Thirdly, flat CTL is investigated, i.e., formulas with very low temporal operator nesting depth. A sharp dichotomy is shown in terms of complexity and minimal models: Temporal depth one has low expressive power, while temporal depth two is equivalent to full CTL.

We use classes of Hilbert lattice equations for an alternative representation of Hilbert lattices and Hilbert spaces of arbitrary quantum systems that might enable a direct introduction of the states of the systems into quantum computers. More specifically, we look for a way to feed a quantum computer with algebraic equations of n-th order underlying an infinite dimensional Hilbert space description of quantum systems. A number of new results on states defined on Hilbert lattices are presented and discussed and a number of recently obtained results in the field of Hilbert space equations are reviewed.