First-Order Theorem Proving and Vampire

... http://www.tptp.org contains a large collection of first-order problems. For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax this group theory problem can be written down as follows: %---- 1 * x = 1 fof(lef ...

... http://www.tptp.org contains a large collection of first-order problems. For representing these problems it uses the TPTP syntax, which is understood by all modern theorem provers, including Vampire. In the TPTP syntax this group theory problem can be written down as follows: %---- 1 * x = 1 fof(lef ...

a-logic - Digital Commons@Wayne State University

... set of theorems, the same semantical foundations, and use the same concepts of validity and logical truth though they differ in notation, choices of primitives and axioms, diagramatic devices, modes of introduction and explication, etc. This standard logic is an enormous advance over any preceding s ...

... set of theorems, the same semantical foundations, and use the same concepts of validity and logical truth though they differ in notation, choices of primitives and axioms, diagramatic devices, modes of introduction and explication, etc. This standard logic is an enormous advance over any preceding s ...

PowerPoint

... by applying substitution {Manuf -> compaq, Pers -> C} • The second resolvent is more general than the first; the second substitution (unifier) is more general than the first. • A most general unifier is one that would not be a unifier if any substitutions were removed. • mgus are unique, apart from ...

... by applying substitution {Manuf -> compaq, Pers -> C} • The second resolvent is more general than the first; the second substitution (unifier) is more general than the first. • A most general unifier is one that would not be a unifier if any substitutions were removed. • mgus are unique, apart from ...

QUASI-MV ALGEBRAS. PART III

... at least one ﬁxpoint for ′ , namely, the unique regular member of C. If any other ﬁxpoints exist, they also belong to C. We choose arbitrarily some maximal set S of clouds that contains at most one of each pair of twin clouds. In particular, the median cloud is not a member of S, but, by maximality, ...

... at least one ﬁxpoint for ′ , namely, the unique regular member of C. If any other ﬁxpoints exist, they also belong to C. We choose arbitrarily some maximal set S of clouds that contains at most one of each pair of twin clouds. In particular, the median cloud is not a member of S, but, by maximality, ...

Hilbert`s Program Then and Now

... contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory). Had it been successful, Hilbert’s program would perhaps not only have eclipsed in subsequent influence other foundational projects of the time, such as the logic ...

... contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory). Had it been successful, Hilbert’s program would perhaps not only have eclipsed in subsequent influence other foundational projects of the time, such as the logic ...

A tableau-based decision procedure for LTL

... procedure for LTL For the sake of clarity, among the various existing tableau systems for LTL, we selected Manna and Pnueli’s implicit declarative one Z. Manna, A. Pnueli, Temporal Verification of Reactive Systems: Safety, ...

... procedure for LTL For the sake of clarity, among the various existing tableau systems for LTL, we selected Manna and Pnueli’s implicit declarative one Z. Manna, A. Pnueli, Temporal Verification of Reactive Systems: Safety, ...

Hilbert`s Program Then and Now - Philsci

... term 1920, he concluded that “the aim of reducing set theory, and with it the usual methods of analysis, to logic, has not been achieved today and maybe cannot be achieved at all” [Hilbert, 1920]. At the same time, Brouwer’s intuitionist mathematics gained currency. In particular, Hilbert’s former s ...

... term 1920, he concluded that “the aim of reducing set theory, and with it the usual methods of analysis, to logic, has not been achieved today and maybe cannot be achieved at all” [Hilbert, 1920]. At the same time, Brouwer’s intuitionist mathematics gained currency. In particular, Hilbert’s former s ...

TR-14-06 - Ynot - Harvard University

... Importantly, Q can depend not only upon the return value x, but also the initial and final stores making it possible to capture relational properties of stateful computations in the style of Standard Relational Semantics well-known in the research on Hoare Logic for first-order languages [12]. In o ...

... Importantly, Q can depend not only upon the return value x, but also the initial and final stores making it possible to capture relational properties of stateful computations in the style of Standard Relational Semantics well-known in the research on Hoare Logic for first-order languages [12]. In o ...

How to Write a 21st Century Proof

... let’s turn them into steps 3 and 4 of the structured proof. The final sentence could also be viewed as two statements, but I find it more natural to turn it into the assertion f (b) > f (a) and its proof. Adding these statements and their proofs, we obtain the simple structured proof of Figure 2. Ex ...

... let’s turn them into steps 3 and 4 of the structured proof. The final sentence could also be viewed as two statements, but I find it more natural to turn it into the assertion f (b) > f (a) and its proof. Adding these statements and their proofs, we obtain the simple structured proof of Figure 2. Ex ...

A counterexample-guided abstraction

... (5) A sub-logic of our safe-PCTL, which we call weak safety, does indeed admit counterexamples that have a tree-like structure. For this case, we present an onthe-fly algorithm to unroll the minimal counterexample that we generate and check validity. This algorithm may perform better than the algori ...

... (5) A sub-logic of our safe-PCTL, which we call weak safety, does indeed admit counterexamples that have a tree-like structure. For this case, we present an onthe-fly algorithm to unroll the minimal counterexample that we generate and check validity. This algorithm may perform better than the algori ...

Deductive Databases with Universally Quantified Conditions

... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...

... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...

On Weak Ground

... Nevertheless, the question of whether any of these characterizations can be used to specify the notion is worth exploring. The arguments that follow, if sound, provide reasons not only for rejecting the full explications of the notion of weak ground suggested by the characterizations, but also for f ...

... Nevertheless, the question of whether any of these characterizations can be used to specify the notion is worth exploring. The arguments that follow, if sound, provide reasons not only for rejecting the full explications of the notion of weak ground suggested by the characterizations, but also for f ...

Die Grundlagen der Arithmetik §§82–83

... from (50 ) and (3) from (4*). From (3) derive (1). Prove (2). Then, finally, infer (00 ) from (2) and (1), by a similar appeal to the definition of P∗ . However, it will turn out that this precise strategy cannot succeed. It cannot be (4*) and (3) that Frege wishes to derive—(3), e.g., is false if a ...

... from (50 ) and (3) from (4*). From (3) derive (1). Prove (2). Then, finally, infer (00 ) from (2) and (1), by a similar appeal to the definition of P∗ . However, it will turn out that this precise strategy cannot succeed. It cannot be (4*) and (3) that Frege wishes to derive—(3), e.g., is false if a ...

A Course in Modal Logic - Sun Yat

... concepts, whereas these differences are actually supported by different intuitive semantics (even philosophical background). The reader who is interested in this can refer to the related literature. (Ⅱ) The cardinality of At is finite or countable infinite, but, in fact, most of results given in thi ...

... concepts, whereas these differences are actually supported by different intuitive semantics (even philosophical background). The reader who is interested in this can refer to the related literature. (Ⅱ) The cardinality of At is finite or countable infinite, but, in fact, most of results given in thi ...

Chapter 13 BOOLEAN ALGEBRA

... Notice that the two definitions above refer to "...a greatest lower bound" and "a least upper bound." Any time you define an object like these you need to have an open mind as to whether more than one such object can exist. In fact, we now can prove that there can't be two greatest lower bounds or t ...

... Notice that the two definitions above refer to "...a greatest lower bound" and "a least upper bound." Any time you define an object like these you need to have an open mind as to whether more than one such object can exist. In fact, we now can prove that there can't be two greatest lower bounds or t ...

Functional Dependencies in a Relational Database and

... to A, B, and D, and truth value 0 to C. It is very simple to check that each implicational statement in DEP has truth value 1 under t , whereas a has truth value 0 under t . So, since we have exhibited a counterexample to truth assignment t such that each statement in DEP has truth value 1 under t b ...

... to A, B, and D, and truth value 0 to C. It is very simple to check that each implicational statement in DEP has truth value 1 under t , whereas a has truth value 0 under t . So, since we have exhibited a counterexample to truth assignment t such that each statement in DEP has truth value 1 under t b ...

Predicate Logic

... Other forms of quantification Other Quantifiers The most important quantifiers are ∀ and ∃, but we could define many different quantifiers: “there is a unique”, “there are exactly two”, “there are no more than three”, “there are at least 100”, etc. A common one is the uniqueness quantifier, denoted ...

... Other forms of quantification Other Quantifiers The most important quantifiers are ∀ and ∃, but we could define many different quantifiers: “there is a unique”, “there are exactly two”, “there are no more than three”, “there are at least 100”, etc. A common one is the uniqueness quantifier, denoted ...

A Unified View of Induction Reasoning for First-Order Logic

... schemas. They explicitly define the induction hypotheses which are linked to the induction conclusions in order to be used in further derivations. The information justifying the soundness property is locally embedded inside the induction schemas, hence their natural integration into deductive, seque ...

... schemas. They explicitly define the induction hypotheses which are linked to the induction conclusions in order to be used in further derivations. The information justifying the soundness property is locally embedded inside the induction schemas, hence their natural integration into deductive, seque ...

Discrete Mathematics

... Hint: The Appendix of our text Isabelle/HOL – A Proof Assistant for Higher-Order Logic lists the logical connectives, such as −→ and ¬, and for each of them gives two ways to input them as ASCI text. If you use Emacs (or XEmacs) to edit your Isabelle files, then the x-symbol package can be used to d ...

... Hint: The Appendix of our text Isabelle/HOL – A Proof Assistant for Higher-Order Logic lists the logical connectives, such as −→ and ¬, and for each of them gives two ways to input them as ASCI text. If you use Emacs (or XEmacs) to edit your Isabelle files, then the x-symbol package can be used to d ...

Interpretability formalized

... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...

... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...