Theory-Specific Automated Reasoning

  title={Theory-Specific Automated Reasoning},
  author={Andrea Formisano and Eugenio G. Omodeo},
  booktitle={25 Years GULP},
In designing a large-scale computerized proof system, one is often confronted with issues of two kinds: issues regarding an underlying logical calculus, and issues that refer to theories, either specified axiomatically or characterized by indication of either a privileged model or a family of intended models. Proof services related to the theories most often take the form of satisfiability decision or semi-decision procedures (in a sense, polyadic inference rules), while some of the services… 
Decision Procedures for Elementary Sublanguages of Set Theory. XVII. Commonly Occurring Decidable Extensions of Multi-level Syllogistic
A technique based on a syntactic translation of formulae with the special function and predicate symbols above into multilevel syllogistic that yields nondeterministic polynomial-time decision procedures can be quite useful for tool developers who aim at providing assistance to common mathematical reasoning.
Solvable (and unsolvable) cases of the decision problem for fragments of analysis
We survey two series of results concerning the decidability of fragments of Tarksi's elementary algebra extended with one-argument functions which meet significant properties such as continuity,
More on the Structure of the Verifier System
This chapter describes in detail the set-based proof-verification system Ref, developed by the authors, and its underlying design. The chapter falls into two parts: (i) An account of the general
Membership and Edge Relations
The aim is not to uphold or study a specific axiomatization of Set Theory, but to give enough information to convince the reader of the viability of a fully formal approach; in fact, concerning non-secondary issues such as whether infinite sets should enter the game or not, or acyclicity vs. non-well foundedness of membership, the author will indicate antithetical options without enforcing any particular choice beforehand.
Counting and Encoding Sets
The Ackermann order is reconsidering and it is proved that it can be applied also to the universe of hypersets, which is to come, addressing natural questions about sets for whose treatment the set-to-graph correspondence can be of use.
The Representation of Boolean Algebras in the Spotlight of a Proof Checker
This pretty-printed scenario reflects an early phase in the formal development of the proof of Stone’s theorem on the representation of Boolean algebras: only the algebraic version of that theorem is proved here.
Infinite Sets and Finite Combinatorics
In tackling the set-satisfiability problem in Chap. 4, we have not gone beyond the analysis of formulae with a single prefixed universal quantifier: we have seen how to determine whether or not a
Sets, Graphs, and Set Universes
The simple combinatorial character of the arguments discussed in this book allows us to opt for a simplified semantic approach. Most often, in fact, our work will regard the specific model of
Graphs as Transitive Sets
This chapter will represent connected claw-free graphs as specially constrained transitive sets: each element x′ of the set T that represents one such graph will act as a corresponding vertex x and the edge relationship will be mimicked by membership over T.
Random Generation of Sets
Algorithms that produce a well-founded set of “size” n, uniformly at random, so that each set of size n has equal probability to occur.


Automated Deduction in Classical and Non-Classical Logics
Theorem Proving in First-Order Logic Modulo: On the Difference between Type Theory and Set Theory and Completeness and Redundancy in Constrained Clause Logic.
Automated deduction by theory resolution
  • M. Stickel
  • Mathematics
    Journal of Automated Reasoning
  • 2004
Theory resolution constitutes a set of complete procedures for incorporating theories into a resolution theorem-proving program, thereby making it unnecessary to resolve directly upon axioms of the theory, and is a generalization of numerous previously known resolution refinements.
Resolution in Type Theory
In [8] J. A. Robinson introduced a complete refutation pro­cedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand’s Theorem, and provides a very
Deciding Combinations of Theories
A method is given for deciding formulas in combinations of unquatified first-order theories that makes use of a single, uniform procedure that minimizes the code needed to accommodate each additional theory.
Self-applied proof verication (Extended abstract) ?
Here a group of THEORYs, one of the simplest being the one in Fig. 1, are presented which grew as digressions with respect to the most challenging scenario out of the willingness to assess.
A Machine-Oriented Logic Based on the Resolution Principle
The paper concludes with a discussion of several principles which are applicable to the design of efficient proof-procedures employing resolution as the basle logical process.
Automated Reasoning with Analytic Tableaux and Related Methods
  • R. Dyckhoff
  • Computer Science
    Lecture Notes in Computer Science
  • 2000
Tableau Algorithms for Description Logics and Towards Tableau-Based Decision Procedures for Non-Well-Founded Fragments of Set Theory.
A Language for Programming in Logic with Finite Sets
Solvable Set/Hyperset Contexts: II. A Goal-Driven Unification Algorithm for the Blended Case
A goal-driven algorithm for solving the corresponding unification problem is provided, it is proved to be totally correct, and exploited to show that the problem itself is NP-complete.
T-Theorem Proving I
A theoretical basis justifying the incorporation of decidability results for a first-order theory T into an automated theorem prover for T is presented and rules which extend resolution using decidable results relative to T in both the ground and the non-ground case are state.