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April 2011 - April 2015

July 2001 - July 2003

## Publications

Publications (93)

Proofs are a key feature of modern propositional and first-order theorem provers. Proofs generated by such tools serve as explanations for unsatisfiability of statements. However, these explanations are complicated by proofs which are not necessarily as concise as possible. There are a wide variety of compression techniques for propositional resolu...

Automated reasoning tools for the verification and synthesis of software often produce proofs to allow independent certification of the correctness of the produced solutions. As proofs can be large, this paper considers the problem of compressing proofs with respect to their space, which is approximately proportional to the memory necessary to chec...

Resolution and sequent calculus are two well-known formal proof systems. Their differences make them suitable for distinct tasks. Resolution and its variants are very efficient for automated reasoning and are in fact the theoretical basis of many theorem provers. However, being intentionally machine oriented, the resolution calculus is not as natur...

Resolution and superposition are common techniques which have seen widespread use with propositional and first-order logic in modern theorem provers. In these cases, resolution proof production is a key feature of such tools; however, the proofs that they produce are not necessarily as concise as possible. For propositional resolution proofs, there...

In the original publication of the article, the reference 11 was omitted inadvertently.

This paper defines the (first-order) conflict resolution calculus: an extension of the resolution calculus inspired by techniques used in modern SAT-solvers. The resolution inference is restricted to (first-order) unit-propagation and the calculus is extended with a mechanism for assuming decision literals and a new inference rule for clause learni...

On closer inspection many apparent contradictions turn out to be mere disagreements between distinct sources of information. For example, if a source \(s_1\) says P and a source \(s_2\) says \(\lnot P\), their disagreement would only become an actual contradiction if we naively merged what they say into our own knowledge base.

The efficiency of satisfiability modulo theories (SMT) solvers is dependent on the capability of theory reasoners to provide small conflict sets, i.e. small unsatisfiable subsets from unsatisfiable sets of literals. Decision procedures for uninterpreted symbols (i.e. congruence closure algorithms) date back from the very early days of SMT. Neverthe...

The conflict resolution calculus CR shown in Fig. 1 is a first-order hybrid resolution and clausal natural deduction calculus inspired by (and extending to first-order logic) the main ideas used in modern propositional sat-solvers. The calculus restricts the resolution inference rule to unit propagation and provides a mechanism for assuming decisio...

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving...

The modal collapse that afflicts Gödel’s modal ontological argument for God’s existence is discussed from the perspective of the modal square of opposition.

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Ande...

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving...

This paper defines the (first-order) conflict resolution calculus: an extension of the resolution calculus inspired by techniques used in modern Sat-solvers. The resolution inference rule is restricted to (first-order) unit propagation and the calculus is extended with a mechanism for assuming de- cision literals and with a new inference rule for c...

This paper presents detailed formalizations of ontological arguments in a simple modal natural deduction calculus. The first formal proof closely follows the hints in Scott's manuscript about Gödel's argument and fills in the gaps, thus verifying its correctness. The second formal proof improves the first one, by relying on the weaker modal logic K...

This was the second last talk in a seminar where much had already been said about the present and future of universality of proofs. To complement that, I decided to talk briefly about the distant past, sharing interesting facts about Leibniz, which I learned during a historical research triggered by the 300th anniversary of his death. The talk was...

In this talk I present the new (first-order) conflict resolution calculus: an extension of the resolution calculus inspired by techniques used in modern SAT-solvers. The resolution inference rule is restricted to (first-order) unit propagation and the calculus is extended with a mechanism for assuming decision literals and with a new inference rule...

This paper discusses the inconsistency in Gödel's ontological argument. Despite the popularity of Gödel's argument, this inconsistency remained unnoticed until 2013, when it was detected automatically by the higher-order theorem prover Leo-II. Complementing the meta-logic explanation for the inconsistency available in our IJCAI 2016 paper [6], we p...

On closer inspection many apparent contradictions turn out to be mere disagreements between distinct sources of knowledge. For example, if a source s 1 says P and a source s 2 says ¬P , their disagreement would only become an actual contradiction if, in our own knowledge base, we naively merged what they say. In this case, our own knowledge base wo...

This paper discusses the discovery of the inconsistency in Gödel's ontological argument as a success story for artificial intelligence. Despite the popularity of the argument since the appearance of Gödel's manuscript in the early 1970's, the inconsistency of the axioms used in the argument remained unnoticed until 2013, when it was detected automa...

The contextual natural deduction calculus (\({\mathbf{ND }^\mathbf{c }}\)) extends the usual natural deduction calculus (\(\mathbf{ND }\)) by allowing the implication introduction and elimination rules to operate on formulas that occur inside contexts. It has been shown that, asymptotically in the best case, \({\mathbf{ND }^\mathbf{c }}\)-proofs ca...

We have applied an elegant and flexible logic embedding approach to verify and automate a prominent philosophical argument: the ontological argument for the existence of God. In our ongoing computer-assisted study, higher-order automated reasoning tools have made some interesting observations, some of which were previously unknown.

The recently developed LowerUnits algorithm compresses propositional resolution proofs generated by SAT- and SMT-solvers by postponing and lowering resolution inferences involving unit clauses, which have exactly one literal. This paper describes a generalization of this algorithm to the case of first-order resolution proofs generated by automated...

These are the lecture notes of a tutorial on higher-order modal logics held at the 11th Reasoning Web Summer School. After defining the syntax and (possible worlds) semantics of some higher-order modal logics, we show that they can be embedded into classical higher-order logic by systematically lifting the types of propositions, making them depend...

This paper describes an embedding of higher-order modal logics in the Coq proof assistant. Coq's capabilities are used to implement modal logics in a minimalistic manner, which is nevertheless sufficient for the formalization of significant, non-trivial modal logic proofs. The elegance, flexibility and convenience of this approach, from a user pers...

The axioms in Gödel's ontological proof (Scott, 2004) entail what is called modal collapse (Sobel, 1987, 2004): the formula holds for any formula and not just for as intended. This fact has led to strong criticism of the argument and stimulated attempts to remedy the problem. One of those attempts (Anderson, 1990) sparked a controversy between Háje...

CERes is a method of cut-elimination that uses resolution proof search to avoid some kinds of redundancies that affect reductive cut-elimination methods. This paper shows that, unfortunately, there are also cases where CERes can produce proofs that are more redundant and even exponentially larger than the proofs produced by reductive cut-eliminatio...

Computer scientists prove the existence of God "-variants of this headline appeared in the international press in autumn 2013. Unfortunately, many media reports had only moderate success in communicating to the wider public what had actually been achieved and what not. This article outlines the main findings of the authors' joint work in computatio...

The UITP workshop series brings together researchers interested in designing,
developing and evaluating user interfaces for automated reasoning tools, such
as interactive proof assistants, automated theorem provers, model finders,
tools for formal methods, and tools for visualising and manipulating logical
formulas and proofs. The eleventh edition...

Kurt Gödel's ontological argument for God's existence has been formalized and automated on a computer with higher-order automated theorem provers. From Gödel's premises, the computer proved: necessarily, there exists God. On the other hand, the theorem provers have also confirmed prominent criticism on Gödel's ontolog-ical argument, and they found...

This paper introduces Skeptik: a system for checking, compressing and improving proofs obtained by SAT- and SMT-solvers.

The modal collapse that afflicts Gödel's modal ontological argument for God's existence is discussed from the perspective of the modal square of opposition. Mathematics Subject Classification (2010). Prim. 03A02; Sec. 68T02 .

This paper describes a generalization of the LowerUnits algorithm [8] for the compression of propositional resolution proofs. The generalized algorithm, called LowerUnivalents, is able to lower not only units but also subproofs of non-unit clauses, provided that they satisfy some additional conditions. This new algorithm is particularly suited to b...

Goedel's ontological proof has been analysed for the first-time with an
unprecedent degree of detail and formality with the help of higher-order
theorem provers. The following has been done (and in this order): A detailed
natural deduction proof. A formalization of the axioms, definitions and
theorems in the TPTP THF syntax. Automatic verification...

This paper introduces PROOFTOOL, the graphical user interface for the General
Architecture for Proof Theory (GAPT) framework. Its features are described with
a focus not only on the visualization but also on the analysis and
transformation of proofs and related tree-like structures, and its
implementation is explained. Finally, PROOFTOOL is compare...

This paper defines the contextual natural deduction calculus \(\textbf{ND}^\textbf{c}\) for the implicational fragment of intuitionistic logic. \(\textbf{ND}^\textbf{c}\) extends the usual natural deduction calculus (here called \(\textbf{ND}\)) by allowing the implication introduction and elimination rules to operate on formulas that occur inside...

This paper describes a generalization of the LowerUnits algorithm [9] for the compression of propositional resolution proofs. The generalized algorithm, called LowerUnivalents, is able to lower not only units but also subproofs of non-unit clauses, provided that they satisfy some additional conditions. This new algorithm is particularly suited to b...

Cut-elimination, introduced by Gentzen, plays an important role in automating the analysis of mathematical proofs. The removal of cuts corresponds to the elimination of intermediate statements (lemmas), resulting in an analytic proof. CERes is a method of cut-elimination by resolution that relies on global proof transformations, in contrast to redu...

This paper describes a new feature of the GAPT framework, namely the ability to import refutations obtained from external automated theorem provers. To cope with coarse-grained, under-specified and non-standard inference rules used by various theorem provers, the technique of proof replaying is employed. The refutations provided by external theorem...

CERES, HLK and ProofTool form together a system for the computer-aided analysis of mathematical proofs. This analysis is based on a proof transformation known as cut-elimination, which corresponds to the elimination of lemmas in the corresponding informal proofs. Consequently, the resulting formal proof in atomic-cut normal form corresponds to a di...

This paper discusses advantages and disadvantages of some possible alternatives for inference rules that handle quantifiers in the proof format of the SMT-solver veriT. The quantifier-handling modules in veriT being fairly standard, we hope this will motivate the discussion among the PxTP audience around proof production for quantifier handling. Th...

Methods exploiting problem symmetries have been very successful in several areas including constraint programming and SAT
solving. We here recast a technique to enhance the performance of SMT-solvers by detecting symmetries in the input formulas
and use them to prune the search space of the SMT algorithm. This technique is based on the concept of (...

This paper describes two algorithms for the compression of propositional resolution proofs. The first algorithm, RecyclePivots-WithIntersection, performs partial regularization, removing an inference η when it is redundant in the sense that its pivot literal already occurs as the pivot of another inference located below in
the path from η to the ro...

This paper defines a (deep) natural deduction calculus ND d for the implicational fragment of intuitionistic logic. ND d extends the usual (shallow) natural deduction calculus ND by allowing the implication introduction and elimination rules to be applied deeply inside formulas. In analogy to the Curry-Howard isomorphism between ND and the simply-t...

Axiomatization of Physics (and science in general) has many drawbacks that are correctly criticized by opposing philosophical views of science. This paper shows that, by giving formal proofs a more prominent role in the formalization, many of the drawbacks can be solved and many of the opposing views are naturally conciliated. Moreover, this approa...

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds
to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination
by resolution) works by extracting a set of clauses from a proof with cuts. Any resolution refu...

Defining natural language discourse semantics compositionally has been one of the main challenges in the areas of natural language semantics and pragmatics. In [Mon70], Montague argued that there are no essential differences between natural and formal programming languages. He developed a semantics that assigns a lambda term (i.e. a functional prog...

It is well-known that eliminating cuts from sequent calculus proofs frequently increases the size and length of proofs. In the worst case, cut-elimination can produce non-elementarily larger and longer proofs [3, 2]. Given this fact, it is natural to attempt to devise methods that could introduce cuts and compress cut-free proofs. However, this has...

The careful introduction of cut inferences can be used to structure and possibly compress formal sequent calculus proofs.
This paper presents CIRes, an algorithm for the introduction of atomic cuts based on various modifications and improvements
of the CERes method, which was originally devised for efficient cut-elimination. It is also demonstrated...

Axiomatization of Physics (and Science in general) has many drawbacks that are correctly criticized by opposing philosophical views of Science. This paper shows that, by giving formal proofs a more prominent role in the formalization, many of the drawbacks can be solved and many of the opposing views are naturally conciliated. Moreover, this approa...

This work defines an extension CERES2 of the first-order cut-elimination method CERES to the subclass of sequent calculus proofs in second-order logic using quantifier-free
comprehension. This extension is motivated by the fact that cut-elimination can be used as a tool to extract information from
real mathematical proofs, and often a crucial part...

Traditional reductive cut-elimination and CERES seem to be methods of entirely different nature and hence hard to compare. This short paper describes ongoing research that aims at comparing and possibly combining them in ways that retain that best features of each method.

Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information,
and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed.
In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentze...

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathe- matical proofs. Cut-elimination can be applied to mine real mathematical proofs, i.e. for extracting explicit and algorithmic information. The system CERES (cu...

Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentze...

Poster for the ESSLLI Student Session

GATE is an architecture that allows the use and development of useful plugins for typical Natural Language Processing tasks. However, there is currently no plugin capable of annotating a document that contains words that only approximately match words specified in a list of words to be searched. Here we describe the development of such a plugin, ba...

In recent years, governmental agencies have been looking for the application of automated tools for risk analysis as a way to enhance investigation and detection of frauds. One of the difficulties encountered is due to the existence of similar but different records in the databases, which actually refer to the same entity. These redundant records p...

The Schenberg gravitational wave detector is almost completed for operation at its site in the Physics Institute of the University of São Paulo, under the full support of FAPESP (the São Paulo State Foundation for Research Support). We have been working on the development of a transducer system, which will be installed after the arrival of all the...

Different studies show that dark matter of non-baryonic origin might exist. There have been experimental evidences that at least one form of dark matter has been detected through microlensing effects. This form of dark matter is named MACHOs (Massive Astrophysical Compact Halo Objects). The MACHO collaboration estimated that the masses of these obj...

We are building the Schenberg gravitational wave detector at the Physics Institute of the University of São Paulo as programmed by the Brazilian Graviton Project. The antenna and its vibration isolation system are already built, and we have made a first cryogenic run for an overall test, in which we measured the antenna mechanical Q (figure of meri...

This work defines an extension CERES 2 of the first-order cut-elimination method CERES to the subclass of sequent calculus proofs in second-order logic using quantifier-free comprehension. This extension is motivated by the fact that cut-elimination can be used as a tool to extract information from real mathematical proofs, and often a crucial part...