It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.

We survey fundamental concept in inverse programming and present the Universal Resolving Algorithm (URA), an algorithm for inverse computation in a first-order, functional programming language. We discusst he principles behind the algorithm, including a three-step approach based on the notion of a perfect process tree, and demonstrate our implementation with several examples. We explaint he idea of a semantics modifier for inverse computation which allows us to perform inverse computation in other programming languages via interpreters.

Abstract 2 Most autonomous agents are situated in a social context and need to interact with other agents (both human and artificial) to complete their problem solving objectives. Such agents are usually capable of performing a wide range of actions and engaging in a variety of social interactions. Faced with this variety of options, an agent must decide what to do. There are many potential decision making functions that could be employed to make the choice. Each such function will have a different effect on the success of the individual agent and of the overall system in which it is situated. To this end, this thesis examines agents ’ decision making functions to ascertain their likely properties and attributes. A novel framework for characterising social decision making is presented which provides explicit reasoning about the potential benefits of the individual agent, particular sub-groups of agents or the overall system. This framework enables multi-farious social interaction attitudes to be identified and defined; ranging from the purely self-interested to the purely altruistic. In particular, however, the focus is on the spectrum of socially responsible agent behaviours in which agents attempt to balance their own needs with those of the overall system. Such behaviour aims to ensure that both the agent and the overall system perform well.

In this paper, we continue to develop our approach to theorem proof search in the EA-style, that is theorem proving in the framework of integrated processing mathematical texts written in a 1st-order formal language close to the natural language used in mathematical papers. This framework enables constructing a sound and complete goal-oriented sequent-type calculus with "large-block " inference rules. In particular, it contains the formal analogs of such natural proof search techniques as definition handling and auxiliary proposition application. The calculus allows to incorporate symbolic computations in an inference search.

In this paper we address the problem of enriching an interactive theorem prover with complex proof procedures. We show that the approach of building complex proof procedures out of deciders for (decidable) quantifier-free theories has many advantages: (i) deciders for quantifier-free theories provide powerful, high level functionalities which greatly simplify the activity of designing and implementing complex and higher level proof procedures; (ii) this approach is of wide applicability since most of the proof procedures are composed by steps of propositional reasoning intermixed with steps carrying out higher level strategical functionalities; (iii) decidability and efficiency are retained on important (decidable) subclasses, while they are often sacrificed by uniform proof strategies for the sake of generality; finally (iv), from a software engineering perspective, the modularity of the procedures guarantees that any modification in the implementation can be accomplished ...

Lee and Plaisted recently developed a new automated theorem proving strategy called hyper-linking. As part of his dissertation, Lee developed a round-by-round implementation of the hyper-linking strategy, which competes well with other automated theorem provers on a wide range of theorem proving problems. However, Lee's round-by-round implementation of hyper-linking is not particularly well suited for the addition of special methods in support of equality. In this dissertation, we describe, as alternative to the round-by-round hyper-linking implementation of Lee, a smallest instance first implementation of hyper-linking which addresses many of the inefficiencies of round-by-round hyper-linking encountered when adding special methods in support of equality. Smallest instance first hyper-linking is based on the formalization of generating smallest clauses first, a heuristic widely used in other automated theorem provers. We prove both the soundness and logical completeness of smallest instance first hyper-linking and show that it always generates smallest clauses first under

In this paper we present a complete decider for the subclass of first order logic (FOL) of the prenex universal-existential formulas not containing function symbols. The decider is composed of a set of decision procedures for simpler subclasses of FOL. By looking at the structure of the formula to be proved, different decision procedures (with different time complexity) are applied. Each decision procedure is built on top of one another, down to the propositional decider which is the core of the whole system. Any procedure (except the propositional one) rewrites the input formula onto a logically equivalent formula belonging to a simpler class. A claim is made that the underlying hypotheses have allowed to construct a highly structured, very efficient decider.

Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for

The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.

A propositionalization of a theory in First-Order Logic (FOL) is a set of propositional sentences that is satisfiable iff the original theory is satisfiable. We cannot translate arbitrary